From Wikipedia, the free encyclopedia
.^ 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.
.^ Thus, every complex number has a real part, a , and an imaginary part, b i . 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]
^ 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. 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]
.^ Thus, every complex number has a real part, a , and an imaginary part, b i . 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]
^ 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. 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]
This is in order to form a
closed field, where any
polynomial equation has a
root, including examples such as
x^{2} = −1.
.^ 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]
^ 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]} .^ 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]
.^ 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]} .^ 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]
.^ 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]
.^ 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.
Definitions
Notation
.^ 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
where
.^ 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. 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]
^ 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]
^ 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). 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]
.^ Thus, every complex number has a real part, a , and an imaginary part, b i . 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]
^ 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]
.^ Thus, every complex number has a real part, a , and an imaginary part, b i . 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]
^ 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. 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]
.^ 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]
.^ 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]
.^ 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 fourdimension 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]
.^ 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 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]
.^ Thus, every complex number has a real part, a , and an imaginary part, b i . 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]
^ If a = b , the number is equal parts real and 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]
^ 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]
Formal development
.^ 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). 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]
Firstly, write
C for
R^{2}, the set of
ordered pairs of real numbers, and define operations on complex numbers in
C according to
 (a, b) + (c, d) = (a + c, b + d)
 (a, b)·(c, d) = (a·c − b·d, b·c + a·d).
It is then just a matter of notation to express (
a,
b) 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
R^{2}, it is easy to check that we satisfy the
field axioms (associativity, commutativity, identity, inverses, distributivity). In particular,
R is a subfield of
C.
.^ 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]
.^ Also, I like what you said about math being a language that is selfdescribing 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): = (ac − bd) + (bc + ad)i. 
[multiplication] 
The notation is welldesigned, since proceeding to manipulate the symbols purely intuitively gives
 (a + bi)(c + di) = ac + bci + adi + bdi^{2} = (ac − bd) + (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, u − w means "u plus the number which cancels out w (its additive inverse)", giving

(a + bi) − (c + di) = (a − c) + (b − d)i. 
[subtraction] 
Similarly for multiplication, the inverse of c + di is found to be
when c and d are not both zero (by solving for z in z(c + di) = 1), which then yields


[division] 
If the symbols are manipulated intuitively, so that
the formula agrees with the one just derived.
.^ It is nice to have an explanation on why they work instead of the usually “Here is the rule. A Visual, Intuitive Guide to Imaginary Numbers  BetterExplained 10 February 2010 13:39 UTC betterexplained.com [Source type: General]
^ 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]
^ 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]
.^ 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]
.^ 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]
Where the radical symbol (√) is used, it is normally used to denote the root with smallest nonnegative phase (see section below on polar representation).
Conjugation
Define the
conjugate of
z = x + iy to be
x − iy, written as
or
z * .
.^ 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]
^ 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]
.^ 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
is not complexdifferentiable (see
Holomorphic function).
Elementary functions
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.
.^ 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]
The inverse of exp is shown to be
Exponentiation
Raising numbers to positive integer powers is done using the operation of multiplication:
.^ 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 (nonrepeating 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]
.^ 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 (8
^{1/3})
^{4} ≠ 16, as there are three cube roots of 8, so the given expression, often shortened to 8
^{4/3}, is the simplest possible.
.^ 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
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 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]
.^ 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]
.^ 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.

. = 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]
.^ 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]


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. 

. = 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]

.^ 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.
.^ 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
re^{iφ} 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,
.^ 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]
.^ 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 (nonrepeating 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]
.^ 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:
 (taking the sign appropriately so that x + iy = r(cosφ + isinφ).
The value of
φ can change by any multiple of 2
π 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).
.^ 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
In angle notation
θ may be in either radians or degrees.
.^ 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:
and
.^ 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]
.^ 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 counterclockwise 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:

Finally, polar forms are also useful for finding roots.
.^ 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 nonzero, there are exactly
n distinct
nth roots of
c (by the
fundamental theorem of algebra). Let
c =
re^{ iφ} with
r > 0; then the set of
nth roots of
c is
where
.^ 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.
.^ 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
.^ 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]
.^ 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
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 nonzero matrix of this form is invertible, and its inverse is again of this form.
.^ 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
which suggests that we should identify the real number 1 with the identity matrix
and the imaginary unit i with
a counterclockwise rotation by 90 degrees. Note that the square of this latter matrix is indeed equal to the 2×2 matrix that represents −1.
.^ 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
Kalgebra 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 2dimensional
Calgebra, 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.
.^ 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]
.^ 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 counterclockwise 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.
.^ 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]
has the property that its square is the negative of the identity matrix:
J^{2} = − I. Then
is also isomorphic to the field
C, and gives an alternative complex structure on
R^{2}. This is generalized by the notion of a
linear complex structure.
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]
.^ 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]
.^ 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]
Rlinear maps
C →
C have the general form
with complex coefficients
a and
b.
.^ 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
corresponds to rotations combined with scaling, while the function
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.
.^ 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
x^{2}+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
x^{2}+1 is
irreducible over
R, the
quotient ring R[
x]/(
x^{2}+1) will be a field.
.^ 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.
.^ 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 wellknown 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.
.^ 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 xy or yx 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
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
Dedekindcomplete 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.
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
threedimensional 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 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
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 realvalued signal as a sum of periodic functions, these periodic functions are often written as complex valued functions of the form
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, frequencydependent resistances for the latter two and combining all three in a single complex number called the
impedance.
.^ 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
Quantum mechanics
Relativity
Applied mathematics
Fluid dynamics
Fractals
History
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).
.^ Negative numbers complete the “real” numbers in a onedimensional 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
x^{3} −
x = 0:
At first glance this looks like nonsense. However formal calculations with complex numbers show that the equation
z^{3} =
i has solutions
–i,
and
. Substituting these in turn for
in Tartaglia's cubic formula and simplifying, one gets 0, 1 and −1 as the solutions of
x^{3} –
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.
.^ 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
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. The incorrect use of this identity (and the related identity
) 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
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 wellknown formula which bears his name,
de Moivre's formula:
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
should represent a unit line, and its negative, perpendicular to the real axis. Buée's paper was not published until 1806, in which year
JeanRobert Argand also issued a pamphlet on the same subject.
.^ 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
the
modulus; Cauchy (1828) called
cosφ + isinφ the
reduced form (l'expression réduite); Gauss used
i for
, introduced the term
complex number for
a +
bi, and called
a^{2} +
b^{2} 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
x^{2} + 1 = 0). His student,
Ferdinand Eisenstein, studied the type
a + bω, where
ω is a complex root of
x^{3} − 1 = 0. Other such classes (called
cyclotomic fields) of complex numbers are derived from the
roots of unity x^{k} − 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.
See also
Notes
References
Mathematical references
 Ahlfors, Lars (1979), Complex analysis (3rd ed.), McGrawHill, ISBN 9780070006577
 Conway, John B. (1986), Functions of One Complex Variable I, Springer, ISBN 0387903283
 Joshi, Kapil D. (1989), Foundations of Discrete Mathematics, New York: John Wiley & Sons, ISBN 9780470211526
 Pedoe, Dan (1988), Geometry: A comprehensive course, Dover, ISBN 0486658120
 Solomentsev, E.D. (2001), "Complex number", in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Springer, ISBN 9781556080104, http://eom.springer.de/c/c024140.htm
Historical references
 Burton, David M. (1995), The History of Mathematics (3rd ed.), New York: McGrawHill, ISBN 9780070094659
 Katz, Victor J. (2004), A History of Mathematics, Brief Version, AddisonWesley, ISBN 9780321161932
 Nahin, Paul J. (1998), An Imaginary Tale: The Story of (hardcover ed.), Princeton University Press, ISBN 0691027951
 A gentle introduction to the history of complex numbers and the beginnings of complex analysis.
 H.D. Ebbinghaus ... (1991), Numbers (hardcover ed.), Springer, ISBN 0387974970
 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 0679454438. Chapters 47 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 030909657X (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 0198534477 (hardcover, 1997). History of complex numbers and complex analysis with compelling and useful visual interpretations.
External links
Number systems 

Basic 


Real numbers and
their extensions 


Other number systems 

