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.
The plane of the picture represents the complex numbers, with points in the Mandelbrot set (a fractal) coloured blue.
^ A real number is a point on the number line, a complex number represents a vector on the two dimension plane.
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

^ I have occasionally pointed people towards http://mathforum.org/johnandbetty/ which is good for the early stages of complex numbers.
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

^ Awesome, so… if real numbers are represented in one dimension and complex numbers are represented in two dimensions, what’s represented in three, four, n dimensions?
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

The points in the set are those which stay near the origin after repeated squaring and adding.
.A complex number is a number comprising a real and imaginary part.^ Thus, every complex number has a real part, a , and an imaginary part, b i .
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^ A complex number is the fancy name for numbers with both real and imaginary parts.
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

^ If a = b , the number is equal parts real and imaginary.
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.It can be written in the form a + bi, where a and b are real numbers, and i is the standard imaginary unit with the property i 2 = −1.[1] The complex numbers contain the ordinary real numbers, but extend them by adding in extra numbers and correspondingly expanding the understanding of addition and multiplication.^ Thus, every complex number has a real part, a , and an imaginary part, b i .
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^ A complex number is the fancy name for numbers with both real and imaginary parts.
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

^ If a = b , the number is equal parts real and imaginary.
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This is in order to form a closed field, where any polynomial equation has a root, including examples such as x2 = −1.
.Complex numbers were first conceived and defined by the Italian mathematician Gerolamo Cardano, who called them "fictitious", during his attempts to find solutions to cubic equations.^ Complex Numbers A complex number is expressed in the standard form a + b i , where a and b are real numbers and i is defined by i ^2 = -1 (that is, i is the square root of -1).
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^ Yep, there’s something called quaternions which are complex numbers extended to 4 dimensions, and are used in graphics programming for 3d rotations (like complex numbers can be used for 2d rotations).
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

[2] .The solution of a general cubic equation in radicals (without trigonometric functions) may require intermediate calculations containing the square roots of negative numbers, even when the final solutions are real numbers, a situation known as casus irreducibilis.^ Notice how negative numbers “keep track of the sign” — we can throw -1^47 into a calculator without having to count (” Week 1 is good, week 2 is bad… week 3 is good… “).
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

^ Having no real knowledge of what ‘i’ is, never knowing more than it’s the result of the square root of negative one, I had the thought of drawing a geometric plane with one axis being the real number set and the other axis being the imaginary number set, as you showed above.
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

^ We are looking for that length y that gives y^2=-1, that number y is square root of -1 but that doesn’t mean anything since square root is only defined for positive numbers so we need a new symbol for that unit.
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

.This ultimately led to the fundamental theorem of algebra, which shows that with complex numbers, a solution exists to every polynomial equation of degree one or higher.^ Geez, his theorem shows up everywhere , even in numbers invented 2000 years after his time.
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

^ Awesome, so… if real numbers are represented in one dimension and complex numbers are represented in two dimensions, what’s represented in three, four, n dimensions?
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

^ But there’s one last question: how “big” is a complex number?
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

The rules for addition, subtraction, multiplication, and division of complex numbers were developed by the Italian mathematician Rafael Bombelli.[3] .A more abstract formalism for the complex numbers was further developed by the Irish mathematician William Rowan Hamilton, who extended this abstraction to the theory of quaternions.^ There’s much more complex numbers: check out the details of complex arithmetic .
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

^ Yep, there’s something called quaternions which are complex numbers extended to 4 dimensions, and are used in graphics programming for 3d rotations (like complex numbers can be used for 2d rotations).
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

^ Yes, I want to learn more about quaternions , imaginary numbers extended to more dimensions .
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

.Complex numbers are used in a number of fields, including: engineering, electromagnetism, quantum physics, applied mathematics, and chaos theory.^ You can model imaginary numbers using vectors and matrices (more generally in the field of linear algebra).
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

^ In Physics and Electrical Engineering, imaginary numbers are used to describe electric current and other things that can have cyclical patterns.
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

^ We’re not going to wait until college physics to use imaginary numbers.
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

.When the underlying field of numbers for a mathematical construct is the field of complex numbers, the name usually reflects that fact.^ A complex number is the fancy name for numbers with both real and imaginary parts.
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

^ I share share your frustration at the fact that most high school mathematics courses do not explain complex numbers adequately.
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

Examples are complex analysis, complex matrix, complex polynomial, and complex Lie algebra.

Contents

Definitions

Notation

.The set of all complex numbers is usually denoted by C, or in blackboard bold by \mathbb{C}.^ David gave detailed geometrical interpretations on all complex number arithmatic.
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

Although other notations can be used, complex numbers are usually written in the form
 a + bi \,
where .a and b are real numbers, and i is the imaginary unit, which has the property i 2 = −1. The real number a is called the real part of the complex number, and the real number b is the imaginary part.^ Complex numbers are often represented on a graph known as the "complex plane," where the horizontal axis represents the infinity of real numbers, and the vertical axis represents the infinity of imaginary numbers.
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^ Complex Numbers A complex number is expressed in the standard form a + b i , where a and b are real numbers and i is defined by i ^2 = -1 (that is, i is the square root of -1).
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^ The b i term is often referred to as an imaginary number (though this may be misleading, as it is no more "imaginary" than the symbolic abstractions we know as the "real" numbers).
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.For example, 3 + 2i is a complex number, with real part 3 and imaginary part 2. If z = a + bi, the real part a is denoted Re(z) or ℜ(z), and the imaginary part b is denoted Im(z) or ℑ(z).^ Thus, every complex number has a real part, a , and an imaginary part, b i .
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^ We can’t measure the real part or imaginary parts in isolation, because that would miss the big picture.
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

^ Complex refers to something made from more than one part (in this case the real and imaginary parts)Think of a complex of buildings.What you mean is that they aren’t complicated.
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

.The complex numbers (C) are regarded as an extension of the real numbers (R) by considering every real number as a complex number with an imaginary part of zero.^ Thus, every complex number has a real part, a , and an imaginary part, b i .
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^ A complex number is the fancy name for numbers with both real and imaginary parts.
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

^ If a = b , the number is equal parts real and imaginary.
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.The real number a is identified with the complex number a + 0i.^ Awesome, so… if real numbers are represented in one dimension and complex numbers are represented in two dimensions, what’s represented in three, four, n dimensions?
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

^ A real number is a point on the number line, a complex number represents a vector on the two dimension plane.
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

^ A complex number is the fancy name for numbers with both real and imaginary parts.
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

.Complex numbers with a real part of zero (Re(z)=0) are called imaginary numbers.^ A complex number is the fancy name for numbers with both real and imaginary parts.
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

^ We’re at a 45 degree angle, with equal parts in the real and imaginary (1 + i).
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

^ We can’t measure the real part or imaginary parts in isolation, because that would miss the big picture.
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

.Instead of writing 0 + bi, that imaginary number is usually denoted as just bi.^ Imaginary numbers have the rotation rules baked in: it just works.
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

^ The number i is just as normal as other numbers, but the name “imaginary” stuck so we’ll use it.
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

^ Well, just an idea to discuss on: now we could think about a+bi+cj numbers Or we could think about four-dimension numbers too: a+bi+cj+dk And so on… .
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

.If b equals 1, instead of using 0 + 1i or 1i, the number is denoted as i.^ One comment just for fun: Did you know that engineers (at least electrical engineers) use “j” instead of “i” to denote sqrt(-1)?
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

.In some disciplines (in particular, electrical engineering, where i is a symbol for current), the imaginary unit i is instead written as j, so complex numbers are sometimes written as a + bj or a + jb.^ In Physics and Electrical Engineering, imaginary numbers are used to describe electric current and other things that can have cyclical patterns.
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

^ We have the symbol “1″ for a positive unit, we have the symbol “-1″ for the negative unit but now we need a symbol for the lateral unit (the “imaginary” unit) which is i and it’s opposite “-i” which basically is laterally moving away from the horizontal number line into another dimension.
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

^ We are looking for that length y that gives y^2=-1, that number y is square root of -1 but that doesn’t mean anything since square root is only defined for positive numbers so we need a new symbol for that unit.
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

.Two complex numbers are said to be equal if and only if their real parts are equal and their imaginary parts are equal.^ Thus, every complex number has a real part, a , and an imaginary part, b i .
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^ If a = b , the number is equal parts real and imaginary.
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^ Complex Numbers A complex number is expressed in the standard form a + b i , where a and b are real numbers and i is defined by i ^2 = -1 (that is, i is the square root of -1).
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Formal development

.In a rigorous setting, it is not acceptable to simply assume that there exists a number whose square is −1. There are various ways of defining C, building on the knowledge of real numbers.^ Having no real knowledge of what ‘i’ is, never knowing more than it’s the result of the square root of negative one, I had the thought of drawing a geometric plane with one axis being the real number set and the other axis being the imaginary number set, as you showed above.
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

^ Yes, I really love teaching and sharing knowledge, but coming out of college I wasn’t sure I wanted to dedicate my life to it in such a way.
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

^ Complex Numbers A complex number is expressed in the standard form a + b i , where a and b are real numbers and i is defined by i ^2 = -1 (that is, i is the square root of -1).
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Firstly, write C for R2, the set of ordered pairs of real numbers, and define operations on complex numbers in C according to
(ab) + (cd) = (a + cb + d)
(ab)·(cd) = (a·c − b·db·c + a·d).
It is then just a matter of notation to express (ab) as a + ib. This means we can associate the numbers (a, 0) with the real numbers, and write i = (0, 1). Since (0, 1)·(0, 1) = (−1, 0), we have found i by constructing it, not postulating it. Using these formal operations on R2, it is easy to check that we satisfy the field axioms (associativity, commutativity, identity, inverses, distributivity). In particular, R is a subfield of C.
.Though this low-level construction does accurately describe the structure of the complex numbers, the definitions seem arbitrary, so secondly C can be considered algebraically.^ Convince you that complex numbers were considered “crazy” but can be useful (just like negative numbers were) Show how complex numbers can make certain problems easier, like rotations If I seem hot and bothered about this topic, there’s a reason.
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

^ Complex numbers beat you to it, instantly, accurately, and without a calculator.
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

.In algebra (the theory of group-like structures), this explicit definition of operations in fact turns out to be the mechanism behind the idea of constructing the algebraic closure of the reals, that is, adding in some elements to R to make a new field, of which R is a subfield, where every non-constant polynomial has a root.^ Also, I like what you said about math being a language that is self-describing to some extent; you can communicate with others *and* discover new ideas by using it.
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

^ This was a very fun experience, you really do learn something new every day!
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

^ I’m not sure where I first saw the seeds of it, but I recommend Hestenes’ Oersted lecture on geometric algebra for how to extend this idea and where to find lots more like this.
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

Finally, yet another way of characterising C is in terms of its topological properties. Details of these are given below.

Operations

Complex numbers can be intuitively added, subtracted, multiplied, and divided by applying the associative, commutative and distributive laws of algebra, together with the equation i 2 = −1. Start by defining the two basic operations:
(a + bi) + (c + di): = (a + c) + (b + d)i [addition]
(a + bi)(c + di): = (acbd) + (bc + ad)i. [multiplication]
The notation is well-designed, since proceeding to manipulate the symbols purely intuitively gives
(a + bi)(c + di) = ac + bci + adi + bdi2 = (acbd) + (bc + ad)i.
The fact that this agrees with the definition justifies this sort of manipulation, by indirectly deriving the distributive law, and so on.
Subtraction and division are found from these two by working backwards. In the case of subtraction, uw means "u plus the number which cancels out w (its additive inverse)", giving
(a + bi) − (c + di) = (ac) + (bd)i. [subtraction]
Similarly for multiplication, the inverse of c + di is found to be
\left(\frac{c}{c^2+d^2}\right)-\left(\frac{d}{c^2+d^2}\right)i
when c and d are not both zero (by solving for z in z(c + di) = 1), which then yields
\frac{a + bi}{c + di} =\left({ac + bd \over c^2 + d^2}\right) + \left( {bc - ad \over c^2 + d^2} \right)i. [division]
If the symbols are manipulated intuitively, so that
\begin{align} \frac{a + bi}{c + di} &= \frac{(a+bi)(c-di)}{(c+di)(c-di)} = \frac{ac + bci - adi + bd}{(c+di)(c-di)} \\&=\left({ac + bd \over c^2 + d^2}\right) + \left( {bc - ad \over c^2 + d^2} \right)i, \end{align}
the formula agrees with the one just derived. .In fact, once the gaps are filled in, all the usual rules of simple arithmetic on real numbers can be shown to work when the symbols are appropriately swapped for complex numbers.^ It is nice to have an explanation on why they work instead of the usually “Here is the rule.
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^ It is Euler’s formula that links trigonometry to arithmetic (and allows for a geometric interpretation of complex numbers as a result).
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^ Trigonometry is great, but complex numbers can make ugly calculations simple (like calculating cosine(a+b) ).
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

.Finally, square (and higher) roots can now be defined.^ Complex Numbers A complex number is expressed in the standard form a + b i , where a and b are real numbers and i is defined by i ^2 = -1 (that is, i is the square root of -1).
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.However, every non-zero number has two square roots, so there is an ambiguity.^ Complex Numbers A complex number is expressed in the standard form a + b i , where a and b are real numbers and i is defined by i ^2 = -1 (that is, i is the square root of -1).
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Where the radical symbol (√) is used, it is normally used to denote the root with smallest non-negative phase (see section below on polar representation).

Conjugation

Define the conjugate of z = x + iy to be xiy, written as \bar{z} or z * . .Both z+\bar{z} and z\cdot\bar{z} are real numbers.^ A complex number is the fancy name for numbers with both real and imaginary parts.
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^ There’s another detail to cover: can a number be both “real” and “imaginary”?
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

.Conjugation distributes over all the algebraic operations and many derived functions, for example \sin\bar z=\overline{\sin z}; this is as expected, since the labelling of one of the two roots of −1 as i is arbitrary, so many situations should exhibit symmetry about the real axis.^ Having no real knowledge of what ‘i’ is, never knowing more than it’s the result of the square root of negative one, I had the thought of drawing a geometric plane with one axis being the real number set and the other axis being the imaginary number set, as you showed above.
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

^ Awesome, so… if real numbers are represented in one dimension and complex numbers are represented in two dimensions, what’s represented in three, four, n dimensions?
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

^ I like that point about the beauty of math and physics — many people scoff that notion, but the beauty really is there!
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

It is important to note, however, that the function f(z) = \bar{z} is not complex-differentiable (see Holomorphic function).

Elementary functions

One of the most important functions on the complex numbers is perhaps the exponential function exp(z), also written ez, defined in terms of the infinite series
\exp(z):=\sum_{n=0}^{\infty} \frac{z^n}{n!} = 1+z+\frac{z^2}{2\cdot 1}+\frac{z^3}{3\cdot 2\cdot 1}+\cdots.
The elementary functions are those which can be finitely built using exp and the arithmetic operations given above, as well as taking inverses; in particular, the inverse of the exponential function, the logarithm. .The real-valued logarithm over the positive reals is well-defined, and the complex logarithm generalises this idea.^ Complex Numbers A complex number is expressed in the standard form a + b i , where a and b are real numbers and i is defined by i ^2 = -1 (that is, i is the square root of -1).
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The inverse of exp is shown to be
\log(x+iy)=	frac{1}{2}\ln(x^2+y^2)+i\arg(x+iy),
where arg is the argument defined below, and ln the real logarithm. As arg is a multivalued function, unique only up to a multiple of 2π, log is also multivalued. The principal value of log is often taken by restricting the imaginary part to the interval (−π,π].
The familiar trigonometric functions are composed of these, so they are also elementary. For example,
\sin(z)=\frac{e^{iz} - e^{-iz}}{2i}.
Hyperbolic functions such as sinh are similarly constructed.

Exponentiation

Raising numbers to positive integer powers is done using the operation of multiplication:
z^n = \underbrace{z\cdot z \cdots z}_{n	ext{ factors}}. \,
.Negative integer powers also are defined just as for real numbers, since 1/zn is the only way of interpreting zn such that the familiar rules of indices still work (zn = zn(zn/zn) = zn+n/zn = 1/zn).^ Really Understanding Negative Numbers .
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

^ It seems crazy, just like negatives, zero, and irrationals (non-repeating numbers) must have seemed crazy at first.
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

^ Yes, I really love teaching and sharing knowledge, but coming out of college I wasn’t sure I wanted to dedicate my life to it in such a way.
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

.Similar considerations show that we can define rational real powers just as for the reals, so z1/n is the nth root of z.^ Complex Numbers A complex number is expressed in the standard form a + b i , where a and b are real numbers and i is defined by i ^2 = -1 (that is, i is the square root of -1).
  • Writing.Com: Complex Numbers (Book) 10 February 2010 13:39 UTC www.writing.com [Source type: FILTERED WITH BAYES]
  • Writing.Com: Complex Numbers (Book) 10 February 2010 13:39 UTC www.writing.com [Source type: FILTERED WITH BAYES]

Roots are not unique, so it is already clear that complex powers are multivalued, thus careful treatment of powers is needed; for example (81/3)4 ≠ 16, as there are three cube roots of 8, so the given expression, often shortened to 84/3, is the simplest possible.
.For arbitrary complex powers, the general meaning of zω must be multi-valued, since it is in the case of ω rational.^ Complex refers to something made from more than one part (in this case the real and imaginary parts)Think of a complex of buildings.What you mean is that they aren’t complicated.
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

To agree with the definitions so far, this suggests
z^\omega := \exp(\omega \log z), \,
which is the general extension of exponentiation to the complex numbers.

The complex plane

Figure 1: A complex number plotted as a point (red) and position vector (blue) on an Argand diagram; a + bi is the rectangular expression of the point.
.A complex number can be viewed as a point or position vector in a two-dimensional Cartesian coordinate system called the complex plane or Argand diagram (see Pedoe 1988 and Solomentsev 2001), named after Jean-Robert Argand.^ A point on saying that numbers have two dimensions.
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

^ A real number is a point on the number line, a complex number represents a vector on the two dimension plane.
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

^ Yep that’s right, I was being paid and didn’t even realise that a complex number was just a number with two dimensions.
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

.The numbers are conventionally plotted using the real part as the horizontal component, and imaginary part as vertical (see Figure 1).^ A complex number is the fancy name for numbers with both real and imaginary parts.
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

^ But why were imaginary numbers first used?
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

^ You can model imaginary numbers using vectors and matrices (more generally in the field of linear algebra).
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

.These two values used to identify a given complex number are therefore called its Cartesian-, rectangular-, or algebraic form.^ Yep that’s right, I was being paid and didn’t even realise that a complex number was just a number with two dimensions.
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

^ Dave: a pair (z,s) of two complex numbers would “live” in 4 dimensional space.
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

^ Awesome, so… if real numbers are represented in one dimension and complex numbers are represented in two dimensions, what’s represented in three, four, n dimensions?
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

Geometric interpretation of the operations

The operations described algebraically above can be visualised using Argand diagrams.
Complex numbers addition.png .X = A + B: The sum of two points A and B of the complex plane is the point X = A + B such that the triangles with vertices 0, A, B, and X, B, A, are congruent.^ A real number is a point on the number line, a complex number represents a vector on the two dimension plane.
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

.Thus the addition of two complex numbers is the same as vector addition of two vectors.^ But for complex numbers, how do we measure two components at 90 degree angles?
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

^ Yep that’s right, I was being paid and didn’t even realise that a complex number was just a number with two dimensions.
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

^ Dave: a pair (z,s) of two complex numbers would “live” in 4 dimensional space.
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

Complex numbers multiplication.png X = AB: The product of two points A and B is the point X = AB such that the triangles with vertices 0, 1, A, and 0, B, X, are similar.
Complex numbers conjugation.png .X = A*: The complex conjugate of a point A is the point X = A* such that the triangles with vertices 0, 1, A, and 0, 1, X, are mirror images of each other.^ Yeah, if negatives are “mirror images”, then complex numbers are “rotations”.
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

.These geometric interpretations allow problems of algebra to be translated into geometry.^ It is Euler’s formula that links trigonometry to arithmetic (and allows for a geometric interpretation of complex numbers as a result).
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

And, conversely, geometric problems can be examined algebraically. .For example, the problem of the geometric construction of the 17-gon was by Gauss translated into the analysis of the algebraic equation x17 = 1 (see Heptadecagon).^ Stary: You’re welcome — yes, it’d be nice if students got to see the geometric viewpoint along with the pure algebra approach.
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

Polar form

Figure 2: The argument φ and modulus r locate a point on an Argand diagram; r(cosφ + isinφ) or reiφ are polar expressions of the point.
The diagrams suggest various properties. Firstly, the distance of a point z from the origin (shown as r in Figure 2) is known as the modulus, absolute value, or magnitude, and written | z | . By Pythagoras' theorem,
|x+iy|=\sqrt{x^2+y^2}.
.In general, distances between complex numbers are given by the distance function d(z,w) = | zw | , which turns the complex numbers into a metric space and introduces the ideas of limits and continuity.^ Khalid, now I could make a pretty difference between ‘Complex Number’ & Trigs.
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

^ A relevant lecture on complex numbers with a little diversion into Eulers formula .
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

^ Dave: a pair (z,s) of two complex numbers would “live” in 4 dimensional space.
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

.All of the standard properties of two dimensional space therefore hold for the complex numbers, including important properties of the modulus such as non-negativity, and the triangle inequality (| z + w | \leq | z | + | w | for all z, w).^ Dave: a pair (z,s) of two complex numbers would “live” in 4 dimensional space.
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

^ It seems crazy, just like negatives, zero, and irrationals (non-repeating numbers) must have seemed crazy at first.
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

^ But for complex numbers, how do we measure two components at 90 degree angles?
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

.Secondly, the argument or phase of a complex number z = x + yi is the angle to the real axis (shown as φ in Figure 2), and is written as arg(z).^ Awesome, so… if real numbers are represented in one dimension and complex numbers are represented in two dimensions, what’s represented in three, four, n dimensions?
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

^ A real number is a point on the number line, a complex number represents a vector on the two dimension plane.
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

^ But for complex numbers, how do we measure two components at 90 degree angles?
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

As with the modulus, the argument can be found from the rectangular form x + iy:
\varphi = \pm\arctan\frac{y}{x} (taking the sign appropriately so that x + iy = r(cosφ + isinφ).
The value of φ can change by any multiple of 2π and still give the same angle (note that radians are being used). Hence, the arg function is sometimes considered as multivalued, but often the value is chosen to lie in the interval ( − π,π], or [0,2π) (this is the principal value).
.Together, these give another way of representing complex numbers, the polar form, as the combination of modulus and argument fully specify the position of a point on the plane (confirmed by recovering the original rectangular co-ordinates (x,y)=(r \cos\varphi,r\sin\varphi) from the polar pair (r,φ)).^ A real number is a point on the number line, a complex number represents a vector on the two dimension plane.
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

^ Complex numbers are similar — it’s a new way of thinking.
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

^ I have occasionally pointed people towards http://mathforum.org/johnandbetty/ which is good for the early stages of complex numbers.
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

This can be notated in various ways, including
 z = r(\cos \varphi + i\sin \varphi )\,
called trigonometric form, and sometimes abbreviated r cis φ, or using Euler's formula
 z = r e^{i \varphi},
which is called exponential form. In electronics it is common to use angle notation to represent a phasor with amplitude A and phase θ as
 A \ang 	heta = A e ^ {j 	heta }.
In angle notation θ may be in either radians or degrees. .In electronics it is also common to use j instead of i, as not to create confusion with the electric current which is usually called i.^ In Physics and Electrical Engineering, imaginary numbers are used to describe electric current and other things that can have cyclical patterns.
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

^ One comment just for fun: Did you know that engineers (at least electrical engineers) use “j” instead of “i” to denote sqrt(-1)?
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

Operations in polar form

Multiplication and division have simple formulas in polar form:
(r_1e^{i\varphi_1}) \cdot (r_2e^{i\varphi_2}) = r_1 r_2 e^{i(\varphi_1 + \varphi_2)}
and
\frac{r_1\,e^{i\varphi_1}}{r_2\,e^{i\varphi_2}} = \left(\frac{r_1}{r_2}\right)\,e^{i (\varphi_1 - \varphi_2)}.
.This form demonstrates that multiplication can be visualised as a simultaneous stretching and rotation of one of the multiplicands, adding to its angle the phase of the other and scaling its length.^ If you only want to rotate by 45 and not to scale, you have to multiply by a complex value with length 1.
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

^ Any set of x, y and z coordinates can be represented in a matrix, and other matrices can represent transformations like rotation and scaling.
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

.For example, multiplying by i corresponds to a quarter-rotation counter-clockwise, from which it is clear why i 2 = −1. In particular, multiplication by any number on the unit circle around the origin is a pure rotation.^ Then the resultant force is a + ib.if we want to rotate this resultant force 45 degree in counter clockwise direcion, then multiplying the number a+ib with 1+i will change the direction of the resultant force to 45 degree in counter clockwise direction.but what about the magnitude of the force?.it will not be same as sqrt(a^2 + b^2).but it will be 1.414*sqrt(a^2 + b^2).because the magnitude of 1+i is not equal to 1 .it is equal to sqrt(1^2 + 1^2)=1.414.multiplying a+ib with 1+i actually increasing the original magnitude of the force along with rotating it 45 degree so to rotate the force vector we have to multiply it by unit vector.unit vector can be obtained just by dividing any vector with its own magnitude.In our case it is (1+i)/1.414 so our answer will be (a+ib)*(1+i)/1.414 which is same as a+ib offset 45 degree from current position with same magnitude.
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

^ Original heading: 3 units East, 4 units North = 3 + 4i Rotate counter-clockwise by 45 degrees = multiply by 1 + i If we multiply them together we get: .
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

^ We can’t multiply by a positive twice, because the result stays positive We can’t multiply by a negative twice, because the result will flip back to positive on the second multiplication But what about… a rotation !
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

Division is the same, in reverse.
Exponentiation is also simple; with integer exponents:
 (r(\cos\varphi + i\sin\varphi))^n = r^n\,(\cos n\varphi + i \sin n \varphi). [De Moivre's formula]
Arbitrary complex exponents are discussed in Exponentiation.
Finally, polar forms are also useful for finding roots. .Any complex number z satisfying zn = c (for n a positive integer) is called an nth root of c.^ We are looking for that length y that gives y^2=-1, that number y is square root of -1 but that doesn’t mean anything since square root is only defined for positive numbers so we need a new symbol for that unit.
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

^ Complex Number -> Given an angle, solves what would be the new position .
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

^ Yep, there’s something called quaternions which are complex numbers extended to 4 dimensions, and are used in graphics programming for 3d rotations (like complex numbers can be used for 2d rotations).
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

If c is non-zero, there are exactly n distinct nth roots of c (by the fundamental theorem of algebra). Let c = re  with r > 0; then the set of nth roots of c is
 \left\{ \sqrt[n]r\,e^{i\left(\frac{\varphi+2k\pi}{n}\right)} \mid k\in\{0,1,\ldots,n-1\} \, \right\},
where .\sqrt[n]{r} represents the usual (positive) nth root of the positive real number r.^ Awesome, so… if real numbers are represented in one dimension and complex numbers are represented in two dimensions, what’s represented in three, four, n dimensions?
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

^ A real number is a point on the number line, a complex number represents a vector on the two dimension plane.
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

^ We are looking for that length y that gives y^2=-1, that number y is square root of -1 but that doesn’t mean anything since square root is only defined for positive numbers so we need a new symbol for that unit.
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

If c = 0, then the only nth root of c is 0 itself, which as nth root of 0 is considered to have multiplicity n, hence these do represent all the n roots. .Note that the roots differ only by the rotations e2kπi/n, the nth roots of unity, so all the roots of c lie on a circle about the origin.^ I too am shocked that the “rotation” analogy wasn’t shown when I originally learned about i (in high school).
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

^ It’s only a difference in the use of a symbol, but I think it’s a rather interesting “cultural” difference to know about.
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

Some properties

Matrix representation of complex numbers

.While usually not useful, alternative representations of the complex field can give some insight into its nature.^ Reading your articles gives deep insights into maths (fills the gaps and links things together).
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

.One particularly elegant representation interprets each complex number as a 2×2 matrix with real entries which stretches and rotates the points of the plane.^ A real number is a point on the number line, a complex number represents a vector on the two dimension plane.
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

^ In general, imaginary numbers are good for things that move in cycles (since i can be seen as rotations about a center point).
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

^ Having no real knowledge of what ‘i’ is, never knowing more than it’s the result of the square root of negative one, I had the thought of drawing a geometric plane with one axis being the real number set and the other axis being the imaginary number set, as you showed above.
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

Every such matrix has the form
 \begin{bmatrix} a & -b \ b & \;\; a \end{bmatrix}
where a and b are real numbers. The sum and product of two such matrices is again of this form, and the product operation on matrices of this form is commutative. Every non-zero matrix of this form is invertible, and its inverse is again of this form. .Therefore, the matrices of this form are a field, isomorphic to the field of complex numbers.^ You can model imaginary numbers using vectors and matrices (more generally in the field of linear algebra).
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

Every such matrix can be written as
 \begin{bmatrix} a & -b \ b & \;\; a \end{bmatrix} = a \begin{bmatrix} 1 & \;\; 0 \ 0 & \;\; 1 \end{bmatrix} + b \begin{bmatrix} 0 & -1 \ 1 & \;\; 0 \end{bmatrix}
which suggests that we should identify the real number 1 with the identity matrix
 \begin{bmatrix} 1 & \;\; 0 \ 0 & \;\; 1 \end{bmatrix},
and the imaginary unit i with
 \begin{bmatrix} 0 & -1 \ 1 & \;\; 0 \end{bmatrix},
a counter-clockwise rotation by 90 degrees. Note that the square of this latter matrix is indeed equal to the 2×2 matrix that represents −1.
.More formally, this matrix representation is the regular representation of the complex numbers, thought of as an R-algebra (an R-vector space with a multiplication), with respect to the basis 1,i: the complex numbers are a 2-dimensional vector space over the real numbers, and multiplication by a complex number is a linear map (by distributivity) of the complex numbers to themselves, which is thus represented by a 2×2 matrix once a basis has been chosen.^ A real number is a point on the number line, a complex number represents a vector on the two dimension plane.
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

^ You can model imaginary numbers using vectors and matrices (more generally in the field of linear algebra).
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

^ Regular multiplication (times 2) increases the magnitude (size) of a number.
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

Thus this is not an ad hoc construction, but can be applied to any K-algebra over a field. For example, if the matrix elements are themselves complex numbers, the resulting algebra is that of the quaternions; stated alternatively, the quaternions are a 2-dimensional C-algebra, and hence their regular representation is as 2×2 complex matrices. Generalizing alternatively, this matrix representation is one way of expressing the Cayley–Dickson construction of algebras.
The square of the absolute value of a complex number expressed as a matrix is equal to the determinant of that matrix.
 |z|^2 = \begin{vmatrix} a & -b \ b & a \end{vmatrix} = (a^2) - ((-b)(b)) = a^2 + b^2.
.If the matrix is viewed as a transformation of the plane, then the transformation rotates points through an angle equal to the argument of the complex number and scales by a factor equal to the complex number's absolute value.^ Multiplying by a complex number rotates by its angle Let’s take a look.
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

^ In general, imaginary numbers are good for things that move in cycles (since i can be seen as rotations about a center point).
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

^ I have occasionally pointed people towards http://mathforum.org/johnandbetty/ which is good for the early stages of complex numbers.
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

.The conjugate of the complex number z corresponds to the transformation which rotates through the same angle as z but in the opposite direction, and scales in the same manner as z; this can be represented by the transpose of the matrix corresponding to z.^ Any set of x, y and z coordinates can be represented in a matrix, and other matrices can represent transformations like rotation and scaling.
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

^ Multiplying by a complex number rotates by its angle Let’s take a look.
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

^ Multiplying i is a rotation by 90 degrees counter-clockwise Multiplying by -i is a rotation of 90 degrees clockwise Two rotations in either direction is -1: it brings us back into the “regular” dimensions of positive and negative numbers.
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

This is generalized in the polar decomposition of matrices.
.It should also be noted that the two eigenvalues of the 2x2 matrix representing a complex number are the complex number itself and its conjugate.^ Yep that’s right, I was being paid and didn’t even realise that a complex number was just a number with two dimensions.
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

^ Dave: a pair (z,s) of two complex numbers would “live” in 4 dimensional space.
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

^ Awesome, so… if real numbers are represented in one dimension and complex numbers are represented in two dimensions, what’s represented in three, four, n dimensions?
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

While the above is a linear representation of C in the 2 × 2 real matrices, it is not the only one. Any matrix
J = \begin{pmatrix}p & q \\ r & -p \end{pmatrix}, \quad p^2 + qr + 1 = 0
has the property that its square is the negative of the identity matrix: J2 = − I. Then \{ z = a I + b J : a,b \in R \} is also isomorphic to the field C, and gives an alternative complex structure on R2. This is generalized by the notion of a linear complex structure.

Real vector space

.C is a two-dimensional real vector space.^ Dave: a pair (z,s) of two complex numbers would “live” in 4 dimensional space.
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

^ A real number is a point on the number line, a complex number represents a vector on the two dimension plane.
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

^ Imaginary nu,mbers open that out into a two dimensional complex number space.
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

.Unlike the reals, the set of complex numbers cannot be totally ordered in any way that is compatible with its arithmetic operations: C cannot be turned into an ordered field.^ Complex numbers are similar — it’s a new way of thinking.
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

^ A relevant lecture on complex numbers with a little diversion into Eulers formula .
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

^ Awesome, so… if real numbers are represented in one dimension and complex numbers are represented in two dimensions, what’s represented in three, four, n dimensions?
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

.More generally, no field containing a square root of −1 can be ordered.^ You can model imaginary numbers using vectors and matrices (more generally in the field of linear algebra).
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

^ Having no real knowledge of what ‘i’ is, never knowing more than it’s the result of the square root of negative one, I had the thought of drawing a geometric plane with one axis being the real number set and the other axis being the imaginary number set, as you showed above.
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

R-linear maps CC have the general form
f(z)=az+b\overline{z}
with complex coefficients a and b. .Only the first term is C-linear, and only the first term is holomorphic; the second term is real-differentiable, but does not satisfy the Cauchy-Riemann equations.^ I did the equations but never really felt satisfied, it was after I read your article on finding pi that I got more interested in math.
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

The function
f(z)=az\,
corresponds to rotations combined with scaling, while the function
f(z)=b\overline{z}
corresponds to reflections combined with scaling.

Solutions of polynomial equations

A root of the polynomial p is a complex number z such that p(z) = 0. A surprising result in complex analysis is that all polynomials of degree n with real or complex coefficients have exactly n complex roots (counting multiple roots according to their multiplicity). This is known as the fundamental theorem of algebra, and it shows that the complex numbers are an algebraically closed field. .Indeed, the complex numbers are the algebraic closure of the real numbers, as described below.^ Awesome, so… if real numbers are represented in one dimension and complex numbers are represented in two dimensions, what’s represented in three, four, n dimensions?
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

^ A real number is a point on the number line, a complex number represents a vector on the two dimension plane.
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

^ A complex number is the fancy name for numbers with both real and imaginary parts.
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

Construction and algebraic characterization

One construction of C is as a field extension of the field R of real numbers, in which a root of x2+1 is added. To construct this extension, begin with the polynomial ring R[x] of the real numbers in the variable x. Because the polynomial x2+1 is irreducible over R, the quotient ring R[x]/(x2+1) will be a field. .This extension field will contain two square roots of -1; one of them is selected and denoted i.^ So there’s really two square roots of -1: i and -i .
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

The set {1, i} will form a basis for the extension field over the reals, which means that each element of the extension field can be written in the form a+ b·i. Equivalently, elements of the extension field can be written as ordered pairs (a,b) of real numbers.
.Although only roots of x2+1 were explicitly added, the resulting complex field is actually algebraically closed – every polynomial with coefficients in C factors into linear polynomials with coefficients in C.^ You can model imaginary numbers using vectors and matrices (more generally in the field of linear algebra).
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

^ Believe it or not, I only understood this when I was actually working on complex power as an electrical engineer.
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

Because each field has only one algebraic closure, up to field isomorphism, the complex numbers can be characterized as the algebraic closure of the real numbers.
The field extension does yield the well-known complex plane, but it only characterizes it algebraically. The field C is characterized up to field isomorphism by the following three properties:
One consequence of this characterization is that C contains many proper subfields which are isomorphic to C (the same is true of R, which contains many subfields isomorphic to itself[citation needed]). As described below, topological considerations are needed to distinguish these subfields from the fields C and R themselves.

Characterization as a topological field

As just noted, the algebraic characterization of C fails to capture some of its most important topological properties. .These properties are key for the study of complex analysis, where the complex numbers are studied as a topological field.^ Nice guide, I was just studying the (inverse) euler formulas and didn’t really understand complex numbers..
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

The following properties characterize C as a topological field:[citation needed]
  • C is a field.
  • C contains a subset P of nonzero elements satisfying:
    • P is closed under addition, multiplication and taking inverses.
    • If x and y are distinct elements of P, then either x-y or y-x is in P
    • If S is any nonempty subset of P, then S+P=x+P for some x in C.
  • C has a nontrivial involutive automorphism x→x*, fixing P and such that xx* is in P for any nonzero x in C.
Given a field with these properties, one can define a topology by taking the sets
  • B(x,p) = \{y | p - (y-x)(y-x)^*\in P\}
as a base, where x ranges over the field and p ranges over P.
To see that these properties characterize C as a topological field, one notes that P ∪ {0} ∪ -P is an ordered Dedekind-complete field and thus can be identified with the real numbers R by a unique field isomorphism. The last property is easily seen to imply that the Galois group over the real numbers is of order two, completing the characterization.
Pontryagin has shown that the only connected locally compact topological fields are R and C. This gives another characterization of C as a topological field, since C can be distinguished from R by noting that the nonzero complex numbers are connected, while the nonzero real numbers are not.

Complex analysis

The study of functions of a complex variable is known as complex analysis and has enormous practical use in applied mathematics as well as in other branches of mathematics. Often, the most natural proofs for statements in real analysis or even number theory employ techniques from complex analysis (see prime number theorem for an example). Unlike real functions which are commonly represented as two dimensional graphs, complex functions have four dimensional graphs and may usefully be illustrated by color coding a three-dimensional graph to suggest four dimensions, or by animating the complex function's dynamic transformation of the complex plane.

Applications

Some applications of complex numbers are:

Control theory

In control theory, systems are often transformed from the time domain to the frequency domain using the Laplace transform. The system's poles and zeros are then analyzed in the complex plane. The root locus, Nyquist plot, and Nichols plot techniques all make use of the complex plane.
In the root locus method, it is especially important whether the poles and zeros are in the left or right half planes, i.e. have real part greater than or less than zero. If a system has poles that are
If a system has zeros in the right half plane, it is a nonminimum phase system.

Signal analysis

Complex numbers are used in signal analysis and other fields for a convenient description for periodically varying signals. For given real functions representing actual physical quantities, often in terms of sines and cosines, corresponding complex functions are considered of which the real parts are the original quantities. For a sine wave of a given frequency, the absolute value |z| of the corresponding z is the amplitude and the argument arg(z) the phase.
If Fourier analysis is employed to write a given real-valued signal as a sum of periodic functions, these periodic functions are often written as complex valued functions of the form
 f ( t ) = z e^{i\omega t} \,
where ω represents the angular frequency and the complex number z encodes the phase and amplitude as explained above.
In electrical engineering, the Fourier transform is used to analyze varying voltages and currents. The treatment of resistors, capacitors, and inductors can then be unified by introducing imaginary, frequency-dependent resistances for the latter two and combining all three in a single complex number called the impedance. .(Electrical engineers and some physicists use the letter j for the imaginary unit since i is typically reserved for varying currents and may come into conflict with i.^ In Physics and Electrical Engineering, imaginary numbers are used to describe electric current and other things that can have cyclical patterns.
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

^ We have the symbol “1″ for a positive unit, we have the symbol “-1″ for the negative unit but now we need a symbol for the lateral unit (the “imaginary” unit) which is i and it’s opposite “-i” which basically is laterally moving away from the horizontal number line into another dimension.
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

^ BTW, electrical engineering makes very *heavy* use of complex math.
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

) This approach is called phasor calculus. This use is also extended into digital signal processing and digital image processing, which utilize digital versions of Fourier analysis (and wavelet analysis) to transmit, compress, restore, and otherwise process digital audio signals, still images, and video signals.

Improper integrals

In applied fields, complex numbers are often used to compute certain real-valued improper integrals, by means of complex-valued functions. Several methods exist to do this; see methods of contour integration.

Quantum mechanics

The complex number field is relevant in the mathematical formulation of quantum mechanics, where complex Hilbert spaces provide the context for one such formulation that is convenient and perhaps most standard. The original foundation formulas of quantum mechanics – the Schrödinger equation and Heisenberg's matrix mechanics – make use of complex numbers.

Relativity

In special and general relativity, some formulas for the metric on spacetime become simpler if one takes the time variable to be imaginary. (This is no longer standard in classical relativity, but is used in an essential way in quantum field theory.) Complex numbers are essential to spinors, which are a generalization of the tensors used in relativity.

Applied mathematics

In differential equations, it is common to first find all complex roots r of the characteristic equation of a linear differential equation and then attempt to solve the system in terms of base functions of the form f(t) = ert.

Fluid dynamics

In fluid dynamics, complex functions are used to describe potential flow in two dimensions.

Fractals

Certain fractals are plotted in the complex plane, e.g. the Mandelbrot set and Julia sets.

History

The earliest fleeting reference to square roots of negative numbers perhaps occurred in the work of the Greek mathematician and inventor Heron of Alexandria in the 1st century AD, when, apparently inadvertently, he considered the volume of an impossible frustum of a pyramid,[4] though negative numbers were not conceived in the Hellenistic world.
Complex numbers became more prominent in the 16th century, when closed formulas for the roots of cubic and quartic polynomials were discovered by Italian mathematicians (see Niccolo Fontana Tartaglia, Gerolamo Cardano). .It was soon realized that these formulas, even if one was only interested in real solutions, sometimes required the manipulation of square roots of negative numbers.^ Negative numbers complete the “real” numbers in a one-dimensional number line.
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

^ Having no real knowledge of what ‘i’ is, never knowing more than it’s the result of the square root of negative one, I had the thought of drawing a geometric plane with one axis being the real number set and the other axis being the imaginary number set, as you showed above.
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

^ Okay, even the slightest application of critical thought (that is, provided one has not been taking these pills) will lead to the conclusion that this, like so much of the internet, is bogus.
  • Writing.Com: Complex Numbers (Book) 10 February 2010 13:39 UTC www.writing.com [Source type: FILTERED WITH BAYES]
  • Writing.Com: Complex Numbers (Book) 10 February 2010 13:39 UTC www.writing.com [Source type: FILTERED WITH BAYES]

For example, Tartaglia's cubic formula gives the following solution to the equation x3 − x = 0:
\frac{1}{\sqrt{3}}\left(\sqrt{-1}^{1/3}+\frac{1}{\sqrt{-1}^{1/3}}\right).
At first glance this looks like nonsense. However formal calculations with complex numbers show that the equation z3 = i has solutions –i, {\scriptstyle\frac{\sqrt{3}}{2}}+{\scriptstyle\frac{1}{2}}i and {\scriptstyle\frac{-\sqrt{3}}{2}}+{\scriptstyle\frac{1}{2}}i. Substituting these in turn for {\scriptstyle\sqrt{-1}^{1/3}} in Tartaglia's cubic formula and simplifying, one gets 0, 1 and −1 as the solutions of x3 – x = 0. Rafael Bombelli was the first to explicitly address these seemingly paradoxical solutions of cubic equations and developed the rules for complex arithmetic trying to resolve these issues.
.This was doubly unsettling since not even negative numbers were considered to be on firm ground at the time.^ Geez, his theorem shows up everywhere , even in numbers invented 2000 years after his time.
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

The term "imaginary" for these quantities was coined by René Descartes in 1637 and was meant to be derogatory[citation needed] (see imaginary number for a discussion of the "reality" of complex numbers). A further source of confusion was that the equation \sqrt{-1}^2=\sqrt{-1}\sqrt{-1}=-1 seemed to be capriciously inconsistent with the algebraic identity \sqrt{a}\sqrt{b}=\sqrt{ab}, which is valid for positive real numbers a and b, and which was also used in complex number calculations with one of a, b positive and the other negative. The incorrect use of this identity (and the related identity \scriptstyle 1/\sqrt{a}=\sqrt{1/a}) in the case when both a and b are negative even bedeviled Euler. This difficulty eventually led to the convention of using the special symbol i in place of \sqrt{-1} to guard against this mistake. Even so Euler considered it natural to introduce students to complex numbers much earlier than we do today. In his elementary algebra text book, Elements of Algebra, he introduces these numbers almost at once and then uses them in a natural way throughout.
In the 18th century complex numbers gained wider use, as it was noticed that formal manipulation of complex expressions could be used to simplify calculations involving trigonometric functions. For instance, in 1730 Abraham de Moivre noted that the complicated identities relating trigonometric functions of an integer multiple of an angle to powers of trigonometric functions of that angle could be simply reexpressed by the following well-known formula which bears his name, de Moivre's formula:
(\cos 	heta + i\sin 	heta)^{n} = \cos n 	heta + i\sin n 	heta. \,
In 1748 Leonhard Euler went further and obtained Euler's formula of complex analysis:
\cos 	heta + i\sin 	heta = e ^{i	heta } \,
by formally manipulating complex power series and observed that this formula could be used to reduce any trigonometric identity to much simpler exponential identities.
The existence of complex numbers was not completely accepted until the geometrical interpretation (see above) had been described by Caspar Wessel in 1799; it was rediscovered several years later and popularized by Carl Friedrich Gauss, and as a result the theory of complex numbers received a notable expansion. The idea of the graphic representation of complex numbers had appeared, however, as early as 1685, in Wallis's De Algebra tractatus.
Wessel's memoir appeared in the Proceedings of the Copenhagen Academy for 1799, and is exceedingly clear and complete, even in comparison with modern works. He also considers the sphere, and gives a quaternion theory from which he develops a complete spherical trigonometry. In 1804 the Abbé Buée independently came upon the same idea which Wallis had suggested, that \pm\sqrt{-1} should represent a unit line, and its negative, perpendicular to the real axis. Buée's paper was not published until 1806, in which year Jean-Robert Argand also issued a pamphlet on the same subject. .It is to Argand's essay that the scientific foundation for the graphic representation of complex numbers is now generally referred.^ Khalid, now I could make a pretty difference between ‘Complex Number’ & Trigs.
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

^ Thus, each complex number has a unique representation on the complex plane: some closer to real; others, more imaginary.
  • Writing.Com: Complex Numbers (Book) 10 February 2010 13:39 UTC www.writing.com [Source type: FILTERED WITH BAYES]
  • Writing.Com: Complex Numbers (Book) 10 February 2010 13:39 UTC www.writing.com [Source type: FILTERED WITH BAYES]

^ Yep, there’s something called quaternions which are complex numbers extended to 4 dimensions, and are used in graphics programming for 3d rotations (like complex numbers can be used for 2d rotations).
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

Nevertheless, in 1831 Gauss found the theory quite unknown, and in 1832 published his chief memoir on the subject, thus bringing it prominently before the mathematical world. Mention should also be made of an excellent little treatise by Mourey (1828), in which the foundations for the theory of directional numbers are scientifically laid. The general acceptance of the theory is not a little due to the labors of Augustin Louis Cauchy and Niels Henrik Abel, and especially the latter, who was the first to boldly use complex numbers with a success that is well known.
The common terms used in the theory are chiefly due to the founders. Argand called cosφ + isinφ the direction factor, and r = \sqrt{a^2+b^2} the modulus; Cauchy (1828) called cosφ + isinφ the reduced form (l'expression réduite); Gauss used i for \sqrt{-1}, introduced the term complex number for a + bi, and called a2 + b2 the norm.
The expression direction coefficient, often used for cosφ + isinφ, is due to Hankel (1867), and absolute value, for modulus, is due to Weierstrass.
Following Cauchy and Gauss have come a number of contributors of high rank, of whom the following may be especially mentioned: Kummer (1844), Leopold Kronecker (1845), Scheffler (1845, 1851, 1880), Bellavitis (1835, 1852), Peacock (1845), and De Morgan (1849). Möbius must also be mentioned for his numerous memoirs on the geometric applications of complex numbers, and Dirichlet for the expansion of the theory to include primes, congruences, reciprocity, etc., as in the case of real numbers.
A complex ring or field is a set of complex numbers which is closed under addition, subtraction, and multiplication. Gauss studied complex numbers of the form a + bi, where a and b are integral, or rational (and i is one of the two roots of x2 + 1 = 0). His student, Ferdinand Eisenstein, studied the type a + bω, where ω is a complex root of x3 − 1 = 0. Other such classes (called cyclotomic fields) of complex numbers are derived from the roots of unity xk − 1 = 0 for higher values of k. This generalization is largely due to Kummer, who also invented ideal numbers, which were expressed as geometrical entities by Felix Klein in 1893. The general theory of fields was created by Évariste Galois, who studied the fields generated by the roots of any polynomial equation in one variable.
The late writers (from 1884) on the general theory include Weierstrass, Schwarz, Richard Dedekind, Otto Hölder, Henri Poincaré, Eduard Study, and Alexander MacFarlane.

See also

Notes

References

Mathematical references

Historical references

  • Burton, David M. (1995), The History of Mathematics (3rd ed.), New York: McGraw-Hill, ISBN 978-0-07-009465-9 
  • Katz, Victor J. (2004), A History of Mathematics, Brief Version, Addison-Wesley, ISBN 978-0-321-16193-2 
  • Nahin, Paul J. (1998), An Imaginary Tale: The Story of \sqrt{-1} (hardcover ed.), Princeton University Press, ISBN 0-691-02795-1 
    A gentle introduction to the history of complex numbers and the beginnings of complex analysis.
  • H.-D. Ebbinghaus ... (1991), Numbers (hardcover ed.), Springer, ISBN 0-387-97497-0 
    An advanced perspective on the historical development of the concept of number.

Further reading

  • The Road to Reality: A Complete Guide to the Laws of the Universe, by Roger Penrose; Alfred A. Knopf, 2005; ISBN 0-679-45443-8. Chapters 4-7 in particular deal extensively (and enthusiastically) with complex numbers.
  • Unknown Quantity: A Real and Imaginary History of Algebra, by John Derbyshire; Joseph Henry Press; ISBN 0-309-09657-X (hardcover 2006). A very readable history with emphasis on solving polynomial equations and the structures of modern algebra.
  • Visual Complex Analysis, by Tristan Needham; Clarendon Press; ISBN 0-198-53447-7 (hardcover, 1997). History of complex numbers and complex analysis with compelling and useful visual interpretations.

External links


Study guide

Up to date as of January 14, 2010

From Wikiversity

Complex Numbers
Nuvola apps edu mathematics-p.svg Subject classification: this is a mathematics resource .
Sciences humaines.svg Educational level: this is a tertiary (university) resource.
SYawning.svg .Completion status: this resource has been started but most of the work is still to be done.^ Even so, the "simpler" proof is still daunting when worked out in complete formal detail, involving some 39 lemmas.
  • Real and Complex Numbers - Metamath Proof Explorer 10 February 2010 13:39 UTC us.metamath.org [Source type: FILTERED WITH BAYES]

.Complex numbers arise from dealing with the square root of negative numbers, such as the solutions to x^2 = -1\,, which are x=\pm\sqrt{-1}.^ Returns the "negative" of a Complex number.

^ Returns the square of this complex number.
  • Complex 10 February 2010 13:39 UTC jscience.org [Source type: Reference]

^ We next need to address an issue on dealing with square roots of negative numbers.
  • http://tutorial.math.lamar.edu/Classes/Alg/ComplexNumbers.aspx 10 February 2010 13:39 UTC tutorial.math.lamar.edu [Source type: FILTERED WITH BAYES]

.The foundation of complex number theory is the definition i = \sqrt{-1}, where i\, is referred to as an 'imaginary number'.^ Definition of nth Root of a Complex Number .

^ Suffix is the suffix for the imaginary component of the complex number ( I or j).

^ The pure imaginary numbers are also complex numbers, because b i = 0 + b i.
  • Intermediate Algebra/Complex Numbers - Wikibooks, collection of open-content textbooks 10 February 2010 13:39 UTC en.wikibooks.org [Source type: Reference]

A complex number \mathbf{z} is the sum of a real part a\, and an imaginary part b\,,
i.e. \mathbf{z} = a + \sqrt{-1}b = a + ib .
If b=0\, then z\in\mathbb{R}. .That is, z\, is a real number, and can be called 'pure real'.^ A purely real number is represented by a point on the real axis.
  • Intermediate Algebra/Complex Numbers - Wikibooks, collection of open-content textbooks 10 February 2010 13:39 UTC en.wikibooks.org [Source type: Reference]

^ In the complex number z = (x, y) = x + i y, x is called the real part of z and y is called the imaginary part.

^ Complex Number: An ordered pair of real numbers a, b, written as (a, b) is called a complex number If we write z = (a, b), then a is called the real part of z and b the imaginary part of z.

Conversely, if a=0\, then z\in\mathbb{I}, and z\, is 'pure imaginary'. .We can refer specifically to the real and imaginary parts (a\, and b\,) of z\, respectively as follows: \operatorname{Re}(z) \equiv a and \operatorname{Im}(z) \equiv b.^ The abscissa and the ordinate are then referred to as the real part and the imaginary part of the complex number.
  • Complex numbers 10 February 2010 13:39 UTC www.math.grin.edu [Source type: Reference]

^ Returns a Complex from real and imaginary parts.

^ [On equating real and imaginary parts.


.Whilst the real numbers are readily visualised by considering a straight "number line", complex numbers are best seen as being positions on a plane.^ We have that horizontal number line from negative and positive numbers.
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

^ A complex number z is said to be purely real if; .
  • Assignment help :: Math :: Algebra :: COMPLEX NUMBER 10 February 2010 13:39 UTC www.assignmenthelp.net [Source type: Reference]

^ Subtracts a complex and a real number.

.This plane would have one axis considered to be the real axis, the equvalent to the x axis on the cartesian plane (where \operatorname{Im}(z)=0), and the other an imaginary axis, the equvalent to the y axis on the cartesian plane (where \operatorname{Re}(z)=0).^ The argument is the angle phi between the positive real axis and the point representing this number in the complex plane.
  • Complex (Commons Math 2.1-SNAPSHOT API) 10 February 2010 13:39 UTC commons.apache.org [Source type: Reference]

^ The Cartesian coordinates of the complex number are the real part x = Re( z ) and the imaginary part y = Im( z ).
  • WikiSlice 10 February 2010 13:39 UTC dev.laptop.org [Source type: Reference]

^ Having no real knowledge of what ‘i’ is, never knowing more than it’s the result of the square root of negative one, I had the thought of drawing a geometric plane with one axis being the real number set and the other axis being the imaginary number set, as you showed above.
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

.A typical example of complex numbers is the w:quadratic equation for finding the roots of a second order polynomial: ax^2 + bx +c = 0\,.^ Example 4 We have two complex numbers: z 1 = 4 j 6 and z 2 = 5 e j 45 .
  • Complex numbers 10 February 2010 13:39 UTC www.tina.com [Source type: Reference]

^ The equation for finding a complex exponent of a complex number looks like this: .
  • Complex Numbers - Winamp Developer Wiki 10 February 2010 13:39 UTC dev.winamp.com [Source type: FILTERED WITH BAYES]

^ BUT I can not find away of of using complex numbers in c++.
  • Complex numbers and equation optimization using GA - CodeGuru Forums 10 February 2010 13:39 UTC www.codeguru.com [Source type: FILTERED WITH BAYES]


The roots are found by the formula:\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}[1].

Contents

Basics

Imagine the equation x^2=-1\,
What are the roots?
There is no number that multiplies by itself to give -1, so we make up a number called the imaginary number,i so that
\,i=\sqrt{-1},
(letting i = the "positive" root-though in fact it is irrelevant as this is defined as such)
So in our second equation, \,x=i or \,x=-i
And also any negative square root can now be found, for example:
\,x=\sqrt{-64}
\,x=\sqrt{64}	imes\sqrt{-1}
\,x=8	imes\sqrt{-1}
so \,x=8i
Any number that is a multiple of i is an imaginary number.
But, if \,a=i, and \,b=10, what does \,a+b equal?
So we say \,z = a+b
and \,z = 10+i
so z is a complex number, that is a number that has a real part and an imaginary part.

Manipulation

.In this section we will learn how to use basic maths on complex numbers.^ BUT I can not find away of of using complex numbers in c++.
  • Complex numbers and equation optimization using GA - CodeGuru Forums 10 February 2010 13:39 UTC www.codeguru.com [Source type: FILTERED WITH BAYES]

^ Excel can be used to deal with complex numbers.

^ So this number can simply be used anywhere any complex can be used.
  • Complex numbers and equation optimization using GA - CodeGuru Forums 10 February 2010 13:39 UTC www.codeguru.com [Source type: FILTERED WITH BAYES]

Addition and Subtraction

.Here we treat a complex number like any other piece of algebra.^ Other properties of complex numbers .
  • An introduction to complex numbers 10 February 2010 13:39 UTC www.ping.be [Source type: Academic]

^ Here is a presentation of the theory of complex numbers.

^ Suppose here’s a complex number.
  • Pre-Calculus: Complex Numbers - Trig or Polar Form | MindBites.com 10 February 2010 13:39 UTC www.mindbites.com [Source type: FILTERED WITH BAYES]

We add the real parts and the imaginary parts seperately, just like numbers and constants:
(a+ib) ~+~ (c+id) = ((a+c) + i(b+d))
and
(a+ib)~-~(c+id) ~+~ (e+if) = ( ( a-c+e ) ~+~ i( b-d+f ) )

Example

We know that \,(3 + 7x) + (1 + x) = 4 + 8x
In the same way \,(3 + 7i) + (1 + i) = 4 + 8i
The same is for subtraction , if \,z=4+3i
then \,z-2i = 4+i

Multiplication

.The same procedure to multiply polynomials is applicable to multiplying complex numbers: term by term.^ A root of the polynomial p is a complex number z such that p ( z ) = 0.
  • WikiSlice 10 February 2010 13:39 UTC dev.laptop.org [Source type: Reference]

^ Polynomials and complex numbers .
  • An introduction to complex numbers 10 February 2010 13:39 UTC www.ping.be [Source type: Academic]

^ The result of multiplying two complex numbers.
  • Complex.js 10 February 2010 13:39 UTC thejit.org [Source type: Reference]

Simply keep in mind that:
 i * i ~=~ i^2 ~=~ \sqrt{-1} ~ * ~ \sqrt{-1} ~=~-1

So:
 (a + ib)~*~(c + id) ~~=~~ ac ~+~ iad ~+~ ibc ~+~ i^2bd

Group the terms (remember: i^2=-1\,)

 ~~=~~ ac ~+~ iad ~+~ ibc ~+~ (-1)bd
 ~~=~~ ac ~+~ iad ~+~ ibc ~-~ bd
 ~~=~~ ac ~-~ bd ~+~ iad ~+~ ibc
 ~~=~~ (ac - bd)~+~i(ad + bc)

Scalar Multiplication

.Multiplying by a real number (w:scalar), again uses the same rules as algebra.^ Then, of course, these are not the same numbers that we are used to.

^ A further source of confusion was that the equation seemed to be capriciously inconsistent with the algebraic identity , which is valid for positive real numbers a and b , and which was also used in complex number calculations with one of a , b positive and the other negative.
  • WikiSlice 10 February 2010 13:39 UTC dev.laptop.org [Source type: Reference]

^ You can model imaginary numbers using vectors and matrices (more generally in the field of linear algebra).
  • A Visual, Intuitive Guide to Imaginary Numbers | BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]

So that \,3	imes(1+i) = 3+3i

Example

We know that \,(2+3x)	imes(1+x) = 2 + 5x + 3x^2

So to \,(2+3i)	imes(1+i) = 2 + 5i + 3i^2
Group the terms (remember: i^2=-1\,)

\,~~=~~ 2 + 5i + 3i^2
\,~~=~~ 2 + 5i - 3
\,~~=~~ -1 + 5i

Division and the "Complex Conjugate"

.We do not have a way to divide by a complex number, so we need to elminate any complex numbers we have in the denominator.^ We can divide a complex number ( a + bi ) by another complex number ( c + di ) ≠ 0 in two ways.
  • WikiSlice 10 February 2010 13:39 UTC dev.laptop.org [Source type: Reference]

^ Here is where we need to abandon scalar numbers for something better suited: complex numbers .
  • Lessons In Electric Circuits -- Volume II (AC) - Chapter 2 10 February 2010 13:39 UTC openbookproject.net [Source type: FILTERED WITH BAYES]

^ To begin with, there should be ways of constructing a complex number from its coordinates, either rectangular or polar: make-rectangular Inputs: real-part and imaginary-part , both real numbers.
  • Complex numbers 10 February 2010 13:39 UTC www.math.grin.edu [Source type: Reference]

.First step is to use the above methods (addition, subtraction and multiplication to get the denominator into one complex number, \,a+bi.^ First Addition Method , head-to-tail addition.
  • Help for Understanding and Using Complex Numbers 10 February 2010 13:39 UTC whyslopes.com [Source type: Reference]

^ What is the rule for addition of complex numbers?

^ Complex subtract ( Complex z) Returns the subtraction of this complex number by another.
  • Complex 10 February 2010 13:39 UTC jscience.org [Source type: Reference]

We know that \,(2+i)(2-i) = 4+2i-2i-i^2 = 4-(-1)= 5
So too in general \,(a+bi)(a-bi) = a^2+abi-abi-(b^2)(i^2) = a^2 + b^2
.That is to divide, we can get rid of a complex number a+bi on the denominator by multiplying it by a-bi numerator and denominator.^ We can divide a complex number ( a + bi ) by another complex number ( c + di ) ≠ 0 in two ways.
  • WikiSlice 10 February 2010 13:39 UTC dev.laptop.org [Source type: Reference]

^ We realize the denominator by multiplying with the conjugate of the denominator; a complex number times its conjugate is a real number: .
  • Intermediate Algebra/Complex Numbers - Wikibooks, collection of open-content textbooks 10 February 2010 13:39 UTC en.wikibooks.org [Source type: Reference]

^ Complex numbers, written in the form (a + bi), are an extension of the real numbers obtained by adjoining an imaginary unit, denoted by i, which is the square root of negative 1.
  • Pre-Calculus: Complex Numbers - Trig or Polar Form | MindBites.com 10 February 2010 13:39 UTC www.mindbites.com [Source type: FILTERED WITH BAYES]

If : z=a+ib \, then :\,a-ib=\overline(z) which is the complex conjugate.
So the rule is:
Multiply numerator and denominator by the complex conjugate of the denominator, and then result can be simplfied.
e.g. \frac{\,4i}{\,1+i} = \frac{\,4i	imes(1-i)}{\,(1+i)	imes(1-i)} = \frac{\,4i - 4i^2}{\,1-i^2} =\frac{\,4+ 4i}{\,2}= \,2+2i

Polar Notation

.We have already seen the cartesian representation of complex numbers, expressed in terms of real and imaginary parts.^ Real (double real) Returns the addition of this complex number with a real part.
  • Complex 10 February 2010 13:39 UTC jscience.org [Source type: Reference]

^ The Cartesian coordinates of the complex number are the real part x = Re( z ) and the imaginary part y = Im( z ).
  • WikiSlice 10 February 2010 13:39 UTC dev.laptop.org [Source type: Reference]

^ Suffix is the suffix for the imaginary component of the complex number ( I or j).

.Alternatively, we can represent a complex number by a magnitude and direction, \mathbf{r} and \mathbf	heta.^ Complex NaN A complex number representing "NaN + NaNi" .
  • Complex (Commons Math 2.1-SNAPSHOT API) 10 February 2010 13:39 UTC commons.apache.org [Source type: Reference]

^ INF A complex number representing "+INF + INFi" .
  • Complex (Commons Math 2.1-SNAPSHOT API) 10 February 2010 13:39 UTC commons.apache.org [Source type: Reference]

^ NaN A complex number representing "NaN + NaNi" .
  • Complex (Commons Math 2.1-SNAPSHOT API) 10 February 2010 13:39 UTC commons.apache.org [Source type: Reference]

In this representation a complex number z can be represented as,
(r,	heta)\,
Euler's formula shown graphically when r = 1
where,
{r}=\sqrt{Re(z)^2+Im(z)^2}\ (=\lVert{z}\rVert)
	heta=	an^{-1}\frac{Im(z)}{Re(z)}
This also allows one to write,
\mathbf{z}=r(\cos	heta+i\sin	heta)\Leftrightarrow\mathbf{z}=re^{i	heta}
When r = 1 this is known as w:Euler's formula.[2][3]
e^{i	heta} = \cos 	heta + i\sin 	heta \!
.Note: To represent a complex number graphically, simply draw a vector on the X - Y plane with the X axis \mathbf r\cos	heta and the Y axis \mathbf r\sin	heta.^ Jstor.   A note on ordering the complex numbers.

^ Complex NaN A complex number representing "NaN + NaNi" .
  • Complex (Commons Math 2.1-SNAPSHOT API) 10 February 2010 13:39 UTC commons.apache.org [Source type: Reference]

^ Complex numbers are easier to grasp when they're represented graphically.
  • Lessons In Electric Circuits -- Volume II (AC) - Chapter 2 10 February 2010 13:39 UTC openbookproject.net [Source type: FILTERED WITH BAYES]

Complex Plane

.It is useful to be able to represent complex numbers geometrically, just as we are able to represent real numbers geometrically on the number line.^ Complex numbers are the extension of the real numbers, i.e., the number line, into a number plane.
  • Intermediate Algebra/Complex Numbers - Wikibooks, collection of open-content textbooks 10 February 2010 13:39 UTC en.wikibooks.org [Source type: Reference]

^ Complex real (double real) Returns a Complex representing a real number.

^ Complex (double re) Constructs a Complex representing a real number.

.Moreover, we don’t need any genuinely new ideas to do this.^ Before we get on to that, we need a couple of new ideas.
  • nrich.maths.org :: Mathematics Enrichment :: An Introduction to Complex Numbers 10 February 2010 13:39 UTC nrich.maths.org [Source type: FILTERED WITH BAYES]

.We know that each pair of real numbers (x, y) corresponds uniquely to a point P(x, y) in the cartesian plane.^ Since a complex number a + bi is uniquely specified by an ordered pair ( a , b ) of real numbers, the complex numbers are in one-to-one correspondence with points on a plane, called the complex plane.
  • WikiSlice 10 February 2010 13:39 UTC dev.laptop.org [Source type: Reference]

^ A complex plane (or Argand diagram ) is any 2D graph in which the horizontal axis is the real part and the vertical axis is the imaginary part of a complex number or function.

^ The argument is the angle phi between the positive real axis and the point representing this number in the complex plane.
  • Complex (Commons Math 2.1-SNAPSHOT API) 10 February 2010 13:39 UTC commons.apache.org [Source type: Reference]


.To represent complex numbers geometrically, we use the same representation: a complex number z = x+iy is represented by the point P(x, y) in the plane, as in the following diagram: Complexplane.JPG
When the plane is used to represent complex numbers in this way, it is called the Complex Plane or the Argand Diagram.
^ Matrix representation of complex numbers .
  • WikiSlice 10 February 2010 13:39 UTC dev.laptop.org [Source type: Reference]

^ BUT I can not find away of of using complex numbers in c++.
  • Complex numbers and equation optimization using GA - CodeGuru Forums 10 February 2010 13:39 UTC www.codeguru.com [Source type: FILTERED WITH BAYES]

^ Geometric interpretation of the operations on complex numbers .
  • WikiSlice 10 February 2010 13:39 UTC dev.laptop.org [Source type: Reference]

References

See Also


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