# Finding the Square of a Number

• Other Patterns in Squares
• Pythagorean Triplets

## Notes

### Finding the Square of a Number:

A number which produces a specified number when multiplied by itself is known as square numbers.

If a natural number m can be expressed as n2, where n is also a natural number, then m is a square number.

Example, 4, 9, 16, 25....etc.

## Finding square of a number using identity:

Find the square of the following numbers without actual multiplication.

1) 392

= (30 + 9)2

= 30(30 + 9) + 9(30 + 9)

= 302 + 30 × 9 + 9 × 30 + 92

= 900 + 270 + 270 + 81

= 1521.

2) 422

= (40 + 2)2

= 40(40 + 2) + 2(40 + 2)

= 402 + 40 × 2 + 2 × 40 + 22

= 1600 + 80 + 80 + 4

= 1764.

### Other patterns in squares:

1) 752 = 5625 = (7 × 8) hundreds + 25

2) 1252 = 15625 = (12 × 13) hundreds + 25

3) (a5)2 = (10a + 5)2 = 10a(10a + 5) + 5(10a + 5)

= 100a2 + 50a + 50a + 25

= 100a(a + 1) + 25

= a(a + 1) hundred + 25.

#### Pythagorean triplets:

For any natural number m > 1, we have (2m)2 + (m2 – 1)2 = (m2 + 1)2. So, 2m, m2 – 1 and m2 + 1 forms a Pythagorean triplet.

32 + 42 = 9 + 16 = 25 = 52

The collection of numbers 3, 4, and 5 is known as a Pythagorean triplet.

6, 8, 10 is also a Pythagorean triplet, since 62 + 82 = 36 + 64 = 100 = 102.

## Example

Write a Pythagorean triplet whose smallest member is 8.
We can get Pythagorean triplets by using general form 2m, m2 - 1, m2 + 1.
m2 -  1 = 8
m2 = 8 + 1 = 9
m = 3
2m = 6 and m2 + 1 = 10
The triplet is thus 6, 8, 10. But 8 is not the smallest member of this.
2m = 8
m = 4
m2 – 1 = 16 – 1 = 15
m2 + 1 = 16 + 1 = 17
The triplet is 8, 15, 17 with 8 as the smallest member.

## Example

Find a Pythagorean triplet in which one member is 12.

m2 - 1 = 12
m2 = 12 + 1 = 13.
Then the value of m will not be an integer.
2m = 12
m = 6
Thus, m2 - 1 = 36 - 1 = 35 and m2 + 1 = 36 + 1 = 37.
Therefore, the required triplet is 12, 35, 37.
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Calculating the Square of a Number Without Doing Actual Multiplication [00:06:19]
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