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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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