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 r^{2} for some integer r. As 16=4^{2}, the number 16 is an example of a perfect square. In this section, we will learn about perfect square numbers.

Table of Contents

**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.

**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 $X=n^2$ for some natural number $n.$

**•** The square root of a perfect square must have unit digit $0, 1, 4, 5$ 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 $\sqrt{8}=2\sqrt{2}$ is not an integer, $8$ is an example of non-perfect square numbers, and the square root $\sqrt{8}$ is a quadratic surd.

**List of Perfect Squares**

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 |

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.