MATHEMATICS POWER POINT

PRESENTATION

SQUARES & SQUARE ROOTS

BY:ROHIT KUMAR

CONTENTSQUARESPERFECT SQUARESTABLE OF SQUARESPROPERTIES OF SQUARESPROPERTIES OF PERFECT SQUARESPYTHAGOREAN TRIPLET SQUARES OF INTEGERSSQUARE ROOTSREPEATED SUBTRACTIONPRIME FACTORISATIONLONG-DIVISION METHODSQUARE ROOTS OF NUMBERS IN DECIMAL

FORMPATTERN OF SQUARE NUMBERQUICK NOTES

SQUARES In mathematics square of a number is

obtained by multiplying the number by itself.The usual notation for the formula for the

square of a number n is not the product n × n, but the equivalent  exponentiation n2

FOR EXAMPLE: 6 2 =6*6=36On the next slide there is a video clipping by

Adhithan who explains about SQUARES.

Perfect squares A Perfect square is a natural number which

is the square of another natural number . For Example consider two number 84 and

36. The factors of 84 are 2*2*3*7 Factors of 36 are 2*2*3*3. The Factor of 84

cannot be grouped into pairs of identical factors. So, 84 is not a perfect. But the factor of 36 can be grouped into pairs of identical factors , like

36 = 2*2 *3*3 =62

Table of squares NUMBERS(1 TO 10) MULTIPLICATION SQUARE NUMBER

1 1*1=12 1 2 2*2=22 4 3 3*3=32 9

4 4*4=42 16 5 5*5=52 25 6 6*6=62 36 7 7*7=72 49 8 8*8=82 64 9 9*9=92 81 10 10*10=102 100

PROPERTIES OF SQUARESThe number m is a square number if and only if one can compose a square of m equal (lesser) squares:m = 12 = 1 =

m = 22 = 4 =

m = 32 = 9 =

m = 42 = 16 =

m = 52 = 25 =

PROPERTIES OF PERFECT SQUARES A number ending in 2,3,7or 8 is never a

perfect square. A number ending in an odd number of zeros is never a perfect square.

The square of even number is even. The square of odd number is odd. The square of a proper fraction is smaller

than the fraction. The square of a natural number ‘n’ is equal to

the sum of the first ‘n’ odd numbers . For example : n is equal to the sum of the

first ‘n’ odd numbers.

Pythagorean tripletConsider the following:- 32+42=9+16=25=52

The collection of numbers 3,4 and 5 are known as

Pythagorean tripletFor any natural number m>1, we have (2m)2+(m2-1)2 = (m2+1)2

SQUARES OF INTEGERSSquares of negative integers:-The square of a negative integer is always a

positive integer. For example :- -m*-m=m2

-5*-5= 52 = 25 Squares of positive integers:-The square of a positive integer is always a

positive integer. For example :- m*m= m2

5* 5= 52 = 25 On the next slide there is a video clipping by

Maharajan who explains about SQUARES OF INTEGERS

Square Roots In mathematics, a square root of a number

x is a number y such that y2 = x ( symbol - ). For example :

There are 3 methods to find square roots , they are :-

REPEATED SUBTRACTION ( for small squares)PRIME FACTORIZATIONLONG DIVISIONOn the next slide there is a video clipping by

Adhithan who explains about Square Roots

Repeated subtractionRepeated subtraction method e.g.,- √81 Sol.:- 81-1=80 (2) 80-3=77 (3) 77-5=72 (4) 72-7=65 (5) 65-9=56 (6) 56-11=45 (7) 45-13=32 (8)32-15=17 (9) 17-17=0 Result=9 On the next slide there is a video clipping by Tarun Prasad

who explains about Repeated subtraction

PRIME FACTORISATIONPRIME FACTORIZATION METHOD In order to find the

square root of a perfect square , resolve it into prime factors; make pairs of similar factors , and take the product of prime factors , choosing one out of every pair.

On the next slide there is a video clipping by Tarun Prasad who explains about PRIME FACTORISATION

LONG-DIVISION METHODWhen numbers are very large , the method of

finding their square roots by factorization becomes lengthy and difficult .So, we use long-division method.

For example :

On the next slide there is a video clipping by Rohit Kumar who explains about LONG-DIVISION METHOD

SQUARE ROOTS OF NUMBERS IN DECIMAL FORM For finding the square root of a decimal fraction ,

make the number of decimal places even by affixing a zero , if necessary; mark the periods , and find out the square root, putting the decimal point in the square root as soon as the integral part is exhausted.

For example :

On the next slide there is a video clipping by Rohit Kumar who explains about SQUARE ROOTS OF NUMBERS IN DECIMAL FORM

Pattern of square numberPattern of square number 12 =1 112 =121 1112 =12321 11112=1234321 111112 =123454321 1111112 =12345654321 11111112 =1234567654321 111111112 =123456787654321 1111111112 =12345678987654321

QUICK NOTESIf p=m 2 , where m is a natural number, then

p is a perfect square. When the sum of odd numbers is even it is a perfect square of even number and when the sum of odd numbers is odd it is a perfect square of odd numbers. To find a square root of a decimal number correct up to “n” places , we find the square root up to (n+1) places and round it off to “n” places.

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