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square root∗
Wkbj79†
2013-03-21 12:46:17
√
The square root of a nonnegative real number x, written√as x, is the
2
2
unique
√
√nonnegative real number y such that y = x. Thus, ( x) ≡ x. Or,
x × x ≡ x.
√
Example. 9 = 3 because 3 ≥ 0 and 32 = 3 × 3 = 9.
√
Example. x2 + 2x + 1 = |x + 1| (see absolute value and even-even-odd rule)
because (x + 1)2 = (x + 1)(x + 1) = x2 + x + x + 1 = x2 + 2x + 1.
√
it is better to allow two values for x. For example,
√ In some situations
4 = ±2 because 22 = 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.rThat √
is, if
√
√
x
x
√
x and y are nonnegative real numbers, then x × y = x × y and
= √ .
y
y
p
Example. x2 y 2 = xy because (xy)2 = xy × xy = x × x × y × y = x2 × y 2 =
x2 y 2 .
r
2
3
3
32
9
9
Example.
= because
= 2 =
.
25
5
5
5
25
√
√
√
√
√
√
On the other hand, in general, x + y 6= x + y and x − y 6= x − y.
This error is an instance of the freshman’s dream error.
The square root notation is actually an alternative to exponentiation. That
√
1
is, x ≡ x 2 . When it is defined,
√ the asquare√ root operation is commutative
with exponentiation. That is, xa = x 2 = ( x)a whenever both xa > 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, −4 is
not
by contradiction as follows: Suppose
√ a real number. This fact can be proven
−4 = x ∈ R. If x is negative, then x2 is positive, and if x is positive, then
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hDefinitioni h11A25i
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x2 is also positive. Therefore, x cannot be positive
or negative. Moreover, x
√
/ R.
cannot be zero either, because 02 = 0. Hence, −4 ∈
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 numwhich, after cancelling,
bers; x is rational if and only if x is a rational number
√
is a fraction of two perfect squares. In particular, 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:
1
d √ x = √
dx
2 x
It is possible to consider square roots in rings other than the integers or the
rationals. For any ring R, with x, y ∈ 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 y 2 ≡ x (mod n). Rabin’s
cryptosystem is based on the difficulty of finding square roots modulo an integer
n.
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