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Squares

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Har Prasad Patel
Har Prasad Patel

This document provides an overview of squares and square roots including: - The definition of a square as a number multiplied by itself - Perfect squares being numbers where the square root is also a whole number - Patterns involving squares such as even/odd properties and formulas for decomposing squares - Methods for finding squares including the diagonal method and using formulas like (a + b)2 - Pythagorean triplets being sets of three numbers that satisfy the Pythagorean theorem when used as sides of a right triangle

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SQUARES AND SQUARE ROOTS A REVIEW
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CONTENTS ? SQUARES. ? PERFECT SQUARES. ? FACTS ABOUT SQUARES. ? SOME METHODS TO FINDING SQUARES. ? SOME IMPORTANT PATTERNS. ? PYTHAGOREAN TRIPLET.
SQUARES If a whole number is multiplied by itself, the product is called the square 1 of that number. For Examples: 1 x 1 = 1 = 12 1 The square of 1 is 1. 2 2 x 2 = 4 = 22 2 The square of 2 is 4
3 3 3x3=9= 32 4 4 x 4 = 16 = 42 4
PERFECT SQUARE A natural number x is a perfect square, if y2 = x where y is natural number. Examples : 16 and 25 are perfect squares, since 16 = 42 25 = 52
FACTS ABOUT SQUARES ? A number ending with 2, 3, 7 or 8 is never a perfect square. ? The squares of even numbers are even. ? The squares of odd numbers are odd. ? A number ending with an odd number of zeros is never a perfect square. ? The ending digits of a square number is 0, 1, 4, 5, 6 or 9 only. Note : it is not necessary that all numbers ending with digits 0, 1, 4, 5, 6 or 9 are square numbers.
SOME METHODS TO FINDING SQUARES USING THE FORMULA ( a + b )2 = a2 + 2ab + b2 1. (27)2 = (20 + 7 )2 (20 + 7)2 = (20)2 + 2 x 20x 7 + (7)2 = 400 + 280 + 49 = 729 . FIND (32)2
(a C b )2 = a2 C 2ab + b2 1. (39)2 = (40 -1)2 (40 C 1)2 = (40)2 C 2 x 40 x 1 + (1)2 = 1600 C 80 + 1 = 1521 . FIND (48)2.
DIAGONAL METHOD FOR SQUARING Example:- Find (72)2 using the diagonal method. SOLUTION:-
Therefore, (72)2 =5184. FIND (23)2
ALTERNATIVE METHOD ALTERNATIVE METHOD
SOME INTERESTING PATTERNS 1. SQUARES ARE SUM OF CONSECUTIVE ODD NUMBERS. EXAMPLES: 1 + 3 = 4 = 22 1 + 3 + 5 = 9 = 32 1+3+5+7 = 16 = 42 1+3+5+7+9 = 25 = 52 1+3+5+7+9+11 = ------- = -------
2. SQUARES OF NUMBERS ENDING WITH DIGIT 5. (15)2 =1X (1 + 1)X 100 +25 = 1X2X100 + 25 = 200 + 25 = 225 (25)2 = 2X3X100 + 25 = 600 + 25 = 625 (35)2 = (3X4) 25 = 1225 TENS UNITS FIND (45)2
PYTHAGOREAN TRIPLETS If three numbers x, y and z are such that x2 + y2 = z2, then they are called Pythagorean Triplets and they represent the sides of a right triangle. x z y
Examples (i) 3, 4 and 5 form a Pythagorean Triplet. 32 + 42 = 52.( 9 + 16 = 25) (ii) 8, 15 and 17 form a Pythagorean Triplet. 82+152 = 172. (64 +225 = 289)
Find Pythagorean Triplet if one element of a Pythagorean Triplet is given. For any natural number n, (n>1), we have (2n)2 + (n2-1)2 = (n2+1)2. such that 2n, n2-1 and n2+1 are Pythagorean Triplet.
Examples- Write a Pythagorean Triplet whose one member is 12. Since, Pythagorean Triplet are 2n, n2-1 and n2+1. So, 2n = 12, n = 6. n2-1 = (6)2-1 = 36 -1= 35 And n2+1 = (6)2+1= 36+1= 37 Therefore, 12, 35 and 37 are Triplet.
Write a Pythagorean Triplet whose one member is 6.
EVALUATION ? EXCEL QUIZ
My sincere thanks to :- NVS R. O. Bhopal and our Principal Ms. Kavita Singh for providing me an opportunity to prepare a PPT, and also to Mrs. Anju Pandey, TGT Eng, For her cooperation.
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  • 1. SQUARES AND SQUARE ROOTS A REVIEW
  • 5. CONTENTS ? SQUARES. ? PERFECT SQUARES. ? FACTS ABOUT SQUARES. ? SOME METHODS TO FINDING SQUARES. ? SOME IMPORTANT PATTERNS. ? PYTHAGOREAN TRIPLET.
  • 6. SQUARES If a whole number is multiplied by itself, the product is called the square 1 of that number. For Examples: 1 x 1 = 1 = 12 1 The square of 1 is 1. 2 2 x 2 = 4 = 22 2 The square of 2 is 4
  • 7. 3 3 3x3=9= 32 4 4 x 4 = 16 = 42 4
  • 8. PERFECT SQUARE A natural number x is a perfect square, if y2 = x where y is natural number. Examples : 16 and 25 are perfect squares, since 16 = 42 25 = 52
  • 9. FACTS ABOUT SQUARES ? A number ending with 2, 3, 7 or 8 is never a perfect square. ? The squares of even numbers are even. ? The squares of odd numbers are odd. ? A number ending with an odd number of zeros is never a perfect square. ? The ending digits of a square number is 0, 1, 4, 5, 6 or 9 only. Note : it is not necessary that all numbers ending with digits 0, 1, 4, 5, 6 or 9 are square numbers.
  • 10. SOME METHODS TO FINDING SQUARES USING THE FORMULA ( a + b )2 = a2 + 2ab + b2 1. (27)2 = (20 + 7 )2 (20 + 7)2 = (20)2 + 2 x 20x 7 + (7)2 = 400 + 280 + 49 = 729 . FIND (32)2
  • 11. (a C b )2 = a2 C 2ab + b2 1. (39)2 = (40 -1)2 (40 C 1)2 = (40)2 C 2 x 40 x 1 + (1)2 = 1600 C 80 + 1 = 1521 . FIND (48)2.
  • 12. DIAGONAL METHOD FOR SQUARING Example:- Find (72)2 using the diagonal method. SOLUTION:-
  • 13. Therefore, (72)2 =5184. FIND (23)2
  • 14. ALTERNATIVE METHOD ALTERNATIVE METHOD
  • 15. SOME INTERESTING PATTERNS 1. SQUARES ARE SUM OF CONSECUTIVE ODD NUMBERS. EXAMPLES: 1 + 3 = 4 = 22 1 + 3 + 5 = 9 = 32 1+3+5+7 = 16 = 42 1+3+5+7+9 = 25 = 52 1+3+5+7+9+11 = ------- = -------
  • 16. 2. SQUARES OF NUMBERS ENDING WITH DIGIT 5. (15)2 =1X (1 + 1)X 100 +25 = 1X2X100 + 25 = 200 + 25 = 225 (25)2 = 2X3X100 + 25 = 600 + 25 = 625 (35)2 = (3X4) 25 = 1225 TENS UNITS FIND (45)2
  • 17. PYTHAGOREAN TRIPLETS If three numbers x, y and z are such that x2 + y2 = z2, then they are called Pythagorean Triplets and they represent the sides of a right triangle. x z y
  • 18. Examples (i) 3, 4 and 5 form a Pythagorean Triplet. 32 + 42 = 52.( 9 + 16 = 25) (ii) 8, 15 and 17 form a Pythagorean Triplet. 82+152 = 172. (64 +225 = 289)
  • 19. Find Pythagorean Triplet if one element of a Pythagorean Triplet is given. For any natural number n, (n>1), we have (2n)2 + (n2-1)2 = (n2+1)2. such that 2n, n2-1 and n2+1 are Pythagorean Triplet.
  • 20. Examples- Write a Pythagorean Triplet whose one member is 12. Since, Pythagorean Triplet are 2n, n2-1 and n2+1. So, 2n = 12, n = 6. n2-1 = (6)2-1 = 36 -1= 35 And n2+1 = (6)2+1= 36+1= 37 Therefore, 12, 35 and 37 are Triplet.
  • 21. Write a Pythagorean Triplet whose one member is 6.
  • 23. My sincere thanks to :- NVS R. O. Bhopal and our Principal Ms. Kavita Singh for providing me an opportunity to prepare a PPT, and also to Mrs. Anju Pandey, TGT Eng, For her cooperation.