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What Are Square Numbers(Meaning, Formulas, and Examples)

perfect squares

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You know the area of a rectilinear figure square = $\left(\text{side} \right) \times \left(\text{side} \right)$, where $\text{side}$ is the length of edge(or side) of a square. For example, the area of a square of side length $4 \text{ cm}$ is $16 \text{ cm}^{2}$, or the area of a square of side length $7 \text{ in}$ is $49 \text{ in}^{2}$.

The numbers $16$ or $49$ are called squares of the numbers $4$ and $7$ respectively.

Let’s understand what are square numbers or what are perfect squares, how you find a square of a number, and what are the properties of square numbers.

What are Square Numbers?

When an integer is multiplied by itself, the resultant number is known as a square number. For example, when you multiply $6$ by $6$ itself, i.e., $6 \times 6$, you get $36$. Here, $36$ is a square number. 

A square number is always a positive number, it cannot be negative because when a negative number is multiplied by the same negative number, it results in a positive number. For example, $\left(-9 \right) \times \left(-9 \right) = 81$. $-9$ is a negative number, whereas it’s square $81$ is a positive number.

Writing Square Numbers

A square of a number $n$ is written as $n^{2}$ which is equal to $n\times n$.

For example, square of a number $8$ is written as $8^{2}$ and its value is $8 \times 8 = 64$.

Similarly, square of a number $15$ is written as $15^{2}$ and its value is $15 \times 15 = 225$.

perfect squares
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How To Find Square of a Number?

If you get your squares and roots mixed up, keep in mind that squaring a number is as simple as multiplying it by itself. Because of this, it’s important to know how to multiply single digits as well as larger numbers. To square fractions, find the squares of both the numerator and denominator. Then reduce or simplify the result.

Following are the procedures to get a square of numbers.

Square of Numbers From 1 To 10

When you square a number, you simply multiply the number by itself so it’s important to know how to multiply. For example, the square of $4$ is $4^{2} = 4 \times 4 = 16$. To make it easier to square commonly used single digits, try to memorize basic multiplication tables.

perfect squares

Square of Larger Numbers

The basic process of finding the square of large numbers also remains the same, i.e., multiply the number by itself. You can use certain tricks or properties of numbers to find the square of larger numbers.

Method 1: Multiplication By Itself

In this method, the number is multiplied by itself and the resultant product gives us the square of that number. For example, the square of $12^{2} = 12 \times 12 = 144$. Here, the resultant product $144$ gives us the square of the number $12$ This method works well for smaller numbers.

Method 2: Using Basic Algebraic Identities

When the numbers are large, you can use algebraic identities to find the square of a number. The two algebraic identities that are helpful here are

  • $\left(a + b \right)^{2} = a^{2} + 2ab + b^{2}$
  • $\left(a – b \right)^{2} = a^{2} – 2ab + b^{2}$

Examples

Let’s consider some examples to find a square of numbers using these identities.

Ex 1: Square of $43$

Square of $43$ is $43^{2} = \left(40 + 3\right)^{2}$

Observe that $ \left(40 + 3\right)^{2}$ is in the form  $\left(a + b\right)^{2}$, where $a = 40$ and $b = 3$.

Therefore, $ \left(40 + 3\right)^{2}$ becomes $40^{2} + 2 \times 40 \times 3 + 3^{2} = 40 \times 40 + 240 + 3 \ times 3 = 1600 + 240 + 9 = 1849$

Square of $43$ can also be calculated using the identity $\left(a – b \right)^{2} = a^{2} – 2ab + b^{2}$

$43^{2}$ can be written as $\left(50 – 7 \right)^{2}$, where $a = 50$ and $b = 7$.

Therefore, $\left(50 – 7 \right)^{2} = 50^{2} – 2 \times 50 \times 7 + 7^{2} = 50 \times 50 – 700 + 7 \times 7 = 2500 – 700 + 49 = 1849$.

Thus, we see that the square of $49$, i.e., $49^{2}$ is $1849$ using any of the two methods.

Perfect Squares

A perfect square is a positive integer that is obtained by multiplying an integer by itself. In simple words, we can say that perfect squares are numbers that are the products of integers by themselves. Generally, we can express a perfect square as $x^{2}$, where $x$ is an integer and the value of $x^{2}$ is a perfect square. 

Let’s analyze the pattern followed by perfect square numbers. We can form a square with $4$ marbles such that there are $2$ rows, with $2$ marbles in each row. Similarly, we can form a square with $9$ marbles such that there are $3$ rows, with $3$ marbles in each row. And, we can form a square with $16$ marbles such that there are $4$ rows, with $4$ marbles in each row.

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Square Number Patterns

The squares of the numbers follow interesting patterns. Let’s observe the patterns followed by squares of numbers.

Adding Triangular Numbers

The triangular number sequence is the representation of the numbers in the form of an equilateral triangle arranged in a series or sequence. These numbers are in a sequence of $1$, $3$, $6$, $10$, $15$, $21$, $28$, $36$, $45$, and so on. The numbers in the triangular pattern are represented by dots. The sum of the previous number and the order of the succeeding number results in the sequence of triangular numbers.

perfect squares

If you add any two consecutive triangular numbers, you get a square number.

$1 + 3 = 4$ is a square number.

$3 + 6 = 9$ is a square number.

$6 + 10 = 16$ is a square number.

Numbers Between Square Numbers

Let’s observe the pattern followed in this case. 

Numbers between $1^{2} \left(= 1 \right)$ and $2^{2} \left(= 4 \right)$ are $2$ and $3$ ($2$ in number)

Numbers between $2^{2} \left(= 4 \right)$ and $3^{2} \left(= 9 \right)$ are $5$, $6$, $7$ and $8$ (($4$ in number))

Numbers between $3^{2} \left(= 9 \right)$ and $4^{2} \left(= 16 \right)$ are $10$, $11$, $12$, $13$, $14$, and $15$ ($6$ in number)

Numbers between $4^{2} \left(= 16 \right)$ and $5^{2} \left(= 25 \right)$ are $17$, $18$, $19$, $20$, $21$, $22$, $23$, and $24$ ($8$ in number)

What did you observe?

The number of non-square numbers between two consecutive square numbers is $2$, $4$, $6$, $8$, … All these numbers are even numbers and can be written in the form $n$, where $n$ is a positive integer.

Therefore, the number of non-square numbers between $n^{2}$ and $\left(n + 1 \right)^{2}$ is $2n$.

You can check this algebraically also.

$\left(n + 1 \right)^{2} – n^{2} = n^{2} + 2n + 1 – n^{2} = 2n + 1$.

Therefore, the numbers between $\left(n + 1 \right)^{2}$ and $n^{2}$ is $2n + 1 – 1 = 2n$.

Note: The number of numbers between any two numbers $a$ and $b$ is always equal to $b – a – 1$.

Sum of Consecutive Odd Numbers

The odd numbers are the numbers that are not divisible by $2$, i.e., the numbers that leave the remainder $1$ when divided by $2$.

The first few odd numbers are $1$, $3$, $5$, $7$, $9$, $11$, $13$, $15$, $17$, $19$, $21$, …

$1 = 1^{2}$

$1 + 3 = 4 = 2^{2}$

$1 + 3 + 5 = 9 = 3^{2}$ 

$1 + 3 + 5 + 7 = 16 = 4^{2}$ 

$1 + 3 + 5 + 7 + 9 = 25 = 5^{2}$ 

So, we observe that the “Sum of first $n$ odd numbers is always equal to $n^{2}$”.

Sum of Consecutive Natural Numbers

Let’s again consider first few square of odd numbers starting from $3$ – $9 = 3^{2}$, $25 = 5^{2}$, $49 = 7^{2}$, $81 = 9^{2}$, …

Now, observe the pattern.

$9 = 4 + 5 = 3^{2} => 3^{2} = \frac{3^{2} – 1}{2} + \frac{3^{2} + 1}{2}$

$25 = 12 + 13 = 5^{2} = \frac{5^{2} – 1}{2} + \frac{5^{2} + 1}{2}$

$49 = 24 + 25 = 7^{2} = \frac{7^{2} – 1}{2} + \frac{7^{2} + 1}{2}$

$81 = 40 + 41 = 9^{2} = \frac{9^{2} – 1}{2} + \frac{9^{2} + 1}{2}$

Product of Two Consecutive Even or Odd Natural Numbers

The even numbers are the numbers that are divisible by $2$, i.e., the numbers that leave the remainder $0$ when divided by $2$.

The first few even numbers are $2$, $4$, $6$, $8$, $10$, $12$, $14$, $16$, $18$, $20$, $22$, …

Let’s observe the product of two consecutive even natural numbers.

$2 \times 4 = \left(3 – 1 \right) \left( 3 + 1\right) = 8 = 9 – 1 = 3^{2} – 1$

$4 \times 6 = \left(5 – 1 \right) \left( 5 + 1\right) = 24 = 25 – 1 = 5^{2} – 1$

$6 \times 8 = \left(7 – 1 \right) \left( 7 + 1\right) = 48 = 49 – 1 = 7^{2} – 1$

$8 \times 10 = \left(9 – 1 \right) \left( 9 + 1\right) = 80 = 81 – 1 = 9^{2} – 1$

The first few odd numbers are $1$, $3$, $5$, $7$, $9$, $11$, $13$, $15$, $17$, $19$, $21$, …

Let’s observe the product of two consecutive odd natural numbers.

$1 \times 3 = \left(2 – 1 \right) \left( 2 + 1\right) = 3 = 4 – 1 = 2^{2} – 1$

$3 \times 5 = \left(4 – 1 \right) \left( 4 + 1\right) = 15 = 16 – 1 = 4^{2} – 1$

$5 \times 7 = \left(6 – 1 \right) \left( 6 + 1\right) = 35 = 36 – 1 = 6^{2} – 1$

$7 \times 9 = \left(7 – 1 \right) \left( 7 + 1\right) = 63 = 64 – 1 = 8^{2} – 1$

Thus, we observe that for any natural number $n$, $\left(n – 1 \right) \left(n + 1 \right) = n^{2} – 1$.

You can verify this, algebraically also.

$\left(n – 1 \right) \left(n + 1 \right) = n\left(n + 1 \right) – 1\left(n + 1 \right) = n^{2} + n – n – 1  = n^{2} – 1$.

Properties of Square Numbers

The square numbers exhibit the following properties

Property 1: Square numbers will always end with the digits $0$, $1$, $4$, $5$, $6$ and $9$. For example $9$,$64$, $25$, $81$, and $100$ are perfect squares whereas numbers like $17$, $22$, $83$ are not perfect numbers.

Property 2:  A square number always contains an even number of zeros at the end and numbers containing an odd number of zeros cannot be square numbers. For example, $1600$, $3600$,  and $2500$ square numbers whereas $1000$, $250$, and $160$ are not square numbers.

Property 3: A number containing $1$ or $9$ as the last digit will always have $1$ at the end of its square number. For example, the square number of $9$ is $81$ and $11$ is $121$.

Property 4: A number that has $4$ or $6$ as the last digit will have a square number ending with $6$. For example, the square root of $4$ is $16$ and the square of $16$ is $256$.

Property 5: The square of an odd number is always odd and the square of an even number is always the reason. For example, the square of $13$ is $169$ and the square of $12$ is $144$.

Property 6: Square numbers a positive as two negative signs multiplied by each other result in a positive sign. For example $-7\times \left(-7 \right) = 49$.

Property 7: Square root of a square number is a whole number. For example, the square root of $441$ is $21$ so your $441$ is a square number. If the square of a number is a decimal of a fraction number then the number is not a perfect square. For example, $\sqrt{56} = 7.48$, so your $56$ is not a perfect square.

Property 8: For every natural number $n$, $\left(n + 1 \right)^{2} – n^{2} = \left(n + 1 \right)+n$.

Property 9: The sum of first $n$ odd natural numbers is equal to the square of a number $n$.

Property 10: For any natural number $m \gt 1$, $\left(2m, m^{2} – 1, m^{2} + 1 \right)$ is a Pythagorean triplet.

Conclusion

When an integer is multiplied by itself, the resultant number is known as a square number. A perfect square is a positive integer that is obtained by multiplying an integer by itself. It means every square number is also called a perfect square number. The square numbers exhibit very interesting patterns.

Practice Problems

  1. Find the square of following numbers using algebraic expression method.
    • $14$
    • $23$
    • $58$
    • $79$
  2. Which of the digits from $0$ to $9$ can be the ones place of a perfect square number?
  3. Which of the following is True?
    • Sum of consecutive natural numbers is a perfect square
    • Sum of consecutive odd numbers is a perfect square
    • Sum of consecutive even numbers is a perfect square
    • Sum of consecutive prime numbers is a perfect square
  4. The number of non-square numbers between $n^{2}$ and $\left(n + 1 \right)^{2}$ is 
    • $2n + 1$
    • $2n$
    • $2n – 1$

Recommended Reading

FAQs

What is a square number in math?

The number that is obtained by multiplying an integer by itself is known as a square number. Suppose, $n$ is an integer, then the square number of $n$ is $n \times n or $n^{2}$. For example, in $7 \times 7 = 49$, $49$ is a square number.

What are perfect square numbers?

When a whole number is multiplied by itself, the number that is obtained is a perfect square. For example, if we multiply the number $5$ by itself, we get $5 \times 5 = 25$. Here, $25$ is a perfect square. In other words, a perfect square is the product of two same integers.

How you can find the square of a number algebraically?

When the numbers are large, you can use algebraic identities to find the square of a number. The two algebraic identities that are helpful here are
1) $\left(a + b \right)^{2} = a^{2} + 2ab + b^{2}$
2) $\left(a – b \right)^{2} = a^{2} – 2ab + b^{2}$

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