5.1 - Introduction to Squares and Square Root

In this subtopic, we will introduce two important mathematical concepts: Squares and Square Roots. These concepts are foundational for understanding various other topics in mathematics, including algebra, geometry, and number theory.

1. What is a Square?

A square of a number is the result of multiplying that number by itself. It is also referred to as raising a number to the power of 2. In mathematical terms, for a given number $n$, the square of $n$ is written as $n^2$, and it is calculated as:

Example: If $n = 4$, then the square of $4$ is:

$$4^2 = 4 \times 4 = 16$$

The number that results from squaring a number is always non-negative. Squares of integers are called perfect squares. Some examples of perfect squares are:

• 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, etc.

2. What is a Square Root?

The square root of a number is the inverse operation of squaring. It is the number that, when multiplied by itself, gives the original number. The square root of a number $x$ is denoted as $\sqrt{x}$. In mathematical terms, if $y$ is the square root of $x$, then:

$$y = \sqrt{x} \quad \text{implies} \quad y^2 = x$$

In other words, $y$ is the number whose square is $x$. For example:

3. Properties of Square Numbers

Some important properties of square numbers include:

• The square of any positive integer is always a positive integer.
• The square of any negative integer is also a positive integer, because the product of two negative numbers is positive. For example, $(-3)^2 = 9$.
• The square of zero is zero: $0^2 = 0$.
• Square numbers are always non-negative.

4. Finding Square Roots

Finding the square root of perfect squares is simple. For non-perfect squares, the square root may be irrational, and can be approximated using methods such as the long division method or by using a calculator. Examples of non-perfect square roots include:

• $\sqrt{2} \approx 1.414$
• $\sqrt{3} \approx 1.732$
• $\sqrt{5} \approx 2.236$

Square roots of non-perfect squares cannot be expressed as exact fractions, so they are written in decimal form or left in the radical form $\sqrt{x}$.

5. Perfect Squares

A perfect square is an integer that is the square of another integer. In other words, a number is a perfect square if it has an integer square root. Some examples of perfect squares are:

• $1 = 1^2$
• $4 = 2^2$
• $9 = 3^2$
• $16 = 4^2$
• $25 = 5^2$

6. Non-Perfect Squares

Numbers that are not perfect squares are called non-perfect squares. These numbers do not have an integer square root. For example, the square root of $7$ is not an integer, and is approximately $2.645$.

As students progress in mathematics, they will encounter the concept of irrational numbers, which are the square roots of non-perfect squares.