Square 1 to 100

The square of 1 to 100 ranges from 1 (1²) to 100 (100²). Learning these squares is helpful when it comes to solving complex mathematical problems. The concept of squaring numbers comes from geometry, where we calculate the area of squares.

For example, squaring off 25 means multiplying 25 two times, which gives 625 → 25 × 25 = 625. Now let's find the squares of numbers from 1 to 100.

Using the square numbers 1 to 100 chart, we can see how the square number is growing as we move from 1 to 100. Understanding these numbers will help you simply solve problems. Take a look at the chart given below.

This list will show you the list of squares from 1² to 100². From the list of squares, we can understand how multiplying numbers by itself works. Here, we will look at the complete list of squares from 1 to 100.

Square numbers which can be divided by 2 are even square numbers. We get even square numbers when an even number gets multiplied by itself.

Given below is the list of even square numbers

Square numbers which cannot be divided by 2 are odd square numbers, Odd square numbers are obtained when an odd number is multiplied by itself.

Given below is the list of odd square numbers.

We write the square of a number as X², where ‘X’ is multiplied by itself two times → X × X = X². The two methods used to square the number are:

• Multiplication Method
   
• Expansion Method

Now let's discuss them in detail

Here, we simply multiply the number by itself to get the square. We can try finding the square of 45 according to the steps given:-

Step 1: To find the square of 45, apply the formula → X × X = X².

Here, ‘X’ is 45

Step 2: Multiply 45 by itself → 45 × 45 = 2025

Hence, the square of 45 is 2025

We find the squares of 1 to 100 in this manner.

This method helps in finding the square root of a number using an algebraic formula. Instead of directly multiplying the same number by itself, we use a formula to find the square. This rule helps in finding the square of numbers closer to numbers 10, 11, 12, and so on.

The formula is  (a + b)² = a² + 2ab + b², where ‘a’ is the number closest to the number given and ‘b’ is the difference between the number given and ‘a’. 

Let’s find the square of 13 using the formula. Follow the steps given below:-

Step 1: We identify the value of ‘a’ as 12 and ‘b’ as 1. The value of ‘b’ is 1 because we multiplied 12 from the given number 13 → 13 - 12 = 1

Step 2: Now apply the value of ‘a’ and ‘b’ in the equation a² + 2ab + b²
a²  = 144
2ab = 2 × 12 × 1 = 24
b² = 1 × 1 = 1
a² + 2ab + b² = 144 + 24 + 1 = 169

So we find the square of 13 as 169

Rule 1: Multiplication Rule

The most basic rule to calculate squares is the multiplication rule. Here we multiply the number by itself two times to get the square. If a number is multiplied by another number, the result won’t be a square. For example, When 5 is multiplied by 5 we get 25 which is a square. Multiplying 5 by 6, we get 30 which is not a square.

Rule 2: Addition for Progressive Squares

The squares of the numbers are calculated by adding the consecutive odd numbers. Take the example of the first 8 numbers

1² =  1
2² = 1 + 3 = 4
3² = 1 + 3 + 5 = 9
4² = 1 + 3 + 5 + 7 = 16
5² = 1 + 3 + 5 + 7 + 9 = 25
6² = 1 + 3 + 5 + 7 + 9 + 11 = 36
7² = 1 + 3 + 5 + 7 + 9 + 11 + 13 = 49
8² = 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 = 64

Rule 3: Estimation of Large Numbers

For larger numbers, round them to the nearest simple numbers such as 10, 100, or 1000, then adjust the value. For example, to square 56, round it to 60 and adjust:
60² = 3600 

Then subtract the correction factor:

• 3600−(2×60×4) + 4²
   
• 3600 - 480 + 16 = 3136

Thus, 562 = 56 × 56 = 3136

For students to learn the squares from 1 to 100, here are a few tips and tricks. They will help you understand the squares easily.

Squaring numbers that end in 0

For numbers with the last digit as 0, multiply the non-zero part first and then add the remaining zeroes. For example, to find the squares of numbers 50 and 100, we multiply 5 by 5 and 1 by 1 first and then add the remaining zeroes.

• 50 × 50 = 2500
   
• 100 × 100 = 10000

Multiplication of Even Number

Multiplication of even numbers will always result in even perfect squares.
For example, the square of even numbers like 42, 84, 96 will always be even.

• 42 × 42 = 1764
   
• 84 × 84 = 7056
   
• 96 × 96 = 9216

Multiplication of Odd Number

Multiplication of odd numbers will always result in odd perfect squares.
For example, the square of odd numbers like 31, 57, and 69 will always be odd

• 31 × 31 = 961
   
• 57 × 57 = 3249
   
• 69 × 69 = 4761

• Perfect Square: A number expressed as the square of an integer. For example, 400 is a perfect square when 20 is multiplied by itself.

• Square Root: The opposite process of squaring a number. For example, √484 is 24.

• Even Square: A number that results from multiplying an even number by itself. For example, 42 gives 16, 82 gives 64, 122 gives 144.

• Square Chart: A visual representation of perfect square numbers.