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# Perfect Squares- Definition, Formula, Examples, Tips and Tricks, Chart

Everyone of you must have calculated the square of any number, also known as the perfect square. A perfect square is a number that is the result of multiplying an integer by itself. Multiplying a whole number by itself results in a Perfect Square, such as 4 being the result of multiplying 2 by itself (2 × 2 = 4), making it a perfect square. In this article, we will learn about the definition and formula of perfect squares along with some of its tips and tricks.

## Perfect Squares

When a number is multiplied by itself, the result is referred to as a perfect square and can be written as the second exponent of an integer. Therefore, perfect squares refer to the squares of all whole numbers.

Mathematically speaking, the perfect square can be represented as a², called as a superscript 2. They always have a value of zero or above, and are important for comprehending square roots, quadratic equations, and number theory. A perfect square is a number that is the result of multiplying an integer by itself.

## What is a Perfect Square?

A positive integer that is the result of multiplying an integer by itself is called a perfect square. Put simply, perfect squares are numbers that result from multiplying integers by themselves. Typically, we can represent a perfect square as x², where x is a integer and x² is a perfect square.

## Perfect Square Definition

A perfect square is an integer that can be written as the square of another integer. In simpler terms, it is the result of multiplying an integer (both positive and negative values) by itself.

For example, 36 is a perfect square as it can be obtained by multiplying 6 by itself. Another example is 4, which is 2 times 2.

## Perfect Squares Formula

Assuming N is a perfect square of a whole number x, we can express this as N = x times x = x². Therefore, the formula for perfect squares can be written as:

N = x²

## Perfect Square Numbers

It is a known fact that the square of a number is the result of multiplying that number by itself. Put simply, perfect squares are the result of squaring whole numbers like 1 or 12, 4 or 22, 9 or 32, 16 or 42, 25 or 52, and so forth. You can better understand the building concept of perfect square numbers by looking at the pictorial diagram given below.

## Perfect Squares Examples

Check the examples of perfect square numbers below.

9 = 3 * 3

4 = 2 * 2

25 = 5 * 5

121 = 11 * 11

676 = 26 * 26

576 = 24 * 24

625 = 25 * 25

400 = 20 * 20

## Perfect Squares Tips and Tricks

When working with perfect squares, it is essential to remember the following tips and tricks in order to solve questions quickly.

• A square number that finishes with 0 will consistently contain an even amount of zeros at the end.
• Perfect squares are always positive since the product of two negative numbers results in a positive number.
• Perfect squares can have both positive and negative square roots.
• Perfect cubes can be obtained by multiplying a number by itself three times.
• In order to determine if a certain number is a perfect square, we can find the square root of the number. A perfect square is a whole number if its square root is also a whole number. If the square root is a decimal, then the number is not a perfect square.

## Perfect Squares Chart

The chart on perfect squares for candidates is given below. Candidates can consult this chart whenever they want to know the square of any number.

## How to Identify Perfect Square Numbers

Take note of the final digit of the perfect square numbers from 1 to 20 as provided in the table above. You will observe that they finish with one of these numbers: 0, 1, 4, 5, 6, or 9. Upon experimenting with different perfect square numbers, you would have noticed a key characteristic of perfect squares.

Numbers with 2, 3, 7, or 8 in the units place aren’t perfect squares, while those with 0, 1, 4, 5, 6, or 9 might be perfect squares. One can make the following observations to recognize a perfect square number.

• Numbers that end in 3 or 7 will always result in a square number with 9 as the units place digit.
• Any number that ends with 5 will always have 5 as the units digit in its square.
• Numbers that end in 4 or 6 will have 6 as the last digit in their square.
• A number that ends with 2 or 8 will result in having 4 as the last digit in its square.
• Numbers that end in 1 or 9 will always have 1 as the last digit when squared.

## Alternate Method to Identify Perfect Squares

One more method to determine if a number is a perfect square is by finding the square root of the number in question. A perfect square is a whole number if its square root is an integer. If the square root is not an integer, then the number is not a perfect square.

For instance, if we want to determine if 24 is a perfect square, we can find its square root. The square root of 24 is 4.89. It is evident that 4.89 is a decimal, thus, 24 is not a perfect square. Let’s look at another instance of the number 81. The square root of 81 is 9. 9 being a whole number implies that 81 is a perfect square.

## Perfect Squares List 1 – 10000

Candidates can check the perfect squares list that is found from number 1 to 10,000. The square numbers have been segregated in different groups based on their unit digits. Carefully observe the last digits of the perfect square numbers and their constituting integer.

 Perfect Squares with its Integers 1² = 1 11² = 121 21² = 441 31² = 961 41² = 1681 51² = 2601 61² = 3721 71² = 5041 81² = 6561 91² = 8281 2² = 4 12² = 144 22² = 482 32² = 1024 42² = 1764 52² = 2704 62² = 3844 72² = 5184 82² = 6724 92² = 8464 3² = 9 13² = 169 23² = 529 33² = 1089 43² = 1849 53² = 2809 63² = 3969 73² = 5329 83² = 6889 93² = 8649 4² = 16 14² = 196 24² = 576 34² = 1156 44² = 1936 54² = 2916 64² = 4096 74² = 5476 84² = 7056 94² = 8836 5² = 25 15² = 225 25² = 625 35² = 1225 45² = 2025 55² = 3025 65² = 4225 75² = 5625 85² = 7225 95² = 9025 6² = 36 16² = 256 26² = 676 36² = 1296 46² = 2116 56² = 3136 66² = 4356 76² = 5776 86² = 7396 96² = 9216 7² = 49 17² = 289 27² = 729 37² = 1369 47² = 2209 57² = 3249 67² = 4489 77² = 5929 87² = 7569 97² = 9409 8² = 64 18² = 324 28² = 784 38² = 1444 48² = 2304 58² = 3364 68² =4624 78² = 6084 88² = 7744 98² = 9604 9² = 81 19² = 361 29² = 841 39² = 1521 49² = 2401 59² =3481 69² =4761 79² = 6241 89² = 7921 99² = 9801 10² = 100 20² = 400 30² = 900 40² = 1600 50² = 2500 60² =3600 70² =4900 80² = 6400 90² = 8100 100² = 10000

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## FAQs

### Is 1 a perfect square?

Yes, 1 is a perfect square as it can be obtained by multiplying 1 two times.

### How many perfect square numbers are from 1 to 100?

There are 10 perfect squares between 1 and 100. They are: 1, 4, 9, 16, 25, 36, 49, 64, 81, and 100.