Contents

- Definition
- Notation
- Concept
- Perfect Square – Always a Positive Number
- Square Root
- Square Free Number
- Perfect Square in Rational Numbers
- Perfect Square as a Summation
- Recursive Formulas To Compute the Perfect Square
- The “Difference of Squares” Formula
- Perfect Squares in Trinomials
- Example – Identifying the Perfect Square

# Perfect Square|Definition & Meaning

## Definition

A perfect square is defined as an **integer** that results from the **multiplication** of an **integer** by itself. A perfect square is always a **whole** number, not a decimal, as it is produced from the multiplication of a whole number by **itself**. A perfect square is also called a **square number** or a complete square.

**Figure 1** shows a perfect square **49,** which is the product of** 7** with itself.

## Notation

Suppose an **integer** whole number **m** is multiplied by itself. Instead of writing it as **m ✕ m**, it is usually written as $m^2$. The exponent “**2**” is known as the “**square,**” and **m ^{2}** is pronounced as “

**m squared**.” In figure 1,

**7 $\boldsymbol\times$ 7**can also be written as

**7**.

^{2}## Concept

The perfect square has the word “**square**” in it, which is somewhat related to the **shape** of the square.

The **area** is defined as the number of units in one **column** multiplied by the number of **units** in one **row**. A **square** has an **equal** number of units on all its sides.

A **square** with a side “**p**” length will have the area **p $\boldsymbol\times$ p = p ^{2}**. If

**p**units represent a

**perfect square**, the units can be arranged in

**columns**forming a

**square,**with each side having the same number of

**units**as the square root of

**p**.

**Figure 2** shows the illustration of a square with **5 points** on each side. Counting the rows and columns or **multiplying** a row with a column will give the **perfect square** of 5, i.e., **25**.

So, an integer will be a **perfect square** if and only if its number of **points** can be arranged in a **square**.

## Perfect Square – Always a Positive Number

A **perfect square** is always a** non-negative** number as its **square root** is an integer.

The product of a **positive integer** by itself gives a positive perfect square, and the product of a **negative number** by itself also gives a positive square number. This is because positive times positive is **positive** and negative times negative is also positive.

For example, the** perfect square** **81** can be achieved either by two **products**:

9 ✕ 9 = 81

(-9) ✕ (-9) = 81

**Figure 3** shows the perfect squares of integers from 1 to 20.

## Square Root

The **original integer** can be obtained from the perfect square by taking the **square root**. For example, **441** is a perfect square. Its **original integer** can be found by taking its **square root** as follows:

$\sqrt{441}$ = 21

The **square root** of a number may be **positive** or **negative**. In the above example, square **441** can be obtained by taking the square of **21** or **-21**. But the **radical** symbol of the square root allows only the **positive** result.

**Figure 4** shows how **square** and **square root** are inverses of each other.

## Square Free Number

A square-free number is an integer that is not **divisible** by a **perfect square** except one. The **factors** of a square-free number are unique and **different** and don’t appear **twice** to make a perfect square. For example, the number **15** is **square-free** as it **factorizes** as follows:

15 = 3 ✕ 5

Whereas, the number **12** is **not** square-free as its factors include a **perfect square**:

12 = 3 ✕ 4

The perfect square is **4** which is equal to **2 ^{2}**.

## Perfect Square in Rational Numbers

A **rational** number can be written in the form of** a/b**, where **a** and** b** are **integers**. A **perfect square** of a rational number is a square of both the integers **a** and** b**. Also, the **original** rational number can be obtained by taking the **square root**.

For example, **121/100** is a **perfect square** of the rational number **11/10**. It can be written as:

\[ \frac{121}{100} = \left( \frac{11}{10} \right)^2 \]

## Perfect Square as a Summation

The perfect square **m ^{2}** can be written as a

**summation**as follows:

\[ m^2 = \sum _{k=1}^{m}( 2k \ – \ 1) \]

The function** (2k – 1)** represents the **function** for **odd** numbers. So, the perfect square **m ^{2}** is equal to the

**sum**of the first

**m odd**numbers. For example, the perfect square

**36**is equal to:

\[ 36 = 6^2 = \sum _{k=1}^{6}( 2k \ – \ 1 ) \]

**Expanding** the **series** gives the following result:

**36 = 6 ^{2} = 1 + 3 + 5 + 7 + 9 + 11**

## Recursive Formulas To Compute the Perfect Square

The following two **recursive** formulas are used to compute the perfect square **m ^{2}**:

m^{2} = (m – 1)^{2} + (2m – 1)

m^{2} = 2(m – 1)^{2} – (m – 2)^{2} + 2

where m is the **original** number.

For example, if** m = 3**, m^{2} can be calculated from the **first recursive** formula as follows:

3^{2} = (3 – 1)^{2} + [2(3) – 1]

3^{2} = (2)^{2} + [6 – 1]

3^{2} = 4 + 5

3^{2} = 9

## The “Difference of Squares” Formula

The difference of square formula states that the **product** of the **sum** and difference of two numbers, **c** and **d,** will be equal to the **difference** of their squares. It can be written as:

(c + d)(c – d) = c^{2} – d^{2}

From this formula, the **perfect square** m^{2} minus **one** is always equal to:

(m^{2} – 1) = (m + 1)(m – 1)

For example, as we know that:

8^{2} = 64

So:

(64 – 1) = 63 = (8 + 1)(8 – 1) = 9 ✕ 7

## Perfect Squares in Trinomials

The following two **formulas** are used to identify a trinomial as a **perfect square**:

(a + b)^{2} = a^{2} + 2ab + b^{2}

(a – b)^{2} = a^{2} – 2ab + b^{2}

This means a **trinomial** will be a perfect **square** if and only if it factors out as **(a + b) ^{2}** or

**(a – b)**.

^{2}For example, factorizing a **trinomial** (x^{2} + 18x + 81) gives the **factors** as:

x^{2} + 18x + 81 = (x)^{2} + 2(x)(9) + (9)^{2}

x^{2} + 18x + 81 = (x + 9)^{2}

Hence, this trinomial is a **perfect square**.

## Example – Identifying the Perfect Square

Which of the two is a **perfect** **square**?

1035 or 1681

### Solution

To know the perfect square, calculate the **square root** of both numbers:

$\sqrt{1035}$ = 32.17

$\sqrt{1681}$ = 41

As **32.17** is not a whole number, and **41** is a **whole** number, **1035** is **not** a perfect **square,** whereas **1681** is a perfect **square**.

*All the images are created using GeoGebra.*