A perfect square number is obtained by the product of two same integers. More specifically, if we multiply an integer with itself then the resultant number will be a perfect square number. So the general form of a perfect square is r2 for some integer r. As 16=42, the number 16 is an example of a perfect square. In this section, we will learn about perfect square numbers.
Definition of Perfect Square
Perfect square definition: A number is called a perfect square if its square root is a whole number. Note that the square root of a perfect square is an integer.
For example, the square root of 4 is 2. Since 2 is an integer, we can say that 4 is a perfect square. More examples of perfect squares: 9, 16, 25, 36, 49, 64, 81 etc.
Perfect Square Examples
From the definition of a perfect square, a number X is called a perfect square if X=x2 for some natural number x. Thus, examples of perfect squares are given below.
- 25=52, so 25 is an example of a perfect square.
- 36 is a perfect square as 36=62.
- 100 is a perfect square as 100 is a square of 10.
- 125 is not a perfect square as it is not a square of some natural numbers.
Non Perfect Square
What is a non-perfect square?
Non perfect square definition: A number is said to be a non perfect square if it is not a square of some natural numbers. In other words, we cannot express it as n2 for some natural number. For example,
- 8 is a non-perfect square as 8=(2√2)2 and 2√2 is not a natural number.
- Non perfect squares 1-100: The numbers from 1 to 100 except 1, 4, 9, 16, 25, 36, 49, 64, 81, and 100 are examples of non-perfect squares.
Properties of Perfect Squares
• From the definition, the square root of a perfect square is an integer.
• If X is a perfect square, then we can express X as X=n2 for some natural number n.
• The square root of a perfect square must have unit digits 0, 1, 4, 5, or 9.
• The numbers ending with 2, 3, 6, 7, or 8 cannot be a perfect square number. Those numbers are called non-perfect square numbers.
• The square root of a non-perfect square is a quadratic surd. As √8=2√2 is not an integer, 8 is an example of non-perfect square numbers, and the square root √8 is a quadratic surd.
List of Perfect Squares
Perfect square table: We list the perfect squares between 1 and 100 in the table below with proper explanations.
| Integer (n) | n ×n | Perfect square |
| 0 | 0×0=0 | 0 |
| 1 | 1×1=1 | 1 |
| 2 | 2×2=4 | 4 |
| 3 | 3×3=9 | 9 |
| 4 | 4×4=16 | 16 |
| 5 | 5×5=25 | 25 |
| 6 | 6×6=36 | 36 |
| 7 | 7×7=49 | 49 |
| 8 | 8×8=64 | 64 |
| 9 | 9×9=81 | 81 |
| 10 | 10×10=100 | 100 |
Perfect squares from 1 to 100: From the above table, we see that the perfect squares between 1 to 100 are the numbers 1, 4, 9, 16, 25, 36, 49, 64, 81, and 100. So there are 10 perfect squares from 1 to 100.