square root

square root

The

square root

of a nonnegative

real number

x

, written as

x

, is the unique nonnegative real number

y

such that

y

2

=

x

. Thus,

(

x

)

2

≡

x

. Or,

x

×

x

≡

x

.

Example

.

9

=

3

because

3

≥

0

and

3

2

=

3

×

3

=

9

.

Example

.

x

2

+

2

⁢

x

+

1

=

|

x

+

1

|

(see

absolute value

and even-even-odd rule) because

(

x

+

1

)

2

=

(

x

+

1

)

⁢

(

x

+

1

)

=

x

2

+

x

+

x

+

1

=

x

2

+

2

⁢

x

+

1

.

In some situations it is better to allow two values for

x

. For example,

4

=

±

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

x

×

y

=

x

×

y

and

x

y

=

x

y

.

Example

.

x

2

⁢

y

2

=

x

⁢

y

because

(

x

⁢

y

)

2

=

x

⁢

y

×

x

⁢

y

=

x

×

x

×

y

×

y

=

x

2

×

y

2

=

x

2

⁢

y

2

.

Example

.

9

25

=

3

5

because

(

3

5

)

2

=

3

2

5

2

=

9

25

.

On the other hand, in general,

x

+

y

≠

x

+

y

and

x

-

y

≠

x

-

y

. This error is an

instance

of the freshman’s dream error.

The square root notation is actually an alternative to exponentiation. That is,

x

≡

x

1

2

. When it is defined, the square root operation is commutative with exponentiation. That is,

x

a

=

x

a

2

=

(

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,

-

4

is not a real number. This fact can be proven by

contradiction

(

http://planetmath.org/ProofByContradiction

) as follows: Suppose

-

4

=

x

∈

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,

-

4

∉

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

;

x

is rational if and only if

x

is a rational number which, after cancelling, is a

fraction

of two 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:

d

d

⁢

x

⁢

(

x

)

=

1

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

(

http://planetmath.org/MathbbZ_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

.

Title

square root

Canonical

name

SquareRoot

Date of creation

2013-03-22 11:57:19

Last modified on

2013-03-22 11:57:19

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

27

Author

Wkbj79 (1863)

Entry type

Definition

Classification

msc 11A25

Related topic

CubeRoot

Related topic

NthRoot

Related topic

RationalNumber

Related topic

IrrationalNumber

Related topic

RealNumber

Related topic

ComplexNumber

Related topic

Complex

Related topic

DerivativeOfInverseFunction

Related topic

EvenEvenOddRule

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