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Squares and square roots. INTRODUCTION. Numbers like ,1,4,9,16,25 are known as square numbers If a natural number m can be expressed as n 2 where n is also a natural number, then m is a square number. . PROPERTIES OF SQUARE NUMBERS.

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Presentation Transcript
slide3

INTRODUCTION

  • Numbers like ,1,4,9,16,25 are 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.
properties of square numbers
PROPERTIES OF SQUARE NUMBERS
  • If a number to be square has 1 or 9 in the units place then its square ends in 1.
  • When square number ends in 6 the number whose square it is will have either 4 or 6 in its units place.
numbers between square numbers
NUMBERS BETWEEN SQUARE NUMBERS
  • Any natural number n and (n+ 1), then

(n+1)2 – n2 = (n2 + 2n +1)- n2 = 2n +1

  • There are 2n non perfect square numbers between the squares of the numbers n and (n+1).
adding odd numbers
ADDING ODD NUMBERS
  • Sum of the first n odd natural numbers is n2 .
  • If a natural number cannot be expressed as a sum of successive odd numbers starting with 1, then it is not a perfect square.
patterns in square numbers
PATTERNS IN SQUARE NUMBERS
  • 12 = 1

112 = 121

1112 = 123321

11112 = 12344321

111112 = 123454321

1111112 = 1234554321

11111112 = 12345654321

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72 = 49

  • 672 =4489
  • 6672 =444889
  • 66672 =44448889
  • 666672 =4444488889
  • 6666672 =444444888889
finding the square of a number
FINDING THE SQUARE OF A NUMBER
  • We can find out the square of a number without having two multiply the numbers.

For e.g. 232 = (20 +3)2 =20(20+3)+3(20+3)

= 202 + 20*3 + 3*20 + 32

pythagorean triplets
PYTHAGOREAN TRIPLETS
  • For any natural number m > 1, we have (2m)2 + (m2 – 1) = (m2 + 1)2 .
square roots
SQUARE ROOTS
  • In mathematics, a square root of a number a is a number y such that y2= a, or, in other words, a number y whose square (the result of multiplying the number by itself, or y × y) is a. For example, 4 and -4 are square roots of 16 because 42 = (-4)2 = 16.
history
HISTORY
  • In Ancient India, the knowledge of theoretical and applied aspects of square and square root was at least as old as the Sulba Sutras, dated around 800–500 BC (possibly much earlier). A method for finding very good approximations to the square roots of 2 and 3 are given in the Baudhayana Sulba Sutra. Aryabhata in the Aryabhatiya (section 2.4), has given a method for finding the square root of numbers having many digits.
finding square roots
FINDING SQUARE ROOTS
  • Finding square roots through repeated subtraction

81−1=80

80 − 3=77

77− 5=72

72− 7=65

65− 9=56

56− 11=45

45− 13=32

32− 15=17

17− 17=0

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From 81 we have subtracted successive odd numbers starting from 1 and obtained 0 at 9th step. √81 = 9

  • Finding square root through prime factorization

Each prime factor in a prime factorization of the square of a number, occurs twice the number of times it occurs in the prime factorization of a given square number, say 324.

324=2*2*3*3*3*3

By pairing the prime factors, we get

324=2*2*3*3*3*3=22 * 32 * 32 = (2*3*3)2

So,

comprehension check
Comprehension check
  • There are 500 children in a school. For a P.T. drill they have to stand in such a manner that the numbers of rows is equal to number of columns. How many children would be left out in this arrangement?
  • Find the smallest square that is divisible by each of the numbers 4,9 and 10?
  • Find the square root of 2.56?
  • Find the square root of 4489?
  • Find the square root of 100 and 169 by repeated subtraction?
  • Find the square of 71?