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Perfect Squares

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Last updated date: 28th Feb 2024
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What is a Perfect Square?

A number that can be written as the product of two equal integers is called a perfect square. 16 is a perfect square, for example, because it is the product of two equal integers, 4 x 4 = 16. However, because it cannot be written as the product of two equal numbers, 28 is not a perfect square, (7 x 4) = 28. In this lesson, we'll study more about squares and perfect squares. In this article, we’ll define a perfect square, analyse the relationship between square root and a perfect square, and study a list of perfect squares.


Perfect Square Definition

Perfect squares are numbers that can be made by squaring any whole number. Perfect squares are numbers that can be used to fit in a square shape without leaving any extra parts. Perfect squares are the exact opposite of square roots. They are represented with the “√” symbol. When two identical numbers are multiplied together, they make a number which is a Square root.


⇒ For example, 8 x 8 = 64, so the square root of 64 is 8.


Let us look at the activity below to understand the concept of perfect squares. We'll need a set of four balls and a set of six balls for this. Let's start arranging the balls. Were the balls possible to be arranged in the manner indicated below?


Let's take a closer look at this activity. We can make a square with four balls by arranging them in two rows, each with two balls. We can make a rectangle out of 6 balls by making two rows of three balls each. Mathematically, we can understand this as 4 = 2 × 2 and 6 = 3 × 2. Let's only concentrate on the numbers that make up a square. Here we have, 4 = 2 × 2 = 22. "A perfect square is a number that is obtained by squaring a whole number," according to the definition of a perfect square.


⇒ With this activity, we can define the formula for perfect squares as N = X2.


If we substitute the formula with values.

⇒ If X = 4, and N = X2.

⇒ N = 42 = 16.

⇒ Here, 16 is a perfect square because it is the square of a whole number.

Other Perfect Square Examples are: 1, 4, 9, 16, 25, 36, 49,... etc

Examples of few numbers which are not perfect squares:

  • 6 can produce 1 x 6 and 2 x 3 rectangles, but never a square.

  • 8 may make rectangles of 1 x 8 and 2 x 4, but not squares.

  • 12 can form rectangles 1 x 12, 2 x 6, and 3 x 4, but never a square.

  • The number 20 can be used to generate rectangles such as 1 x 20, 2 X 10, and 4 x 5, but never a square.

Note: A square number, which is a perfect square, is the result of squaring a number. We need to be aware of how we deal with whole numbers.


How Many Perfect Squares are between 1 and 100?

The table given below shows a list of perfect squares between 1 and 100. But it’s always good to learn perfect squares between 1-1000, which helps in higher classes.


Perfect Square Numbers From 1 to 100

1

=

$1\times1$

=

$1^{2}$

4

=

$2\times2$

=

$2^{2}$

9

=

$3\times3$

=

$3^{2}$

16

=

$4\times4$

=

$4^{2}$

25

=

$5\times5$

=

$5^{2}$

36

=

$6\times6$

=

$6^{2}$

49

=

$7\times7$

=

$7^{2}$

64

=

$8\times8$

=

$8^{2}$

81

=

$9\times9$

=

$9^{2}$

100

=

$10\times10$

=

$10^{2}$

Identifying Perfect Squares

In the table above, look at the last digit of the perfect squares of digits 1 to 100. You'll notice that they all end in one of the following numbers: 0, 1, 4, 5, 6, or 9. You would have noticed an important property of perfect squares after experimenting with various perfect square numbers. Non-perfect squares are integers with any of the digits 2, 3, 7, or 8 in their units place, whereas perfect squares have any of the digits 0, 1, 4, 5, 6, or 9 in their units place. To identify a perfect square, make the following observations.

  • The numbers that end with 3 and 7 will have 9 as the unit digit in their square number.

  • The numbers that end with 5 will have 5 as the unit digit in their square number.

  • The numbers that end with 4 and 6 will have 6 as the unit digit in their square number.

  • The numbers that end with 2 and 8 will have 4 as the unit digit in their square number.

  • The numbers that end with 1 and 9 will have 1 as the unit digit in their square number.

Important Notes

While working with perfect squares, keep the following key points in mind:

  • A perfect square with a zero at the end will always have an even number of zeros.

  • Perfect squares will always be positive as we know that (-) × (-) = (+).

  • Perfect squares can have positive or negative square roots.

  • Perfect cubes can also be found by multiplying a number three times.

  • Calculating the square root of a given number can be used to determine whether or not it is a perfect square. It's a perfect square if the square root is a whole number. The given number is not a perfect square if the square root is not a whole number.

  • In the list of perfect squares between 1 and 1000, there are 30 perfect squares. 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, 576, 625, 676, 729, 784, 841, 900, and 961 are the numbers.

  • There are 8 perfect squares from 1 and 100 (excluding 1 and 100). They are as follows: 4, 9, 16, 25, 36, 49, 64 and 81. But, there are actually 10 perfect squares between 1 to 100. They are as follows: 1, 4, 9, 16, 25, 36, 49, 64, 81 and 100.

Practice Questions MCQs

1. When a whole number is multiplied by itself, it forms a perfect square.

A) True

B) False

Ans: Option A) True

2. All the given options are perfect squares except:

  1. 169

  2. 100

  3. 72

  4. 81

Ans: Option C) 72


Conclusion

Perfect Squares are the numbers that represent the product of two equal integers. In basic terms, a perfect square is a result of multiplying two identical numbers. It is important to understand perfect squares up to 25 for mathematics exams or competitive exams, as some questions may demand difficult calculations. You can go further into the squares once you've completed this first list of perfect squares. Perfect squares share a lot of the same patterns, thus they have a lot of the same properties.

Competitive Exams after 12th Science

FAQs on Perfect Squares

1. What is a Perfect Square?

A number's square can be calculated by multiplying it by itself. A number that can be written as the product of two equal integers is called a perfect square. 16 is a perfect square, for example, because it is the product of two equal integers, 4 x 4 = 16.

2. How important is the list of perfect squares?

Great skills on perfect squares, up to 30, allows students to figure out complex problems of higher classes. Memorising the charts will help students to solve problems in exams easily. 

3. What are some properties of Perfect Squares?

Following are a few properties of Perfect Squares:

  • Numbers that end with 2, 3, 7, or 8 are never perfect squares, while numbers that end in 1, 4, 5, 6, 9, or 0 are perfect squares.

  • A perfect square is never a number that ends with odd number of zeros.

  • An even number's square is always even, whereas an odd number's square is always odd.