|Table of contents|
|Introduction to Square Numbers|
|Finding the Square of a Number|
|Properties of Square Numbers|
|Finding the square of a number with the unit’s place 5|
|Adding Triangular Numbers|
|Numbers between Square Numbers|
|Addition of Odd Numbers|
|Product of Two Consecutive Even or Odd Natural Numbers|
|1 Crore+ students have signed up on EduRev. Have you?|
If a natural number m can be expressed as n², where n is also a natural number, then m is a square number.
Example: 1, 4, 9, 16, and 25.
If n is a number, then its square is given as n×n = n².
For example Square of 5 is equal to 5×5=25
The square of a number N5 is equal to (N(N+1))×100+25, where N can have one or more than one digit.
For example: If N = 1, then 152=(1×2)×100+25=200+25=225
If N = 20, then 2052=(20×21)×100+25=42000+25=42025
A number that is obtained from the square of the other number is called a perfect square. For 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: It is a sequence of the numbers 1, 3, 6, 10, 15, etc. It is obtained by continued summation of the natural numbers. The dot pattern of a triangular number can be arranged as triangles.
If we add two consecutive triangular numbers, we get a square number.
Example: and .
There are 2n non-perfect square numbers between squares of the numbers n and (n + 1), where n is any natural number.
The sum of first n odd natural numbers is n2.
The product of two even or odd natural number can be calculated as,, where a is a natural number, and , , are the consecutive odd or even numbers.
Patterns in numbers like 1, 11, 111, … :
Patterns in numbers like 6, 67, 667, … :
For any natural number
, and forms a Pythagorean triplet.
For , , and .
So, 3, 4, 5 is the required Pythagorean triplet.
Finding the number whose square is known is known as finding the square root. Finding square root is inverse operation of finding the square of a number.
, square root of 1 is 1.
, square root of 4 is 2.
, square root of 9 is 3.
Estimating the square root of 247:
Since: 100 < 247 < 400
But it is not very close.
256 is much closer to 247 than 225.
Therefore, is approximately equal to 16.
Squares of numbers having two or more digits can easily be found by writing the number as the sum of two numbers.
To know more about Squares and Square Roots, visit here.
Every square number can be expressed as a sum of successive odd natural numbers starting from one.
The square root can be found through repeated subtraction. To find the square root of a number
Step 1: subtract successive odd numbers starting from one.
Step 2: stop when you get zero.
The number of successive odd numbers that are subtracted gives the square root of that number. Suppose we want to find the square root of 36.
Here 6 odd numbers (1, 3, 5, 7, 9, 11) are subtracted to from 36 to get 0.
So, the square root of 36 is 6.
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 left-most single-digit 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.
If we are given any member of a Pythagorean triplet, then we can find the Pythagorean triplet by using general form
For example, If we want to find the Pythagorean triplet whose smallest number is 8.
The triplet is 6, 8 and 10.
But 8 is not the smallest number of this triplet.
So, we substitute
Therefore, 8, 15, 17 is the required triplet.