Table of contents  
Introduction to Square Numbers  
Properties of Square Numbers  
Some More Interesting Patterns  
Finding the Square of a Number  
Square Roots  
Square Roots of Decimals 
A natural number n is a square number if it can be expressed as n², where n is also a natural number.
Example: 1, 4, 9, 16, and 25.
If n is a number, then its square is given as n×n = n².
Example Square of 5 is equal to 5×5=25
Square Numbers
To find the square of a number with the unit's place as 5, we can follow these steps:
Step 1. Separate the number into two parts  the last digit (5) and the remaining part (N). Hence the number becomes N5.
Step 2. Multiply N with N+1.
Step 3. Append 25 at the end of the result obtained in step 2.
Examples:
Square of 15 :
Here N = 1,
Then 15^{2}=(1×2)×100+25=200+25=225
Square of 205
Here N= 20,
Then 205^{2}=(20×21)×100+25=42000+25=42025
A number that is obtained from the square of the other number is called a perfect square.
OR
A perfect square is a number that results from multiplying another number by itself.
Example : 81 is a perfect square number, which is obtained by taking the square of the number 9.
There exist interesting patterns in:
Triangular numbers: Triangular numbers are a sequence created by adding natural numbers continuously. They form a triangular shape when arranged in dots. For example: 1, 3, 6, 10, 15, etc.
1
1+2=3
1+2+3=6
1+2+3+4=10
Diagram:
•
••
•••
Now, if we add two consecutive triangular numbers, we get a square number.
Example : 1+3=4=22 and 3+6=9=32.
Numbers between square numbers are the integers that fall between two perfect squares.
There are 2n nonperfect square numbers between squares of the numbers n and (n + 1), where n is any natural number.
Example:
The sum of first n odd natural numbers is n^{2}.
Example:
1+3=4=2^{2}
1+3+5=9=3^{2}
The sum of two consecutive numbers is always an odd number. For example, consider a consecutive number sequence 1, 2, 3, 4, 5, 6, 7, 8, 9.
Here, the sum of the two consecutive numbers will always be an odd number. 1+2 = 3, 2+3 = 5, 3+4 = 7, and so on.
Square of an odd number n can be expressed as sum of two consecutive positive integers
The product of two even or odd natural number can be calculated as, (a+1)×(a−1)=(a2−1), where a is a natural number, and a−1, a+1, are the consecutive odd or even numbers.
For example:
11 and 13 are two odd consecutive numbers, therefore the multiplication of 11 and 13 can be calculated as:
11×13 = (12−1)×(12+1) = 122−1 = 144−1 = 143
Patterns in numbers like 1, 11, 111, … :
12= 1
112= 1 2 1
1112= 1 2 3 2 1
11112= 1 2 3 4 3 2 1
111112= 1 2 3 4 5 4 3 2 1
111111112=1 2 3 4 5 6 7 8 7 6 5 4 3 2 1
Patterns in numbers like 6, 67, 667, … :
72=49
672=4489
6672=444889
66672=44448889
666672=4444488889
6666672=444444888889
To find the square of a number, you can use the concept of expanding the expression.
1. Let's take the example of finding the square of 23:
23^{2} = (20 + 3)^{2} = (20 + 3) (20 + 3)
Expanding this expression, we get:
= 20 (20 + 3) + 3 (20 + 3)
Simplifying further:
= 20 * 20 + 20 * 3 + 3 * 20 + 3 * 3
Calculating the multiplication:
= 400 + 60 + 60 + 9
Finally, adding up the values:
= 529
Therefore, the square of 23 is 529.
2. Let's find the square of 15 using the same method:
15^{2} = (10 + 5)^{2} = (10 + 5) (10 + 5)
Expanding this expression:
= 10 (10 + 5) + 5 (10 + 5)
Simplifying further:
= 10 * 10 + 10 * 5 + 5 * 10 + 5 * 5
Calculating the multiplication:
= 100 + 50 + 50 + 25
Finally, adding up the values:
= 225
Therefore, the square of 15 is 225
Let's explore other patterns in squares using the given examples:
When we look at the numbers and their corresponding squares, we can observe the following pattern:
In each case, we notice that the square can be expressed as a product of two consecutive numbers, followed by the word "hundreds," and then we add 25 at the end.
This pattern can be generalized as follows:
For any number n, where n is a positive integer:
n^{2} = (n × (n+1)) hundreds + 25
For example:
By following this pattern, we can easily find the squares of various numbers by multiplying two consecutive numbers, adding "hundreds," and then appending 25 at the end.
A Pythagorean triplet is a set of three positive integers a, b, and c that satisfy the Pythagorean theorem, which states that the sum of the squares of the two smaller numbers is equal to the square of the largest number. In other words, a² + b² = c².
In the given formula for generating Pythagorean triplets, we have:
a = 2m
b = (m²  1)
c = (m² + 1)
For any natural number m > 1, these three numbers form a Pythagorean triplet.
This is because (2m)² + (m²  1)² = (m² + 1)², as stated in the formula.
Example :
Let's say m = 3, then
a = 2m = 2(3) = 6
b = (m²  1) = (3²  1) = (9  1) = 8
c = (m² + 1) = (3² + 1) = (9 + 1) = 10
We have the triplet (6, 8, 10). Now, let's check that it satisfies the Pythagorean theorem:
6² + 8² = 36 + 64 = 100
10² = 100
Determining the number whose square is already known is called finding the square root. The process of finding the square root is the opposite of calculating the square of a number.
Example:
12=1, square root of 1 is 1.
22=4, square root of 4 is 2.
32=9, square root of 9 is 3.
Finding the square root of 180:
To find the square root of a number through repeated subtraction, follow these steps:
Step 1: Start with the given number (let's call it n) for which you want to find the square root.
Step 2: Begin subtracting successive odd numbers, starting from 1. Keep track of how many odd numbers you have subtracted.
Step 3: Continue subtracting the next odd number in the sequence until the result of the subtraction is zero.
Step 4: The number of odd numbers you subtracted in the process is the square root of the given number.
Example :
Suppose you want to find the square root of 49.
1. Start with the given number: n = 49
2. Begin subtracting successive odd numbers:
49  1 = 48 (1 odd number subtracted)
48  3 = 45 (2 odd numbers subtracted)
45  5 = 40 (3 odd numbers subtracted)
40  7 = 33 (4 odd numbers subtracted)
33  9 = 24 (5 odd numbers subtracted)
24  11 = 13 (6 odd numbers subtracted)
13  13 = 0 (7 odd numbers subtracted)
3, The result is zero after subtracting 7 odd numbers.
4. The square root of 49 is 7
Consider the prime factorization of various numbers and their squares:
We observe that each prime factor in the prime factorization of the square of a number occurs twice the number of times it occurs in the prime factorization of the original number.
By using this pattern, we can find the square root of a given square number. For example, let's find the square root of 324:
Similarly, we can find the square root of 256:
If we encounter a number that is not a perfect square, like 48, we can determine the smallest multiple of 48 that is a perfect square by completing the pairs of prime factors. In this case, multiplying 48 by 3 gives 144, which is a perfect square.
To find a number by which we should divide 48 to get a perfect square, we look for the factor that doesn't have a pair in the prime factorization. In this case, the factor 3 is not paired, so dividing 48 by 3 gives us 16, which is a perfect square.
Calculating the squares of twodigit numbers can be made easier by expressing the number as a sum of two smaller numbers.
For example:
Example :
23^{2}= (20+3)^{2} = 20(20+3)+3(20+3)
=202+20×3+20×3+32
=400+60+60+9
=529
To find the square root using the Long division method, follow these steps:
Steps involved in finding the square root of 484 by Long division method:
Step 1: Place a bar over every pair of numbers starting from the digit at units place. If the number of digits in it is odd, then the leftmost singledigit too will have a bar.
Step 2: Take the largest number as divisor whose square is less than or equal to the number on the extreme left. Divide and write quotient.
Step 3: Bring down the number which is under the next bar to the right side of the remainder.
Step 4: Double the value of the quotient and enter it with a blank on the right side.
Step 5: Guess the largest possible digit to fill the blank which will also become the new digit in the quotient, such that when the new divisor is multiplied to the new quotient the product is less than or equal to the dividend.
The remainder is 0, therefore, √484=22.
Step 1: To find the square root of a decimal number we put bars on the integral part (i.e., 17) of the number in the usual manner. And place bars on the decimal part (i.e., 64) on every pair of digits beginning with the first decimal place. Proceed as usual. We get bar17.64
Step 2: Now proceed in a similar manner. The left most bar is on 17 and 4^{2} < 17 < 5^{2}
Step 3: Take this number as the divisor and the number under the leftmost bar as the
dividend, i.e., 17. Divide and get the remainder.
Step 4: Double the divisor and enter it with a blank on its right. Since 64 is the decimal part so put a decimal point in the quotient.
Step 5: We know 82 × 2 = 164, therefore, the new digit is 2. Divide and get the remainder
Step 6: Since the remainder is 0 and no bar left, therefore square root of 17.64 is 4.2
79 videos411 docs31 tests

1. What are square numbers? 
2. What are some properties of square numbers? 
3. Can a negative number be a square number? 
4. How can we find the square of a number? 
5. What is a square root? 

Explore Courses for Class 8 exam
