Square root: Wikis

  

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Measured fall time of a small steel sphere falling from various heights. The data is in good agreement with the predicted fall time of \sqrt{2h/g}, where h is the height and g is the acceleration of gravity.
.In mathematics, a square root of a number x is a number r such that r2 = x, or, in other words, a number r whose square (the result of multiplying the number by itself) is x.^ The square root of a negative number is undefined.
  • Squares and Square Roots 23 January 2010 15:27 UTC www.algebra-help.org [Source type: FILTERED WITH BAYES]

^ You can define this number "i" to be the square root of -1.
  • Metaphor for imaginary numbers? - Straight Dope Message Board 23 January 2010 15:27 UTC boards.straightdope.com [Source type: FILTERED WITH BAYES]

^ Algebra Glossary root A value that, multiplied by itself a number of times, results in the value or number wanted.
  • How to Convert Square Roots to Exponents - For Dummies 23 January 2010 15:27 UTC www.dummies.com [Source type: Reference]
  • How to Convert Square Roots to Exponents - For Dummies 23 January 2010 15:27 UTC www.dummies.com [Source type: Reference]

.Every non-negative real number x has a unique non-negative square root, called the principal square root, which is denoted with a radical sign as \scriptstyle \sqrt{x}.^ No real number multiplied by itself will equal a negative number, so -1 cannot have a real square root.
  • Calculate square root without a calculator 23 January 2010 15:27 UTC www.homeschoolmath.net [Source type: FILTERED WITH BAYES]

^ The square number refers to the number from which the square root is to be extracted.
  • Square Roots as solved by Takashi Kojima 23 January 2010 15:27 UTC webhome.idirect.com [Source type: Academic]

^ In mathematics, the principal square root of a non-negative real number is denoted and represents the non-negative real number whose square (the result of multiplying the number by itself) is x\,\!.
  • http://www.hermes-press.com/square_root.htm 23 January 2010 15:27 UTC www.hermes-press.com [Source type: Reference]

.The square root can also be written in exponent notation, as x1/2.^ And if you want the inverse square root, divide the exponent by -2 to flip the sign: .
  • Understanding Quake’s Fast Inverse Square Root | BetterExplained 23 January 2010 15:27 UTC betterexplained.com [Source type: General]

^ Finding square roots and converting them to exponents is a relatively common operation in algebra.
  • How to Convert Square Roots to Exponents - For Dummies 23 January 2010 15:27 UTC www.dummies.com [Source type: Reference]
  • How to Convert Square Roots to Exponents - For Dummies 23 January 2010 15:27 UTC www.dummies.com [Source type: Reference]

^ Finding the integer square root of the (integer) mantissa M is enough; if the exponent E is even, simply divide the exponent of the result by 2; if the exponent E was odd, shift the mantissa to the left one position before calculating the integer square root.
  • Algorithms - Square root 23 January 2010 15:27 UTC forums.sun.com [Source type: FILTERED WITH BAYES]

.For example, the principal square root of 9 is 3, denoted \scriptstyle \sqrt{9} = 3, because 32 = 3 × 3 = 9 and 3 is non-negative.^ No real number multiplied by itself will equal a negative number, so -1 cannot have a real square root.
  • Calculate square root without a calculator 23 January 2010 15:27 UTC www.homeschoolmath.net [Source type: FILTERED WITH BAYES]

^ The Perfect Square Method The goal of this method is to factor the number inside the square root into a product of perfect squares and one non-perfect square.
  • Simplifying Square Roots 23 January 2010 15:27 UTC www.squidoo.com [Source type: FILTERED WITH BAYES]

^ The error comes in taking the principal square root of a square of −1.
  • http://www.hermes-press.com/square_root.htm 23 January 2010 15:27 UTC www.hermes-press.com [Source type: Reference]

.The principal square root of a positive number, however, is only one of its two square roots.^ The square root of a number is one of its two equal factors.
  • Squares and Square Roots 23 January 2010 15:27 UTC www.algebra-help.org [Source type: FILTERED WITH BAYES]

^ The square number refers to the number from which the square root is to be extracted.
  • Square Roots as solved by Takashi Kojima 23 January 2010 15:27 UTC webhome.idirect.com [Source type: Academic]

^ So, in fact, this equals one over two square roots of x and that’s the derivative.
  • Calculus: Derivative of the Square Root Function | MindBites.com 23 January 2010 15:27 UTC www.mindbites.com [Source type: General]

.Every positive number x has two square roots.^ The square root of a negative number is undefined.
  • Squares and Square Roots 23 January 2010 15:27 UTC www.algebra-help.org [Source type: FILTERED WITH BAYES]

^ You can define this number "i" to be the square root of -1.
  • Metaphor for imaginary numbers? - Straight Dope Message Board 23 January 2010 15:27 UTC boards.straightdope.com [Source type: FILTERED WITH BAYES]

^ Square roots of positive integers are often irrational numbers , i.e., numbers not expressible as a quotient of two integers.
  • http://www.hermes-press.com/square_root.htm 23 January 2010 15:27 UTC www.hermes-press.com [Source type: Reference]

.One of them is \scriptstyle \sqrt{x}, which is positive, and the other \scriptstyle -\sqrt{x}, which is negative.^ Tamara Yardley -1 cannot have a square root (at least, not a real one) because any two numbers with the same "sign" (+/- positive or negative), when multiplied, will equal a positive number.
  • Calculate square root without a calculator 23 January 2010 15:27 UTC www.homeschoolmath.net [Source type: FILTERED WITH BAYES]

^ Algebra Glossary negative reciprocals Two numbers, one positive and one negative, whose product is –1.
  • How to Convert Square Roots to Exponents - For Dummies 23 January 2010 15:27 UTC www.dummies.com [Source type: Reference]

^ Algebra Glossary integer A positive or negative whole number or zero; numbers starting with zero and going up or down in increments of one.
  • How to Convert Square Roots to Exponents - For Dummies 23 January 2010 15:27 UTC www.dummies.com [Source type: Reference]

.Together, these two roots are denoted \scriptstyle \pm\sqrt{x}.^ Given these two results we present an algorithm (used by the ENIAC) to obtain a square root .
  • How the ENIAC took a Square Root 23 January 2010 15:27 UTC www4.wittenberg.edu [Source type: Academic]

^ At the moment, Centre is arguably the most pop-based artist on the Square Root label, and these two EPs explore this idea quite deeply.

^ Note that calculating square roots is rather more difficult than either cube roots or fifth roots - you should learn these two procedures first.
  • Instant Mental Calculation of Square Roots 23 January 2010 15:27 UTC www.psychicscience.org [Source type: General]

.Square roots of negative numbers can be discussed within the framework of complex numbers.^ The square number refers to the number from which the square root is to be extracted.
  • Square Roots as solved by Takashi Kojima 23 January 2010 15:27 UTC webhome.idirect.com [Source type: Academic]

^ To: right_to_defend There is no square root of any negative number.
  • What is the square root of pi? 23 January 2010 15:27 UTC www.freerepublic.com [Source type: FILTERED WITH BAYES]

^ The square root of a complex number can be computed as: .
  • Paul Hsieh's Square Root page 23 January 2010 15:27 UTC www.azillionmonkeys.com [Source type: FILTERED WITH BAYES]

.More generally, square roots can be considered in any context in which a notion of "squaring" of some mathematical objects is defined (including algebras of matrices, endomorphism rings, etc).^ You can define this number "i" to be the square root of -1.
  • Metaphor for imaginary numbers? - Straight Dope Message Board 23 January 2010 15:27 UTC boards.straightdope.com [Source type: FILTERED WITH BAYES]

^ The math-buffs' holiday, which only occurs nine times each century, falls on Tuesday 3/3/09 (for the mathematically challenged, three is the square root of nine).
  • KCBS - How to Celebrate Square Root Day 23 January 2010 15:27 UTC www.kcbs.com [Source type: News]

^ Celebrations are as varied: Some cut root vegetables into squares, others make food in the shape of a square root symbol.
  • KCBS - How to Celebrate Square Root Day 23 January 2010 15:27 UTC www.kcbs.com [Source type: News]

.Square roots of integers that are not perfect squares are always irrational numbers: numbers not expressible as a ratio of two integers.^ The square number refers to the number from which the square root is to be extracted.
  • Square Roots as solved by Takashi Kojima 23 January 2010 15:27 UTC webhome.idirect.com [Source type: Academic]

^ But these are integer square roots.
  • Algorithms - Square root 23 January 2010 15:27 UTC forums.sun.com [Source type: FILTERED WITH BAYES]

^ There are two solutions to the square root of a non-zero number.
  • http://www.hermes-press.com/square_root.htm 23 January 2010 15:27 UTC www.hermes-press.com [Source type: Reference]

.For example, \scriptstyle \sqrt{2} cannot be written exactly as m/n, where n and m are integers.^ For example, \sqrt 2 cannot be written exactly as m / n , where n and m are integers.
  • http://www.hermes-press.com/square_root.htm 23 January 2010 15:27 UTC www.hermes-press.com [Source type: Reference]

^ What is the name for a number that cannot be written in the form where a and b are integers?

.Nonetheless, it is exactly the length of the diagonal of a square with side length 1. This has been known since ancient times, with the discovery that \scriptstyle \sqrt{2} is irrational attributed to Hippasus, a disciple of Pythagoras.^ Nonetheless, it is exactly the length of the diagonal of a square with side length 1.
  • http://www.hermes-press.com/square_root.htm 23 January 2010 15:27 UTC www.hermes-press.com [Source type: Reference]

^ The discovery that \sqrt 2 is irrational is attributed to Hippasus, a disciple of Pythagoras.
  • http://www.hermes-press.com/square_root.htm 23 January 2010 15:27 UTC www.hermes-press.com [Source type: Reference]

^ This exposed them to the difference between perfect squares like 9, 16, and 576, which have integers for the lengths of their sides, and squares such as 5, 8, and 17 whose side length can only be expressed exactly using a radical sign.
  • The Square Root of a Fair Share - Volume 23 No. 2 - Winter 2008/2009 - Rethinking Schools Online 23 January 2010 15:27 UTC www.rethinkingschools.org [Source type: General]

The term whose root is being considered is known as the radicand. In the expression \scriptstyle \sqrt[n]{ab+2}, ab + 2 is the radicand. .The radicand is the number or expression underneath the radical sign.^ This rational number can be found by realizing that x also appears under the radical sign, which gives the equation .
  • http://www.hermes-press.com/square_root.htm 23 January 2010 15:27 UTC www.hermes-press.com [Source type: Reference]

Contents

Properties

The graph of the function \scriptstyle f(x) = \sqrt{x}, made up of half a parabola with a vertical directrix.
.The principal square root function \scriptstyle f(x) = \sqrt{x} (usually just referred to as the "square root function") is a function which maps the set of non-negative real numbers onto itself, and, like all functions, always returns a unique value.^ No real number multiplied by itself will equal a negative number, so -1 cannot have a real square root.
  • Calculate square root without a calculator 23 January 2010 15:27 UTC www.homeschoolmath.net [Source type: FILTERED WITH BAYES]

^ In reality the market is always ahead of itself.
  • New Theory: It Will Be A Square Root-Shaped Recovery 23 January 2010 15:27 UTC www.businessinsider.com [Source type: FILTERED WITH BAYES]

^ In mathematics, the principal square root of a non-negative real number is denoted and represents the non-negative real number whose square (the result of multiplying the number by itself) is x\,\!.
  • http://www.hermes-press.com/square_root.htm 23 January 2010 15:27 UTC www.hermes-press.com [Source type: Reference]

.In geometrical terms, the square root function maps the area of a square to its side length.^ A square with a side length of three has an area of nine.
  • The Square Root of a Fair Share - Volume 23 No. 2 - Winter 2008/2009 - Rethinking Schools Online 23 January 2010 15:27 UTC www.rethinkingschools.org [Source type: General]

^ A geometric view of the square root algorithm .
  • Calculate square root without a calculator 23 January 2010 15:27 UTC www.homeschoolmath.net [Source type: FILTERED WITH BAYES]

^ As unconventional as it was, using squares made it easy to both compare the area of dissimilar shapes and see side lengths of squares as square roots.
  • The Square Root of a Fair Share - Volume 23 No. 2 - Winter 2008/2009 - Rethinking Schools Online 23 January 2010 15:27 UTC www.rethinkingschools.org [Source type: General]

.The square root of x is rational if and only if x is a rational number which can be represented as a ratio of two perfect squares.^ Real numbers that are squares of rational numbers are called perfect squares.
  • Roots, Radicals, and Square Root Equations: Simplifying Square Root Expressions 23 January 2010 15:27 UTC cnx.org [Source type: FILTERED WITH BAYES]

^ You can define this number "i" to be the square root of -1.
  • Metaphor for imaginary numbers? - Straight Dope Message Board 23 January 2010 15:27 UTC boards.straightdope.com [Source type: FILTERED WITH BAYES]

^ The square number refers to the number from which the square root is to be extracted.
  • Square Roots as solved by Takashi Kojima 23 January 2010 15:27 UTC webhome.idirect.com [Source type: Academic]

.See square root of 2 for proofs that this is an irrational number, and quadratic irrational for a proof for all non-square natural numbers.^ And so, I see the square root of x plus delta x.
  • Calculus: Derivative of the Square Root Function | MindBites.com 23 January 2010 15:27 UTC www.mindbites.com [Source type: General]

^ The square root of a complex number can be computed as: .
  • Paul Hsieh's Square Root page 23 January 2010 15:27 UTC www.azillionmonkeys.com [Source type: FILTERED WITH BAYES]

^ There are two solutions to the square root of a non-zero number.
  • http://www.hermes-press.com/square_root.htm 23 January 2010 15:27 UTC www.hermes-press.com [Source type: Reference]

.The square root function maps rational numbers into algebraic numbers (a superset of the rational numbers).^ You can define this number "i" to be the square root of -1.
  • Metaphor for imaginary numbers? - Straight Dope Message Board 23 January 2010 15:27 UTC boards.straightdope.com [Source type: FILTERED WITH BAYES]

^ The square root of a positive irrational is always irrational and the square root of a positive algebraic (which includes rationals) is always algebraic.
  • Paul Hsieh's Square Root page 23 January 2010 15:27 UTC www.azillionmonkeys.com [Source type: FILTERED WITH BAYES]

^ The square root of a complex number can be computed as: .
  • Paul Hsieh's Square Root page 23 January 2010 15:27 UTC www.azillionmonkeys.com [Source type: FILTERED WITH BAYES]

For all real numbers x
 \sqrt{x^2} = \left|x\right| = \begin{cases} x, & \mbox{if }x \ge 0 \ -x, & \mbox{if }x < 0. \end{cases}     (see absolute value)
For all non-negative real numbers x and y,
\sqrt{xy} = \sqrt x \sqrt y
and
\sqrt x = x^{1/2}.
.The square root function is continuous for all non-negative x and differentiable for all positive x.^ Now it occurred to us, since the number of square roots appeared to be unlimited, to try to gather them into one class, by which we could henceforth describe all the roots.
  • Square root of 2 is irrational from Interactive Mathematics Miscellany and Puzzles 23 January 2010 15:27 UTC www.cut-the-knot.org [Source type: Academic]

^ So, the solution to a negative square root yields an imaginary number.
  • Metaphor for imaginary numbers? - Straight Dope Message Board 23 January 2010 15:27 UTC boards.straightdope.com [Source type: FILTERED WITH BAYES]

^ Finding the integer square root of the (integer) mantissa M is enough; if the exponent E is even, simply divide the exponent of the result by 2; if the exponent E was odd, shift the mantissa to the left one position before calculating the integer square root.
  • Algorithms - Square root 23 January 2010 15:27 UTC forums.sun.com [Source type: FILTERED WITH BAYES]

Its derivative is
f'(x) = \frac{1}{2\sqrt x}.
The Taylor series of √1 + x about x = 0 converges for | x | < 1 and is given by
\sqrt{1 + x} = \sum_{n=0}^\infty \frac{(-1)^n(2n)!}{(1-2n)(n!)^2(4^n)}x^n = 1 + 	extstyle \frac{1}{2}x - \frac{1}{8}x^2 + \frac{1}{16} x^3 - \frac{5}{128} x^4 + \dots.\!

Computation

.Most pocket calculators have a square root key.^ Describing this process in words, however, can't begin to evoke the sound that the Friden calculator would generate while taking a square-root.
  • Embedded.com - Integer Square Roots (Programmer's Toolbox) 23 January 2010 15:27 UTC www.embedded.com [Source type: FILTERED WITH BAYES]

^ Most computer languages, and certainly the C language and x86 assembly language come with a built-in square root function.
  • Paul Hsieh's Square Root page 23 January 2010 15:27 UTC www.azillionmonkeys.com [Source type: FILTERED WITH BAYES]

^ In this lesson, we will calculate the derivative of the square root of x using the definition of the derivative.
  • Calculus: Derivative of the Square Root Function | MindBites.com 23 January 2010 15:27 UTC www.mindbites.com [Source type: General]

.Computer spreadsheets and other software are also frequently used to calculate square roots.^ The ENIAC used N for twice the square root .
  • How the ENIAC took a Square Root 23 January 2010 15:27 UTC www4.wittenberg.edu [Source type: Academic]

^ Compute a square root now!
  • Paul Hsieh's Square Root page 23 January 2010 15:27 UTC www.azillionmonkeys.com [Source type: FILTERED WITH BAYES]

^ In this lesson, we will calculate the derivative of the square root of x using the definition of the derivative.
  • Calculus: Derivative of the Square Root Function | MindBites.com 23 January 2010 15:27 UTC www.mindbites.com [Source type: General]

Computer software programs typically implement good routines to compute the exponential function and the natural logarithm or logarithm, and then compute the square root of x using the identity
\sqrt{x} = e^{(\ln x)/2} or \sqrt{x} = 10^{(\log x)/2}.
.The same identity is exploited when computing square roots with logarithm tables or slide rules.^ Compute a square root now!
  • Paul Hsieh's Square Root page 23 January 2010 15:27 UTC www.azillionmonkeys.com [Source type: FILTERED WITH BAYES]

^ The same is true in the square root algorithm.
  • Embedded.com - Integer Square Roots (Programmer's Toolbox) 23 January 2010 15:27 UTC www.embedded.com [Source type: FILTERED WITH BAYES]

^ The square root of a complex number can be computed as: .
  • Paul Hsieh's Square Root page 23 January 2010 15:27 UTC www.azillionmonkeys.com [Source type: FILTERED WITH BAYES]

.The most common iterative method of square root calculation by hand is known as the "Babylonian method" or "Heron's method" after the first century Greek philosopher Heron of Alexandria who first described it.^ The first is the perfect square method.
  • Simplifying Square Roots 23 January 2010 15:27 UTC www.squidoo.com [Source type: FILTERED WITH BAYES]

^ The first method finds the square root of "z" .
  • http://www.hermes-press.com/square_root.htm 23 January 2010 15:27 UTC www.hermes-press.com [Source type: Reference]

^ The corresponding square root is the first digit .
  • Instant Mental Calculation of Square Roots 23 January 2010 15:27 UTC www.psychicscience.org [Source type: General]

[1] .It involves a simple algorithm, which results in a number closer to the actual square root each time it is repeated.^ The same is true in the square root algorithm.
  • Embedded.com - Integer Square Roots (Programmer's Toolbox) 23 January 2010 15:27 UTC www.embedded.com [Source type: FILTERED WITH BAYES]

^ You can define this number "i" to be the square root of -1.
  • Metaphor for imaginary numbers? - Straight Dope Message Board 23 January 2010 15:27 UTC boards.straightdope.com [Source type: FILTERED WITH BAYES]

^ The square root of a complex number can be computed as: .
  • Paul Hsieh's Square Root page 23 January 2010 15:27 UTC www.azillionmonkeys.com [Source type: FILTERED WITH BAYES]

To find r, the square root of a real number x:
.
  1. Start with an arbitrary positive start value r (the closer to the square root of x, the better).
  2. Replace r by the average between r and x/r, that is: \scriptstyle (r + x/r) / 2\, (It is sufficient to take an approximate value of the average in order to ensure convergence.^ ENIAC take a square root?
    • How the ENIAC took a Square Root 23 January 2010 15:27 UTC www4.wittenberg.edu [Source type: Academic]

    ^ The error comes in taking the principal square root of a square of −1.
    • http://www.hermes-press.com/square_root.htm 23 January 2010 15:27 UTC www.hermes-press.com [Source type: Reference]

    ^ To obtain both roots of a positive number, take the value given by the principal square root function as the first root (root 1 ) and obtain the second root (root 2 ) by subtracting the first root from zero (ie root 2 = 0 - root 1 ).
    • http://www.hermes-press.com/square_root.htm 23 January 2010 15:27 UTC www.hermes-press.com [Source type: Reference]

    )
  3. Repeat step 2 until r and x/r are as close as desired.
.The time complexity for computing a square root with n digits of precision is equivalent to that of multiplying two n-digit numbers.^ For a positive real number, the two square roots are the principle square root and the negative square root.
  • http://www.hermes-press.com/square_root.htm 23 January 2010 15:27 UTC www.hermes-press.com [Source type: Reference]

^ Compute a square root now!
  • Paul Hsieh's Square Root page 23 January 2010 15:27 UTC www.azillionmonkeys.com [Source type: FILTERED WITH BAYES]

^ You can define this number "i" to be the square root of -1.
  • Metaphor for imaginary numbers? - Straight Dope Message Board 23 January 2010 15:27 UTC boards.straightdope.com [Source type: FILTERED WITH BAYES]

Square roots of negative and complex numbers

Complex square root
Second leaf of the complex square root
Using the Riemann surface of the square root, one can see how the two leaves fit together
.The square of any positive or negative number is positive, and the square of 0 is 0. Therefore, no negative number can have a real square root.^ This is not true of the real numbers --- the polynomial x 2 + 1 has no real root.
  • Metaphor for imaginary numbers? - Straight Dope Message Board 23 January 2010 15:27 UTC boards.straightdope.com [Source type: FILTERED WITH BAYES]

^ So, the solution to a negative square root yields an imaginary number.
  • Metaphor for imaginary numbers? - Straight Dope Message Board 23 January 2010 15:27 UTC boards.straightdope.com [Source type: FILTERED WITH BAYES]

^ You can define this number "i" to be the square root of -1.
  • Metaphor for imaginary numbers? - Straight Dope Message Board 23 January 2010 15:27 UTC boards.straightdope.com [Source type: FILTERED WITH BAYES]

.However, it is possible to work with a more inclusive set of numbers, called the complex numbers, that does contain solutions to the square root of a negative number.^ For a positive real number, the two square roots are the principle square root and the negative square root.
  • http://www.hermes-press.com/square_root.htm 23 January 2010 15:27 UTC www.hermes-press.com [Source type: Reference]

^ So, the solution to a negative square root yields an imaginary number.
  • Metaphor for imaginary numbers? - Straight Dope Message Board 23 January 2010 15:27 UTC boards.straightdope.com [Source type: FILTERED WITH BAYES]

^ You can define this number "i" to be the square root of -1.
  • Metaphor for imaginary numbers? - Straight Dope Message Board 23 January 2010 15:27 UTC boards.straightdope.com [Source type: FILTERED WITH BAYES]

This is done by introducing a new number, denoted by i (sometimes j, especially in the context of electricity where "i" traditionally represents electric current) and called the imaginary unit, which is defined such that i2 = −1. Using this notation, we can think of i as the square root of −1, but notice that we also have (−i)2 = i2 = −1 and so −i is also a square root of −1. By convention, the principal square root of −1 is i, or more generally, if x is any positive number, then the principal square root of −x is
\sqrt{-x} = i \sqrt x.
The right side (as well as its negative) is indeed a square root of −x, since
(i\sqrt x)^2 = i^2(\sqrt x)^2 = (-1)x = -x.
.For every non-zero complex number z there exist precisely two numbers w such that w2 = z: the principal square root of z (defined below), and its negative.^ Precisely, every polynomial with complex coefficients has a complex root.
  • Metaphor for imaginary numbers? - Straight Dope Message Board 23 January 2010 15:27 UTC boards.straightdope.com [Source type: FILTERED WITH BAYES]

^ For a positive real number, the two square roots are the principle square root and the negative square root.
  • http://www.hermes-press.com/square_root.htm 23 January 2010 15:27 UTC www.hermes-press.com [Source type: Reference]

^ So, the solution to a negative square root yields an imaginary number.
  • Metaphor for imaginary numbers? - Straight Dope Message Board 23 January 2010 15:27 UTC boards.straightdope.com [Source type: FILTERED WITH BAYES]

Imaginary square root

The square root of \scriptstyle i \, is given by
\sqrt{i} = \frac{1}{\sqrt{2}}(1+i).
This result can be obtained algebraically by finding a and b such that
i = (a+bi)^2\,\!
or equivalently
i = a^2 + 2abi - b^2.\,\!
This gives the two equations
2ab = 1\,\!
a^2 - b^2 = 0,\,\!
which are easily solved to
a = b = \pm \frac{1}{\sqrt{2}}.
The choice of the principal root then gives
a = b = \frac{1}{\sqrt{2}}.
The result can also be obtained by using De Moivre's theorem and setting
i = \cos\left (\frac{\pi}{2}\right ) + i\sin\left (\frac{\pi}{2}\right )
which produces
\begin{align} \sqrt{i} & = \left ( \cos\left ( \frac{\pi}{2} \right ) + i\sin \left (\frac{\pi}{2} \right ) \right )^{\frac{1}{2}} \ & = \cos\left (\frac{\pi}{4} \right ) + i\sin\left ( \frac{\pi}{4} \right ) \ & = \frac{1}{\sqrt{2}} + i\left ( \frac{1}{\sqrt{2}} \right ) = \frac{1}{\sqrt{2}}(1+i) . \ \end{align}

Principal square root of a complex number

.To find a definition for the square root that allows us to consistently choose a single value, called the principal value, we start by observing that any complex number x + iy can be viewed as a point in the plane, (x, y), expressed using Cartesian coordinates.^ The principal square root function \sqrt{x} always returns a single unique value.
  • http://www.hermes-press.com/square_root.htm 23 January 2010 15:27 UTC www.hermes-press.com [Source type: Reference]

^ The ENIAC used N for twice the square root .
  • How the ENIAC took a Square Root 23 January 2010 15:27 UTC www4.wittenberg.edu [Source type: Academic]

^ The first method finds the square root of "z" .
  • http://www.hermes-press.com/square_root.htm 23 January 2010 15:27 UTC www.hermes-press.com [Source type: Reference]

.The same point may be reinterpreted using polar coordinates as the pair (r, Φ), where r ≥ 0 is the distance of the point from the origin, and Φ is the angle that the line from the origin to the point makes with the positive real (x) axis.^ Like I said, history gets everything wrong and it takes forever to clean up the cruft and damage (e.g., names like "real" vs. Descartes who invented analytic geometry with the realization that pairs of coordinates describe points in the plane)).
  • Metaphor for imaginary numbers? - Straight Dope Message Board 23 January 2010 15:27 UTC boards.straightdope.com [Source type: FILTERED WITH BAYES]

^ The only major point I'd emphasize on top of this is that every combination of scaling and rotation by some angle amounts to a complex number; in fact, complex numbers are simply the same thing as scaling and rotation.
  • Metaphor for imaginary numbers? - Straight Dope Message Board 23 January 2010 15:27 UTC boards.straightdope.com [Source type: FILTERED WITH BAYES]

^ If you have a complex number z on the unit circle, you can measure the angle from the positive real axis to z ; multiplying by z rotates the whole plane by that angle.
  • Metaphor for imaginary numbers? - Straight Dope Message Board 23 January 2010 15:27 UTC boards.straightdope.com [Source type: FILTERED WITH BAYES]

In complex analysis, this value is conventionally written reiΦ. If
 z=r e^{\phi i}\, with  -\pi < \phi \le \pi \,
then we define the principal square root of z as follows:
\sqrt{z} = \sqrt{r} \, e^{i \phi / 2}.
.The principal square root function is thus defined using the nonpositive real axis as a branch cut.^ In this Demonstration, you can take the square root or absolute value of a function and see the effect.
  • Visualizing Square Root and Absolute Value - Wolfram Demonstrations Project 23 January 2010 15:27 UTC demonstrations.wolfram.com [Source type: FILTERED WITH BAYES]

^ As unconventional as it was, using squares made it easy to both compare the area of dissimilar shapes and see side lengths of squares as square roots.
  • The Square Root of a Fair Share - Volume 23 No. 2 - Winter 2008/2009 - Rethinking Schools Online 23 January 2010 15:27 UTC www.rethinkingschools.org [Source type: General]

^ The error comes in taking the principal square root of a square of −1.
  • http://www.hermes-press.com/square_root.htm 23 January 2010 15:27 UTC www.hermes-press.com [Source type: Reference]

.The principal square root function is holomorphic everywhere except on the set of non-positive real numbers (where it isn't even continuous).^ There are two solutions to the square root of a non-zero number.
  • http://www.hermes-press.com/square_root.htm 23 January 2010 15:27 UTC www.hermes-press.com [Source type: Reference]

^ The square root of a complex number can be computed as: .
  • Paul Hsieh's Square Root page 23 January 2010 15:27 UTC www.azillionmonkeys.com [Source type: FILTERED WITH BAYES]

^ For a positive real number, the two square roots are the principle square root and the negative square root.
  • http://www.hermes-press.com/square_root.htm 23 January 2010 15:27 UTC www.hermes-press.com [Source type: Reference]

The above Taylor series for √1 + x remains valid for complex numbers x with |x| < 1.

Formula

When the number is in rectangular form the following formula can be used for the principal value:
\sqrt{x+iy} = \sqrt{\frac{r + x}{2}} + i \frac{y}{\sqrt{2 (r + x)}}
where
r = |x + iy| = \sqrt{x^2+ y^2}
is the absolute value or .modulus of the complex number, unless x = −r and y = 0. Notice that the sign of the imaginary part of the root is the same as the sign of the imaginary part of the original number.^ For negative real numbers, the concept of imaginary and complex numbers has been developed to provide a mathematical framework to deal with the results.
  • http://www.hermes-press.com/square_root.htm 23 January 2010 15:27 UTC www.hermes-press.com [Source type: Reference]

^ However, it's interesting to note that the rules for imaginary numbers were created 1572 but the idea for 2-dimensional complex plane to represent imaginary numbers geometrically wasn't published until 1799.
  • Metaphor for imaginary numbers? - Straight Dope Message Board 23 January 2010 15:27 UTC boards.straightdope.com [Source type: FILTERED WITH BAYES]

^ This analogy is actually remarkably robust, and can be extended to explain just about everything about imaginary numbers and complex numbers (combinations of imaginary and real).
  • Metaphor for imaginary numbers? - Straight Dope Message Board 23 January 2010 15:27 UTC boards.straightdope.com [Source type: FILTERED WITH BAYES]

.The real part of the principal value is always non-negative.^ The principal square root function \sqrt{x} always returns a single unique value.
  • http://www.hermes-press.com/square_root.htm 23 January 2010 15:27 UTC www.hermes-press.com [Source type: Reference]

^ In mathematics, the principal square root of a non-negative real number is denoted and represents the non-negative real number whose square (the result of multiplying the number by itself) is x\,\!.
  • http://www.hermes-press.com/square_root.htm 23 January 2010 15:27 UTC www.hermes-press.com [Source type: Reference]

^ The principal square root function \sqrt{x} is a function which maps the non-negative real "/psypsych/Domain_%28mathematics%29" title="Domain (mathematics)"> domain R + ∪{0} into the non-negative real "/psypsych/Codomain" title="Codomain"> codomain R + ∪{0}.
  • http://www.hermes-press.com/square_root.htm 23 January 2010 15:27 UTC www.hermes-press.com [Source type: Reference]

Notes

.Note that because of the discontinuous nature of the square root function in the complex plane, the law √zw = √zw is in general not true.^ The same is true in the square root algorithm.
  • Embedded.com - Integer Square Roots (Programmer's Toolbox) 23 January 2010 15:27 UTC www.embedded.com [Source type: FILTERED WITH BAYES]

^ Because we are looking for the square root of 7 .
  • http://www.hermes-press.com/square_root.htm 23 January 2010 15:27 UTC www.hermes-press.com [Source type: Reference]

^ Describing this process in words, however, can't begin to evoke the sound that the Friden calculator would generate while taking a square-root.
  • Embedded.com - Integer Square Roots (Programmer's Toolbox) 23 January 2010 15:27 UTC www.embedded.com [Source type: FILTERED WITH BAYES]

(Equivalently, the problem occurs because of the freedom in the choice of branch. .The chosen branch may or may not yield the equality; in fact, the choice of branch for the square root need not contain the value of √zw at all, leading to the equality's failure.^ It was the same algorithm, in fact, as the one programmed into the cam-and-gear "ROM" of the square-root Friden.
  • Embedded.com - Integer Square Roots (Programmer's Toolbox) 23 January 2010 15:27 UTC www.embedded.com [Source type: FILTERED WITH BAYES]

^ The three is all that’s good and right, why must my three keep out of sight Beneath the vicious square root sign, I wish instead I were a nine .
  • “The Square Root of Three” by David Feinberg < Here Comes The Science 23 January 2010 15:27 UTC www.herecomesthescience.com [Source type: General]

^ Our office had only one square root Friden, so we all had to take turns with it when we needed square roots.
  • Embedded.com - Integer Square Roots (Programmer's Toolbox) 23 January 2010 15:27 UTC www.embedded.com [Source type: FILTERED WITH BAYES]

A similar problem appears with the complex logarithm and the relation log z + log w = log(zw).) Wrongly assuming this law underlies several faulty "proofs", for instance the following one showing that −1 = 1:
-1 = i \cdot i = \sqrt{-1} \cdot \sqrt{-1} = \sqrt{-1 \cdot -1} = \sqrt{1} = 1
The third equality cannot be justified (see invalid proof). .It can be made to hold by changing the meaning of √ so that this no longer represents the principal square root (see above) but selects a branch for the square root that contains (√−1)·(√−1).^ And so, I see the square root of x plus delta x.
  • Calculus: Derivative of the Square Root Function | MindBites.com 23 January 2010 15:27 UTC www.mindbites.com [Source type: General]

^ What is this I see, Another square root of a three .
  • “The Square Root of Three” by David Feinberg < Here Comes The Science 23 January 2010 15:27 UTC www.herecomesthescience.com [Source type: General]

^ I see, Another square root of a three .
  • “The Square Root of Three” by David Feinberg < Here Comes The Science 23 January 2010 15:27 UTC www.herecomesthescience.com [Source type: General]

The left hand side becomes either
\sqrt{-1} \cdot \sqrt{-1}=i \cdot i=-1
if the branch includes +i or
\sqrt{-1} \cdot \sqrt{-1}=(-i) \cdot (-i)=-1
if the branch includes −i, while the right hand side becomes
\sqrt{-1 \cdot -1}=\sqrt{1}=-1,
where the last equality, √1 = −1, is a consequence of the choice of branch in the redefinition of √.

Square roots of matrices and operators

.If A is a positive-definite matrix or operator, then there exists precisely one positive definite matrix or operator B with B2 = A; we then define √A = B.^ So, one has to wonder is there an easy way of taking the derivative that doesn’t require us to actually us the definition of the derivative?
  • Calculus: Derivative of the Square Root Function | MindBites.com 23 January 2010 15:27 UTC www.mindbites.com [Source type: General]

^ Can we define the operations so there are no logical difficulties?
  • Metaphor for imaginary numbers? - Straight Dope Message Board 23 January 2010 15:27 UTC boards.straightdope.com [Source type: FILTERED WITH BAYES]

^ The Definite Integral Order of Operations Percents Defining Variables Additive and Multiplicative Inverse...

.More generally, to every normal matrix or operator A there exist normal operators B such that B2 = A.^ In general, it is easy to see that n 1/m is irrational if there exists at least one prime p such that n is not a perfect m th power in Z p .
  • Square root of 2 is irrational from Interactive Mathematics Miscellany and Puzzles 23 January 2010 15:27 UTC www.cut-the-knot.org [Source type: Academic]

^ Let D be a positive integer but not the square of an integer, then there exists a positive integer λ such that .
  • Square root of 2 is irrational from Interactive Mathematics Miscellany and Puzzles 23 January 2010 15:27 UTC www.cut-the-knot.org [Source type: Academic]

^ Assuming x to be rational, there exists an integer n such that nx is also an integer.
  • Square root of 2 is irrational from Interactive Mathematics Miscellany and Puzzles 23 January 2010 15:27 UTC www.cut-the-knot.org [Source type: Academic]

.In general, there are several such operators B for every A and the square root function cannot be defined for normal operators in a satisfactory manner.^ But the square root function is not defined in them.
  • what is the square root of infinity? 23 January 2010 15:27 UTC www.mahalo.com [Source type: General]

^ You can define this number "i" to be the square root of -1.
  • Metaphor for imaginary numbers? - Straight Dope Message Board 23 January 2010 15:27 UTC boards.straightdope.com [Source type: FILTERED WITH BAYES]

^ Identify and sketch the square root function .
  • Objective -- Square Root Functions (2-3 weeks) 23 January 2010 15:27 UTC www.neisd.net [Source type: Academic]

.Positive definite operators are akin to positive real numbers, and normal operators are akin to complex numbers.^ Positive definite operators are akin to positive real numbers, and normal operators are akin to complex numbers.
  • http://www.hermes-press.com/square_root.htm 23 January 2010 15:27 UTC www.hermes-press.com [Source type: Reference]
  • WikiSlice 23 January 2010 15:27 UTC dev.laptop.org [Source type: Reference]

^ For all positive real numbers x and y , and .
  • WikiSlice 23 January 2010 15:27 UTC dev.laptop.org [Source type: Reference]

^ Thus defined, the square root function is holomorphic everywhere except on the non-positive real numbers (where it isn't even continuous).
  • WikiSlice 23 January 2010 15:27 UTC dev.laptop.org [Source type: Reference]

Principal square roots of the positive integers

As decimal expansions

.The square roots of the perfect squares (1, 4, 9, 16, etc.^ A teacher created explanation covering perfect square trinomials, constants, square roots, and...
  • Square Roots | SPIKE 23 January 2010 15:27 UTC www.spike.com [Source type: General]

) are integers. .In all other cases, the square roots are irrational numbers, and therefore their decimal representations are non-repeating decimals.^ Notice that in the second method, the expanded term (the third expression, \ 384 ) may be difficult to factor into a perfect square and some other number.
  • Roots, Radicals, and Square Root Equations: Multiplication of Square Root Expressions 23 January 2010 15:27 UTC cnx.org [Source type: FILTERED WITH BAYES]

^ So, maybe I should just multiply everything through by the square root and hope all those radicals will just go away.
  • Calculus: Derivative of the Square Root Function | MindBites.com 23 January 2010 15:27 UTC www.mindbites.com [Source type: General]

^ Pi is an irrational (and transcendental) number, so it is impossible to express it as a fraction or a finite decimal.
  • What is the square root of pi? 23 January 2010 15:27 UTC www.freerepublic.com [Source type: FILTERED WITH BAYES]

.Each of these roots has two values, positive and negative, of equal magnitude.^ Assuming that B/A in lowest terms, we observe that the fractional parts of B/A and NA/B have the form a/A and b/B, where a, b are positive integers smaller than A, B. But if two numbers are equal, their fractional parts are also equal: .
  • Square root of 2 is irrational from Interactive Mathematics Miscellany and Puzzles 23 January 2010 15:27 UTC www.cut-the-knot.org [Source type: Academic]

\scriptstyle \sqrt {1} \scriptstyle =\, 1
\scriptstyle \sqrt {2} \scriptstyle \approx 1.414213562373095048801688724209698078569671875376948073176679737990732478462 1 million digits, 2 million, 5 million, 10 million
\scriptstyle \sqrt {3} \scriptstyle \approx 1.732050807568877293527446341505872366942805253810380628055806979451933016909 1 million digits
\scriptstyle \sqrt {4} \scriptstyle =\, 2
\scriptstyle \sqrt {5} \scriptstyle \approx 2.236067977499789696409173668731276235440618359611525724270897245410520925638 1 million digits
\scriptstyle \sqrt {6} \scriptstyle \approx 2.449489742783178098197284074705891391965947480656670128432692567250960377457 1 million digits
\scriptstyle \sqrt {7} \scriptstyle \approx 2.645751311064590590501615753639260425710259183082450180368334459201068823230 1 million digits
\scriptstyle \sqrt {8} \scriptstyle \approx 2.828427124746190097603377448419396157139343750753896146353359475981464956924 1 million digits
\scriptstyle \sqrt {9} \scriptstyle =\, 3
\scriptstyle \sqrt {10} \scriptstyle \approx 3.162277660168379331998893544432718533719555139325216826857504852792594438639 1 million digits
\scriptstyle \sqrt {11} \scriptstyle \approx 3.316624790355399849114932736670686683927088545589353597058682146116484642609
\scriptstyle \sqrt {12} \scriptstyle \approx 3.464101615137754587054892683011744733885610507620761256111613958903866033818
\scriptstyle \sqrt {13} \scriptstyle \approx 3.605551275463989293119221267470495946251296573845246212710453056227166948293
\scriptstyle \sqrt {14} \scriptstyle \approx 3.741657386773941385583748732316549301756019807778726946303745467320035156307
\scriptstyle \sqrt {15} \scriptstyle \approx 3.872983346207416885179265399782399610832921705291590826587573766113483091937
\scriptstyle \sqrt {16} \scriptstyle =\, 4
\scriptstyle \sqrt {17} \scriptstyle \approx 4.123105625617660549821409855974077025147199225373620434398633573094954346338
\scriptstyle \sqrt {18} \scriptstyle \approx 4.242640687119285146405066172629094235709015626130844219530039213972197435386
\scriptstyle \sqrt {19} \scriptstyle \approx 4.358898943540673552236981983859615659137003925232444936890344138159557328203
\scriptstyle \sqrt {20} \scriptstyle \approx 4.472135954999579392818347337462552470881236719223051448541794490821041851276

As expansions in other numeral systems

.The square roots of the perfect squares (1, 4, 9, 16, etc.^ A teacher created explanation covering perfect square trinomials, constants, square roots, and...
  • Square Roots | SPIKE 23 January 2010 15:27 UTC www.spike.com [Source type: General]

) are integers. .In all other cases, the square roots are irrational numbers, and therefore their representations in any standard positional notation system are non-repeating.^ Any indicated square root whose radicand is not a perfect square is an irrational number.
  • Roots, Radicals, and Square Root Equations: Simplifying Square Root Expressions 23 January 2010 15:27 UTC cnx.org [Source type: FILTERED WITH BAYES]

^ If a factor of the radicand contains a variable with an odd exponent, the square root is obtained by first factoring the variable factor into two factors so that one has an even exponent and the other has an exponent of 1, then using the product property of square roots.
  • Roots, Radicals, and Square Root Equations: Simplifying Square Root Expressions 23 January 2010 15:27 UTC cnx.org [Source type: FILTERED WITH BAYES]

^ So, maybe I should just multiply everything through by the square root and hope all those radicals will just go away.
  • Calculus: Derivative of the Square Root Function | MindBites.com 23 January 2010 15:27 UTC www.mindbites.com [Source type: General]

.The hexadecimal representations of some square roots are used in some SHA hash function implementations.^ We now know (or, rather, we've been reminded) how to find square roots by hand, using decimal arithmetic.
  • Embedded.com - Integer Square Roots (Programmer's Toolbox) 23 January 2010 15:27 UTC www.embedded.com [Source type: FILTERED WITH BAYES]

^ In this lesson, we will calculate the derivative of the square root of x using the definition of the derivative.
  • Calculus: Derivative of the Square Root Function | MindBites.com 23 January 2010 15:27 UTC www.mindbites.com [Source type: General]

^ So, there you can get a sense of what these numbers actually mean, and also, how to take the derivative of the square root of x, which is a much more elaborate function.
  • Calculus: Derivative of the Square Root Function | MindBites.com 23 January 2010 15:27 UTC www.mindbites.com [Source type: General]

As periodic continued fractions

.One of the most intriguing results from the study of irrational numbers as continued fractions was obtained by Joseph Louis Lagrange circa 1780. Lagrange found that the representation of the square root of any non-square positive integer as a continued fraction is periodic.^ You can define this number "i" to be the square root of -1.
  • Metaphor for imaginary numbers? - Straight Dope Message Board 23 January 2010 15:27 UTC boards.straightdope.com [Source type: FILTERED WITH BAYES]

^ This is precisely the number we seek for the integer square root.
  • Embedded.com - Integer Square Roots (Programmer's Toolbox) 23 January 2010 15:27 UTC www.embedded.com [Source type: FILTERED WITH BAYES]

^ If the square root of x has to equal one that means that x equals one.
  • Calculus: Derivative of the Square Root Function | MindBites.com 23 January 2010 15:27 UTC www.mindbites.com [Source type: General]

That is, a certain pattern of partial denominators repeats indefinitely in the continued fraction. .In a sense these square roots are the very simplest irrational numbers, because they can be represented with a simple repeating pattern of integers.^ You can define this number "i" to be the square root of -1.
  • Metaphor for imaginary numbers? - Straight Dope Message Board 23 January 2010 15:27 UTC boards.straightdope.com [Source type: FILTERED WITH BAYES]

^ So, notice how I migrated the square roots of x that were originally on top—where we can subtract it, which is bad—to square roots on the bottom where they’re being added.
  • Calculus: Derivative of the Square Root Function | MindBites.com 23 January 2010 15:27 UTC www.mindbites.com [Source type: General]

^ So, the solution to a negative square root yields an imaginary number.
  • Metaphor for imaginary numbers? - Straight Dope Message Board 23 January 2010 15:27 UTC boards.straightdope.com [Source type: FILTERED WITH BAYES]

\scriptstyle \sqrt {2} \scriptstyle =\, [1; 2, 2, ...]
\scriptstyle \sqrt {3} \scriptstyle =\, [1; 1, 2, 1, 2, ...]
\scriptstyle \sqrt {4} \scriptstyle =\, [2]
\scriptstyle \sqrt {5} \scriptstyle =\, [2; 4, 4, ...]
\scriptstyle \sqrt {6} \scriptstyle =\, [2; 2, 4, 2, 4, ...]
\scriptstyle \sqrt {7} \scriptstyle =\, [2; 1, 1, 1, 4, 1, 1, 1, 4, ...]
\scriptstyle \sqrt {8} \scriptstyle =\, [2; 1, 4, 1, 4, ...]
\scriptstyle \sqrt {9} \scriptstyle =\, [3]
\scriptstyle \sqrt {10} \scriptstyle =\, [3; 6, 6, ...]
\scriptstyle \sqrt {11} \scriptstyle =\, [3; 3, 6, 3, 6, ...]
\scriptstyle \sqrt {12} \scriptstyle =\, [3; 2, 6, 2, 6, ...]
\scriptstyle \sqrt {13} \scriptstyle =\, [3; 1, 1, 1, 1, 6, 1, 1, 1, 1, 6, ...]
\scriptstyle \sqrt {14} \scriptstyle =\, [3; 1, 2, 1, 6, 1, 2, 1, 6, ...]
\scriptstyle \sqrt {15} \scriptstyle =\, [3; 1, 6, 1, 6, ...]
\scriptstyle \sqrt {16} \scriptstyle =\, [4]
\scriptstyle \sqrt {17} \scriptstyle =\, [4; 8, 8, ...]
\scriptstyle \sqrt {18} \scriptstyle =\, [4; 4, 8, 4, 8, ...]
\scriptstyle \sqrt {19} \scriptstyle =\, [4; 2, 1, 3, 1, 2, 8, 2, 1, 3, 1, 2, 8, ...]
\scriptstyle \sqrt {20} \scriptstyle =\, [4; 2, 8, 2, 8, ...]
The square bracket notation used above is a sort of mathematical shorthand to conserve space. Written in more traditional notation the simple continued fraction for the square root of 11 – [3; 3, 6, 3, 6, ...] – looks like this:
 \sqrt{11} = 3 + \cfrac{1}{3 + \cfrac{1}{6 + \cfrac{1}{3 + \cfrac{1}{6 + \cfrac{1}{3 + \ddots}}}}}\,
where the two-digit pattern {3, 6} repeats over and over and over again in the partial denominators.

Geometric construction of the square root

A square root can be constructed with a compass and straightedge. In his Elements, Euclid (fl. 300 BC) gave the construction of the geometric mean of two quantities in two different places: Proposition II.14 and Proposition VI.13. Since the geometric mean of a and b is \scriptstyle \sqrt {ab}, one can construct \scriptstyle \sqrt {a} simply by taking b = 1.
The construction is also given by Descartes in his La Géométrie, see figure 2 on page 2. However, Descartes made no claim to originality and his audience would have been quite familiar with Euclid.
Another method of geometric construction uses right triangles and induction: \scriptstyle \sqrt {1} can, of course, be constructed, and once \scriptstyle \sqrt {x} has been constructed, the right triangle with 1 and \scriptstyle \sqrt {x} for its legs has a hypotenuse of \scriptstyle \sqrt {x + 1}. .The Spiral of Theodorus is constructed using successive square roots in this manner.^ In this lesson, we will calculate the derivative of the square root of x using the definition of the derivative.
  • Calculus: Derivative of the Square Root Function | MindBites.com 23 January 2010 15:27 UTC www.mindbites.com [Source type: General]

^ This is great lesson if you need to understand using the square root of x in calculus.
  • Calculus: Derivative of the Square Root Function | MindBites.com 23 January 2010 15:27 UTC www.mindbites.com [Source type: General]

History

.The Rhind Mathematical Papyrus is a copy from 1650 BC of an even earlier work and shows us how the Egyptians extracted square roots.^ Tags Mathematics roots square .
  • square roots / Flashcards - Create Free Flashcards 23 January 2010 15:27 UTC www.proprofs.com [Source type: Academic]

^ The papyrus offered a specific numerical example of how to calculate the volume of a truncated square pyramid, which implied the need for i.
  • square root - Lulu.com 23 January 2010 15:27 UTC www.lulu.com [Source type: General]

^ He recreates the baffling mathematical problems that conjured it up, and the colorful characters who tried to solve them.In 1878, when two brothers stole a mathematical papyrus from the ancient Egyptian burial site in the Valley of Kings, they led scholars to the earliest known occurrence of the square root of a negative number.
  • square root - Lulu.com 23 January 2010 15:27 UTC www.lulu.com [Source type: General]

[2]
The Yale Babylonian Collection YBC 7289 clay tablet was created between 1800 BC and 1600 BC, showing \scriptstyle \sqrt{2} and \scriptstyle 30\sqrt{2} as 1;24,51,10 and 42;25,35 base 60 numbers on a square crossed by two diagonals.[3]
.In Ancient India, the knowledge of theoretical and applied aspects of square and square root was at least as old as the Sulba Sutras, dated around 800-500 B.C. (possibly much earlier).^ But, amazingly, the cube root of -1 can be formed by a combination of normal, real numbers, and that square root of -1 that we defined earlier.
  • Metaphor for imaginary numbers? - Straight Dope Message Board 23 January 2010 15:27 UTC boards.straightdope.com [Source type: FILTERED WITH BAYES]

.A method for finding very good approximations to the square roots of 2 and 3 are given in the Baudhayana Sulba Sutra.^ Let us take the example of finding the square root of a number.
  • Premature parameterization is the square root of all evil | ITworld 23 January 2010 15:27 UTC www.itworld.com [Source type: General]

[4] .Aryabhata in the Aryabhatiya (section 2.4), has given a method for finding the square root of numbers having many digits.^ You can define this number "i" to be the square root of -1.
  • Metaphor for imaginary numbers? - Straight Dope Message Board 23 January 2010 15:27 UTC boards.straightdope.com [Source type: FILTERED WITH BAYES]

^ Originally Posted by Chronos Or for a closer analogy, it's even less qualitatively different than the fact that any given number has infinitely many arcsines.
  • Metaphor for imaginary numbers? - Straight Dope Message Board 23 January 2010 15:27 UTC boards.straightdope.com [Source type: FILTERED WITH BAYES]

^ As for every number having infinitely many logarithms, that is not qualitatively different from the fact that every number has two square roots.
  • Metaphor for imaginary numbers? - Straight Dope Message Board 23 January 2010 15:27 UTC boards.straightdope.com [Source type: FILTERED WITH BAYES]

In the Chinese mathematical work Writings on Reckoning, written between 202 BC and 186 BC during the early Han Dynasty, the square root is approximated by using an "excess and deficiency" method which says to "combine the excess and deficiency as the divisor; (taking) the deficiency numerator multiplied by the excess denominator and the excess numerator times the deficiency denominator, combine them as the dividend."[5]
According to historian of mathematics D.E. Smith, Aryabhata's method for finding the square root was first introduced in Europe by Cataneo in 1546.[6]
The symbol √ for the square root was first used in print in 1525 in Christoph Rudolff's Coss, which was also the first to use the then-new signs '+' and '-'.[7]

See also

Notes

  1. ^ Heath, Thomas (1921). A History of Greek Mathematics, Vol. 2. Oxford: Clarendon Press. pp. 323–324. http://books.google.com/books?id=LOA5AAAAMAAJ&pg=PR323. 
  2. ^ Anglin, W.S. (1994). Mathematics: A Concise History and Philosophy. New York: Springer-Verlag.
  3. ^ http://www.math.ubc.ca/~cass/Euclid/ybc/analysis.html
  4. ^ Joseph, ch.8.
  5. ^ Dauben, p. 210.
  6. ^ Smith, p. 148.
  7. ^ Manguel, Alberto (2006), "Done on paper: the dual nature of numbers and the page", The Life of Numbers, ISBN 8486882141 

References

  • Imhausen, Annette (2007). The Mathematics of Egypt, Mesopotamia, China, India, and Islam. Princeton: Princeton University Press. pp. 187–384. ISBN 0691114854. 
  • Joseph, George (2000). The Crest of the Peacock. Princeton: Princeton University Press. ISBN 0691006598. 
  • Smith, David (1958). History of Mathematics. 2. New York: Dover Publications. ISBN 9780486204307. 

External links


Simple English

A square root of a number is the number that is multiplied by itself and gives the first number. For example, 2 is the square root of 4, because 2×2=4. Only numbers bigger than or equal to zero have real square roots, and a number bigger than zero has two square roots. One is positive (bigger than zero) and the other is negative (smaller than zero). There are two square roots because a negative number multiplied by a negative number is a positive number. Zero only has one square root: zero.

Square roots of negative numbers are not real numbers - they are imaginary numbers. Every complex number except 0 has 2 square roots. For example: -1 has two square roots. We call them i and -i.

The sign for a square root is made by putting a bent line over a number, like this: \sqrt 4. We say "the square root of 4" (or whatever number we are taking the square root of).

A whole number with a square root that is also a whole number is called a perfect square. The first few perfect squares are: 0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225...

Symbol

It is not really known where the square root symbol \sqrt{\,\,} comes from, but some people believe that it was from the letter r, which is the first letter of the Latin and German word radix. Radix means square root.


Citable sentences

Up to date as of December 25, 2010

Here are sentences from other pages on Square root, which are similar to those in the above article.








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