The Art of Interface

Article 11 — Appendix A.29

√ or sqrt — square root function

Category. Mathematics.

Abstract. Square root: definition, plot, properties and identities.

Reference. This article is a part of Librow scientific formula calculator project.

1. Definition

Square root function is inverse of the power function with power a = 2

x2

The square root is denoted with radical symbol:

√x

Square root is equivalent to the power of one second:

√x ≡ x1/2

2. Plot

Square root function defined for non-negative part of real axis — so, its domain is [0, +∞). Function plot is depicted below — fig. 1.

Fig. 1. Plot of the square root function y = sqrt x. Fig. 1. Plot of the square root function y = √x.

Function codomain non-negative part of the real axis: [0, +∞).

3. Identities

Take into account, that because of square root defined only for non-negative values, and power of two defined everywhere, the order of these two functions makes difference:

√x2 ≡ x
√(x2) ≡ |x|

and as well

x ≡ signx √(x2)

Reciprocal argument:

√(1/x) = 1 /√x

Product and ratio of arguments:

√(xy) = √|x|√|y|
√(x/y) = √|x| /√|y|

Power of argument:

√(xa) = √|x|a ≡ |x|a/2

4. Solution of quadratic equation

Quadratic equation

ax2 + bx + c = 0

has roots

x = [−b ± √(b2 − 4ac)] /(2a)

For equation with even coefficient for the first power ax2 + 2bx + c = 0

roots have simplified form

x = [−b ± √(b2 − ac)] /a

5. Solution of normalized quadratic equation

Normalized quadratic equation x2 + bx + c = 0

has roots

x = [−b ± √(b2 − 4c)] /2

And equation with even coefficient for the first power x2 + 2bx + c = 0

has the simplest form for its roots

x = −b ± √(b2 − c)

6. Support

Square root function √ or sqrt of the real argument is supported by free version of the Librow calculator.

Square root function √ or sqrt of the complex argument is supported by professional version of the Librow calculator.

7. How to use

To calculate square root of the number:

sqrt(2);

or

√(2);

To calculate square root of the current result:

sqrt(rslt);

or

√(rslt);

To calculate square root of the number x in memory:

sqrt(mem[x]);

or

√(mem[x]);