Square Root

Definition

The square root of a non-negative number, is a non-negative number that when multiplied by itself results in the original number.

The square root of a negative number does not exist in the real numbers.

Example: Since 25 is a non-negative number, there is a non-negative number 5, such that 5^{2} = 25.

"5 squared equals 25" is analogous to "A square with a side of length 5 units has an area of 25 square units".

"The square root of 25 equals 5" is analogous to "A square with an area of 25 square units has a side of length 5 units".

Example: Since -25 is a negative number, there is no number "r", such that r^{2} = -25.

Estimating Square Roots using Benchmarks

When trying to estimate the square root of a number, benchmark values of whole numbers can be used.

Example: The square root of some numbers (for example the number 60) is an irrational number that never terminates or repeats (7.745 966 692 ...).

Benchmark numbers can be used to state that the square root of 60 is between 6 and 7.

The square root of 60, estimated to two decimal places is 7.75.

Example: The square root of some numbers (for example the number 44.89) is a rational number that terminates (6.7).

Benchmark numbers can be used to state that the square root of 44.89 is between 7 and 8.

Demonstration

Image only

Instructions text as in global.js

Important Consequence

The square root of n^{2} is the absolute value of n.

This is a compact, consise way of expressing the following, (using the number 25 as an example):

Example: Square Roots (Simple Equations)

When solving a simple equation such as x^{2} = 25, it must be observed that there are **two** solutions. The two solutions to this equation are 5 and -5, since both 5^{2} = 25 and (-5)^{2} = 25.

The solution above shows all of the steps.

When someone is proficient with square roots, usually only the steps shown below are written out.

Example: Square Roots (The Pythagorean Theorem)

When working with equations involving the Pythagorean Theorem, it is important to remember that the lengths of the sides of the triangle are positive numbers.