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square root (Definition)

The square root of a nonnegative real number $ x$, written as $ \sqrt{x}$, is the unique nonnegative real number $ y$ such that $ y^2=x$. Thus, % latex2html id marker 398 $ (\sqrt{x})^2 \equiv x$. Or, % latex2html id marker 400 $ \sqrt{x}\times\sqrt{x}\equiv x$.

Example   $ \sqrt{9}=3$ because % latex2html id marker 408 $ 3\geq 0$ and % latex2html id marker 410 $ 3^2=3\times 3=9$.
Example   % latex2html id marker 416 $ \sqrt{x^2+2x+1}=\vert x+1\vert$ (see absolute value and even-even-odd rule) because $ (x+1)^2=(x+1)(x+1)=x^2+x+x+1=x^2+2x+1$.

In some situations it is better to allow two values for $ \sqrt{x}$. For example, $ \sqrt{4}=\pm 2$ because $ 2^2=4$ and $ (-2)^2=4$.

Over nonnegative real numbers, the square root operation is left distributive over multiplication and division, but not over addition or subtraction. That is, if $ x$ and $ y$ are nonnegative real numbers, then % latex2html id marker 432 $ \sqrt{x\times y}=\sqrt{x}\times\sqrt{y}$ and % latex2html id marker 434 $ \displaystyle \sqrt{\frac{x}{y}}=\frac{\sqrt{x}}{\sqrt{y}}$.

Example   $ \sqrt{x^{2}y^{2}}=xy$ because % latex2html id marker 442 $ (xy)^2=xy\times xy=x\times x\times y\times y=x^2\times y^2=x^{2}y^{2}$.
Example   % latex2html id marker 448 $ \displaystyle \sqrt{\frac{9}{25}}=\frac{3}{5}$ because % latex2html id marker 450 $ \displaystyle\left(\frac{3}{5}\right)^2=\frac{3^2}{5^2}=\frac{9}{25}$.

On the other hand, in general, % latex2html id marker 452 $ \sqrt{x+y}\neq\sqrt{x}+\sqrt{y}$ and % latex2html id marker 454 $ \sqrt{x-y}\neq\sqrt{x}-\sqrt{y}$. This error is an instance of the freshman's dream error.

The square root notation is actually an alternative to exponentiation. That is, % latex2html id marker 456 $ \sqrt{x} \equiv x^\frac{1}{2}$. When it is defined, the square root operation is commutative with exponentiation. That is, $ \sqrt{x^a}=x^\frac{a}{2}=(\sqrt{x})^a$ whenever both $ x^a>0$ and $ x>0$. The restrictions can be lifted if we extend the domain and codomain of the square root function to the complex numbers.

Negative real numbers do not have real square roots. For example, $ \sqrt{-4}$ is not a real number. This fact can be proven by contradiction as follows: Suppose $ \sqrt{-4}=x\in\mathbb{R}$. If $ x$ is negative, then $ x^2$ is positive, and if $ x$ is positive, then $ x^2$ is also positive. Therefore, $ x$ cannot be positive or negative. Moreover, $ x$ cannot be zero either, because $ 0^2=0$. Hence, $ \sqrt{-4}\notin\mathbb{R}$.

For additional discussion of the square root and negative numbers, see the discussion of complex numbers.

The square root function generally maps rational numbers to algebraic numbers; $ \sqrt{x}$ is rational if and only if $ x$ is a rational number which, after cancelling, is a fraction of two perfect squares. In particular, $ \sqrt{2}$ is irrational.

The function is continuous for all nonnegative $ x$, and differentiable for all positive $ x$ (it is not differentiable for $ x=0$). Its derivative is given by:

% latex2html id marker 496 $\displaystyle \frac{d}{dx}\big(\sqrt{x}\big)=\frac{1}{2\sqrt{x}} $

It is possible to consider square roots in rings other than the integers or the rationals. For any ring $ R$, with $ x,y\in R$, we say that $ y$ is a square root of $ x$ if $ y^2=x$.

When working in the ring of integers modulo $ n$, we give a special name to members of the ring that have a square root. We say $ x$ is a quadratic residue modulo $ n$ if there exists $ y$ coprime to $ x$ such that % latex2html id marker 518 $ y^2\equiv x\pmod{n}$. Rabin's cryptosystem is based on the difficulty of finding square roots modulo an integer $ n$.



"square root" is owned by Wkbj79. [ full author list (4) | owner history (4) ]
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See Also: cube root, nth root, rational number, irrational number, real number, complex number, complex, derivative of inverse function, even-even-odd rule

Keywords:  square root, root, arithmetic operator, operator

Attachments:
square root of square root binomial (Topic) by pahio
existence of square roots of non-negative real numbers (Theorem) by PrimeFan
square root of 5 (Definition) by MathNerd
square root of 2 (Definition) by MathNerd
square root of 3 (Definition) by PrimeFan
table of continued fractions of $\sqrt{n}$ for $1 < n < 102$ (Data Structure) by PrimeFan
squaring condition for square root inequality (Theorem) by pahio
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Cross-references: coprime, quadratic residue, integers, rings, derivative, differentiable, continuous, irrational, squares, fraction, rational number, rational, algebraic numbers, rational numbers, maps, negative numbers, positive, negative, complex numbers, function, codomain, domain, restrictions, commutative, freshman's dream error, subtraction, addition, division, multiplication, left distributive, operation, even-even-odd rule, absolute value, real number
There are 92 references to this entry.

This is version 23 of square root, born on 2001-11-10, modified 2008-02-24.
Object id is 747, canonical name is SquareRoot.
Accessed 41000 times total.

Classification:
AMS MSC: 11A25 (Number theory :: Elementary number theory :: Arithmetic functions; related numbers; inversion formulas)
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"Clearly, there is only one square root of 2" by alozano on 2004-03-04 10:03:27

>> There is no "the" square root of a positive real number, but rather >>two possible square roots (+ or -).
>>
>> Alvaro

>Sure there is. X squared equals 2; what's X? "Plus-minus square root >of 2". Clearly there is only one "square root of 2", and both it and >its negative, when squared, produce 2.

Absolutely not, there are TWO distinct reals numbers such that X^2=2. Why would we give a special relevance to the positive solution?

>Besides, I wrote this with a sixth grade or younger reader in mind; >everyone else knows perfectly well what a square root is. The whole >plus-minus thing is confusing; I thought it best to use only positive >numbers at first and work up to the rest.

Even in that case, there is no loss being precise. You may write "a square root of 2 is a number x such that x^2=2" instead of "the square root". A sixth grader will understand, and mathematicians will not flinch.

Alvaro
[ reply | up ]
okay, fixed now, but... by wberry on 2001-11-10 16:23:20
There's one thing I don't like that I don't know how to fix.

The exponentiated fraction in the middle, (3/5)^2, doesn't render well.

How do you get the fraction to appear in parenthesis?
[ reply | up ]
I have *no idea* why this doesn't work by wberry on 2001-11-10 15:30:18
I'm new to this. Somebody tell me what I'm doing or
not doing to cause this entry to not show up.

I'm just writing plain text with latex stuff like $x$
embedded in the text. Am I supposed to use HTML tags
or something?
[ reply | up ]
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