Perfect Square: Definition, List, Notation, Characteristics, Facts, Examples
Chandni 19 Jun, 2025
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Definition of Perfect Square - Perfect Square is an interesting number that connects to many situations in practical life. Whether you are solving puzzles or identifying geometric patterns, or learning how to find perfect square numbers, the concept of perfect square number helps you find solutions simply and quickly.
This article will discuss the perfect square number, properties of square numbers, characteristics, and ways to identify them, along with solved examples for better understanding and applications. We’ll also explore the perfect square meaning and definition of the perfect number to dive deep.
What is a Perfect Square?
A perfect square is a number obtained when an integer is multiplied by itself. The perfect square is a positive integer, which is the product of the same two positive or negative integers. Let us learn how to find perfect square, for example, when we multiply 6 by 6, we get 36, a perfect square number. Again, the product of (-) 8 and (-) 8 gives another perfect square, 64.Notation of Perfect Square
The perfect square can be represented as the square of a given number when the number is raised to the power of 2. So, a perfect square can be expressed as x 2 , where x is an integer and the value of x 2 is a perfect square. For example, a perfect square 25, which is a product of 5 and 5, can be written as: 25 = 5 2Why is a Perfect Square Always Positive?
A perfect square is always positive because the product is always positive, whether we multiply two positive integers or two negative integers. This applies the following principle:- +ve + ve = +ve
- -ve - ve = +ve
The Square Root Concept to Identify a Perfect Square
To determine what is a perfect square, we calculate the square root of the given number. If the result is a whole number, then the given number is a perfect square number; otherwise it is not. For example, if we take the square root of a given number, say, 49, we get the result as 7, which is a whole number. Therefore, 49 is a perfect square. Again, for a given number 60, we calculate the square root, which is approximately 7.750. Since it is not a whole number, 60 is not a perfect square.
Identify Perfect Square by Prime Factorization
Prime factorization is finding a given number's prime factors so that when these factors are multiplied, the product is the given number. To find a perfect square, we perform prime factorization and determine whether only one prime number occurs even times. For example, if we do prime factorization of 625, we get 625 = 5 x 5 x 5 x 5. Here, 5 is multiplied 4 times, so 625 is a perfect square, and the square root of 625 is 5 x 5 = 25.
Characteristics of Perfect Square Numbers
The last digit, or the digit at one's place of a number, indicates whether a number is a perfect square. If the last digit of a number is 2, 3, 7, or 8, it cannot be a perfect square. Perfect numbers always end with any of the digits 0, 1, 4, 5, 6, and 9. A perfect number must have an even number of zeros at the end. For example, 400 and 900 are even numbers. But there is a deviation to it. The number 300 has even zeros, but it is not a perfect square.Observations Made About the Perfect Square
- The numbers that end with 3 and 7 will have 9 as the unit's place digit in its perfect square.
- The number ending with 5 will have 5 at the unit place in its square number.
- The number ending 4 or 6 will generate an ideal square with 6 as the unit's place digit.
- The numbers that end with 2 or 8 will have 4 at the unit's place digit in its square number.
- The numbers ending with 1 and 9 will generate perfect squares with 1 as a digit in the unit's place.
A List of Perfect Squares from 1 to 10
Here is a list of perfect squares for the numbers 1 to 10 for ready reference.|
A List of Perfect Squares from 1 to 10 |
|
| Natural number | Perfect square |
| 1 | 1 × 1 = 1 |
| 2 | 2 × 2 = 4 |
| 3 | 3 × 3 = 9 |
| 4 | 4 × 4 = 16 |
| 5 | 5 × 5 = 25 |
| 6 | 6 × 6 = 36 |
| 7 | 7 × 7 = 49 |
| 8 | 8 × 8 = 64 |
| 9 | 9 × 9 = 81 |
| 10 | 10 x 10 = 100 |
Perfect Square Important Points
- A perfect square that ends with zero will always have an even number of zeros at the end.
- Perfect squares are always positive.
- The square roots of perfect squares may be positive or negative.
- We can also find perfect cubes by multiplying a perfect square with its square root.
- A perfect square has any digits 0, 1, 4, 5, 6, and 9 at its unit place.
- A perfect square of an even number is always even.
- A perfect square of an odd number is always odd.
- The sum of the first n odd numbers is always a perfect square.
Geometrical Correlation of a Perfect Square
What is a perfect square? In geometry, it can be understood as the area of a squared geometrical shape. A square has equal sides, so by multiplying the value of one side by the value of another side, we get the area of the square, which is a perfect square number. For example, if all sides of a square ABCD are 4 cm, then the area of the square is 4 cm x 4 cm = 16 square cm which is the perfect square of 4.
Trinomial Expression of Perfect Square
A perfect square can be expressed as a trinomial expression. For example, if there are two integers a and b, we can use the following trinomial expressions to find the perfect square.- (a + b) x (a + b) = (a + b) 2 = a 2 + 2ab + b 2
- (a - b) x (a - b) = (a - b) 2 a 2 + 2ab + b 2
Perfect Square Solved Examples
1. If the radius of a circle is 3 cm, find its area.
Solution: The formula for the area of a circle is πr 2 Here, the value of π is 3.14, and r represents the radius. Substituting the values of π and radius ( r ), we get: Area = 3.14 x (3) 2 So, the area of the circle is A = 3.14 x 9 = 28.26 square cm.2. Is the number 196 a perfect square? Find its square root.
Solution: By prime factorization, we get 196 = 2 x 2 x 7 x 7 So, it is a perfect square, and its square root is 2 x 7 = 14. 3. In an assembly, there are 12 rows, and in each row, there are 12 students. Find the total number of students in the assembly. Solution: The number of rows is the same as the number of students in each row, which means the arrangement is perfect square. So, the total number of students will be, (12) 2 = 12 x 12 = 144.4. Find the perfect square of the number 94.
Solution: We will use the trinomial expression to simplify the solution. We can write 94 = 100 - 6 So, (94) 2 = (100 – 6) 2 = (100) 2 - 2 x 100 x 6 + (6) 2 = 10000 - 1200 + 36 = 8836 Therefore, the perfect square of the number 94 is 8836 . Join Online Tuition for Kids Now!!| Related Articles | |
| Corresponding Angles | Isosceles Triangle |
| Average Speed Formula | Differentiation |
| Ratio | Surface Area of Cylinder |
Perfect Square FAQs
Is 1 a perfect square? Ans.
Can the square root of a perfect square be negative?
What minimum number must be added to 18 to make it a perfect square?
What is a typical indoor game that uses the concept of a perfect square?
What is the perfect square number between 10 and 50? Ans.
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