A square number is a number that is obtained by the product of two same numbers. In other words, when we multiply a number by itself, we say that the number has been squared, and the product is called the square of the number. In geometry, the area of a square is the finest example of a square number.

Thus, if \(x\) is a number, then the square of \(x\) is written as \({x^2}.\)
Knowing the properties of square numbers helps us understand the concept better. In this article, we will learn the properties of square numbers.

Square Numbers

If \(a \times a = b\) i.e., \({a^2} = b,\) then we can say that \(b\) is the square of a number \(a.\) Now let us explore numbers that look like a square. \(9\) is one such example.
\(9\) can be shown as a line:

Or as a square like this:

The factors of \(9\) are just \(3\) numbers \(1,\,3\) and \(9.\) Thus, \(9\) is known as a square number. We can also say that \(3\) squared equals \(9\) or \({3^2} = 9.\) Square numbers are also known as perfect squares.
A perfect square is an integer that can be expressed as the product of two equal integers.

For example, \(121\) is a perfect square because it is equal to \(11 \times 11.\) If \(y\) is an integer, then \({y^2}\) is a perfect square. And, because of this definition, perfect squares are always non-negative.

Properties of Square Numbers

Let us learn the properties of square numbers.

1. Property-1: A number having \(2,\,3,\,7\) or \(8\) at the unit’s place is never a perfect square. Remember that the unit digit of perfect squares is either \(1,\,4,\,5,\,6,\,9\) or \(0.\) In other words, no square number ends in \(2,\,3,\,7\) or \(8.\) Then by just looking at the numbers, we can say that whether they are perfect squares or not. For example, \(262,\,553,\,8888\) and \(6793\) are not the perfect square numbers.

2. Property-2: If a number has \(0,\,1,\,4,\,5,\,6\) or \(9\) at its unit’s place, then it may or may not be a perfect square. For example, \(121,\,144,\,169,\,196,\,225,\,900\) are perfect square numbers, but the numbers like \(244,\,321,\,369,\,700\) are not perfect squares.

3. Property-3: Squares of even numbers are always even, and squares of odd numbers are always odd. For example, \({2^2} = 4,\,{3^2} = 9,\,{5^2} = 25,\,{6^2} = 36\) etc.

4. Property-4: The square of a number, whether it is positive or negative, is always positive. For example, \({\left( 3 \right)^2} = 3 \times 3 = 9,\,{\left( { – 3} \right)^2} = \left( { – 3} \right) \times \left( { – 3} \right) = 9.\)

5. Property-5: The number of zeros at the end of a perfect square is always even. In other words, a number ending in an odd number of zeros is never a perfect square. For example, \(1000\) is not a perfect square, whereas \(100\) or \(10000\) are perfect squares.

6. Property-6: For any \(2\) consecutive natural number \(n\) and \(n + 1,\) we have
\({\left( {n + 1} \right)^2} – {n^2} = \left( {n + 1 + n} \right)\left( {n + 1 – n} \right)\)
\( = \left( {n + 1 + n} \right) \times 1\)
\( = \left\{ {\left( {n + 1} \right) + n} \right\}\)
\(\left\{ {{{\left( {n + 1} \right)}^2} – {n^2}} \right\} = \left\{ {\left( {n + 1} \right) + n} \right\}\)
For example, if \(n = 20,\) then \(n + 1 = 20 + 1 = 21\)
Thus, \(\left\{ {{{\left( {21} \right)}^2} – {{\left( {20} \right)}^2}} \right\} = \left( {21 + 20} \right) = 41.\)

7. Property-7: For every natural number n, we have, the sum of the first \(n\) odd natural numbers \( = {n^2}.\) In simple words, we can say that the square of a natural number \(n\) is equal to the sum of first \(n\) odd natural numbers. For example, \(\left( {1 + 3 + 5 + 7 + 9 + 11} \right) = \) Sum of first \(6\) odd numbers \( = {6^2} = 36.\)

8. Property-8: A set of \(3\) natural numbers \(a,\,b\) and \(c\) are called the Pythagorean triplets if it can be written as \({a^2} + {b^2} = {c^2}\) where \(a,\,b\) and \(c\) are positive integers.
Pythagoras theorem: The square of the length of the hypotenuse of a right triangle equals the sum of the squares of the lengths of the other two sides.
For example, \({3^2} + {4^2} = {5^2}\) i.e., \(\left( {9 + 16 = 25} \right)\)
To find the numbers in the Pythagorean triplet, we use the formula \(\left( {2n,\,{n^2} – 1,\,{n^2} + 1} \right),\) where, \(n\) is a natural number.

9. Property-9: The square of a proper fraction is, smaller than the given fraction, i.e., \({\left( {\frac{a}{b}} \right)^2} = \frac{{{a^2}}}{{{b^2}}} < \frac{a}{b}.\) For example, \({\left( {\frac{2}{3}} \right)^2} = \frac{{{2^2}}}{{{3^2}}} < \frac{2}{3} = \frac{4}{9} < \frac{2}{3}.\)

10. Property-10: The number of non-square numbers between any two consecutive square numbers can be found in the following way. Let the first square number be \({a^2}.\) Thus, the next consecutive square number is \({\left( {a + 1} \right)^2}.\)
The number of non-square numbers between these two numbers
\( = {\left( {a + 1} \right)^2} – {a^2} – 1\)
\( = \left( {a + 1} \right) \times \left( {a + 1} \right) – {a^2} – 1\)
\( = a\left( {a + 1} \right) + 1\left( {a + 1} \right) – {a^2} – 1\)
\( = {a^2} + a + a + 1 – {a^2} – 1\)
\( = 2a + 1 – 1\)
\( = 2a\)
Thus, the number of non-square numbers between any two consecutive square numbers \({a^2}\) and \({\left( {a + 1} \right)^2}\) is equal to twice the first natural number, that is, \(2a.\)

Solved Examples – Properties of Square Numbers

Q.1. Show that the following natural numbers are not perfect squares.
(a) 1222
(b) 23000
Ans: We know that the natural numbers ending with the digits \(2,\,3,\,7,\,8\) or an odd number of zeros are not perfect squares.
(a) \(1222\) ends with the digit \(2\) and thus is not a perfect square.
(b) \(23000\) ends with an odd number of zeros and hence, is not a perfect square.

Q.2. Without actual squaring, find the value of \({96^2} – {95^2}.\)
Ans: Using the property for any \(2\) consecutive natural number \(n,\)
\(\left\{ {{{\left( {n + 1} \right)}^2} – {n^2}} \right\} = \left\{ {\left( {n + 1} \right) + n} \right\}\)
Thus, \({96^2} – {95^2} = 96 + 95 = 191\)
Hence, the value of \({96^2} – {95^2}.\) is equal to \(191.\)

Q.3. Is 29 a square number?
Ans: The number \(28\) has \(8\) in its unit’s place. According to the property of squares, we have seen that square numbers end with \(2,\,4,\,5,\,6,\,9\) or \(0.\) Therefore, \(29\) cannot be a square number.

Q.4. How many non-square numbers are there in between \({5^2}\) and \({6^2}?\)
Ans: We know that the number of non-square numbers between any two consecutive square numbers equals twice the first natural number.
Therefore, there will be \(2 \times 5 = 10\) non-square numbers between \({5^2}\) and \({6^2}.\) As \({5^2} = 25\) and \({6^2} = 36,\) the \(10\) non-square numbers between them are \(26,\,27,\,28,\,29,\,30,\,31,\,32,\,33,\,34\) and \(35.\)

Q.5. Find out whether the following set of numbers form a Pythagorean triplet or not. (5,12,13)
Ans: In a general form, it can be written as \({a^2} + {b^2} = {c^2}\) where \(a,\,b\) and \(c\) are \(5,\,12,\,13\) respectively, and form a set of Pythagorean triplets.
Now, LHS \( = 5,\,12\)
So, \({a^2} + {b^2} = {5^2} + {12^2} = 25 + 144 = 169.\)
RHS \( = {c^2} = {13^2} = 169.\)
LHS \( = \) RHS \( = 169.\)
Hence, the given set is a Pythagorean triplet.

Q.6. What is the square of 13? Find out without actual multiplication.
Ans: We can find the square of \(13\) using the property of square numbers that we have learnt as \({a^2} = \) Sum of first \(a\) odd numbers.
Therefore, \( = {13^2} = \) Sum of first \(13\) odd numbers.
\( = 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 + 21 + 23 + 25.\)
Looking at the addends, we can find a pattern that can help us add the numbers quickly.
\(1 + 25 = 26,\,3 + 23 = 26,\,5 + 21 = 26\) and so on. There will be only \(13\) left in the middle.
Thus, \(1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 + 21 + 23 + 25\)
\(= 26 + 26 + 26 + 26 + 26 + 26 + 13 = 169.\)
So, \({13^2} = 169.\)

Summary

This article had a quick view about the square numbers and learned about the perfect squares with examples. We did a detailed discussion about the properties of square numbers and tried to understand them better with the help of examples.

Frequently Asked Questions (FAQs) – Properties of Square Numbers

Q.1. Define the square of a number.
Ans: The square of an integer is the product of an integer with itself. In other words, when we will multiply a number by itself, we say that the number has been squared, and the product is called the square of the number. Thus, if \(x\) is a number, then the square of \(x\) is written as \({x^2}.\)

Q.2. Define a perfect square number with an example?
Ans: A perfect square is an integer that can be expressed as the product of two equal integers. For example, \(144\) is a perfect square because it is equal to \(12 \times 12.\) If \(y\) is an integer, then \({y^2}\) is a perfect square.

Q.3. What are the 10 properties of a square number?
Ans: The properties of a square number is as follows.
1. If a number has \(0,\,1,\,4,\,5,\,6\) or \(9\) at its unit’s place, then it may or may not be a perfect square
2. A number having \(2,\,3,\,7\) or \(8\) at the unit’s place is never a perfect square.
3. Squares of even numbers are always even, and squares of odd numbers are always odd.
4. The square of a number, whether the number is positive or negative, is always positive.
5. The number of zeros at the end of a perfect square is always even. In other words, a number ending in an odd number of zeros is never a perfect square.
6. For any \(2\) consecutive natural number \(n\) and \(n + 1,\) we have \(\left\{ {{{\left( {n + 1} \right)}^2} – {n^2}} \right\} = \left\{ {\left( {n + 1} \right) + n} \right\}.\)
7. For every natural number \(n,\) we have, the sum of the first \(n\) odd natural numbers \( = {n^2}.\)
8. A set of \(3\) natural numbers \(a,\,b\) and \(c\) are called the Pythagorean triplets, then, \({a^2} + {b^2} = {c^2}.\)
9. The square of a proper fraction is smaller than the given fraction.

Q.4. Explain the Pythagorean triplet property.
Ans: A set of \(3\) natural numbers \(a,\,b\) and \(c\) are called the Pythagorean triplets, named after a famous mathematician in ancient Greece if it can be written as \({a^2} + {b^2} = {c^2}.\) where \(a,\,b\) and \(c\) are positive integers and form a set of Pythagorean triplets. For example, \((5,\,12,\,13)\) is a set of a Pythagorean triplet.

Q.5. Which type of number can never be a perfect square?
Ans: A number having \(2,\,3,\,7\) or \(8\) at the unit’s place is never a perfect square. In other words, no square number ends in \(2,\,3,\,7\) or \(8.\) By just looking at the numbers, we can say that whether they are perfect squares or not. For example, \(262,\,553,\,8888\) and \(6793\) are not the perfect square numbers.

Now you are provided with all the necessary information on the properties of square numbers and we hope this detailed article is helpful to you. If you have any queries regarding this article, please ping us through the comment section below and we will get back to you as soon as possible.