Real Number

1, 2, 3, …

Whole Number

Real Numbers

Irrational Number

0.5254578

Rational Number

IntegerNegative (less than 0)

ZeroDivide all number is no answer

Plus all number is all number

Minus all number is all number

Multiply all number is 0

Number divide 0 is 0

Positive (more than 0)

Natural Number

0, 1, 2, 3, …

T.So

Property of

Equality

No. Number Property of equality

1 8 = 8 Reflexive Property

2 9 = 9

3 100 = 100

4 If 6 = 5 + 1 then 5 + 1 = 6 Symmetric Property

5 If 8 = 3 + 5 then 3 + 5 = 8

6 If 10 = 7 + 3 then 7 + 3 = 10

7 If 42 = 16 then 16 = 7 + 9 Â̈oª 42 = 7 + 9

Transitive Property

8 If 52 = 25 then 25 = 20 + 5 Â̈oª 52 = 20 + 5

9 If 72 = 49 then 49 = 40 + 9 Â̈oª 72 = 40 + 9

Reflexive Property

Reflexive Property

Symmetric Property

Symmetric Property

Transitive Property

Transitive Property

10 If 4 5 = 20 then (4 5) + 3 = 20 + 3 Addition Property

11 If (2 6) = 12 then (2 6) + 2 = 12 + 2

12 If 3 2 = 6 then (3 2) + 5 = 6 + 5

13 If (4 3) = 12 then (4 3) 5 = 12 5

Multiplication Property

14 If (5 2) = 10 then (5 2) 6 = 10 6

15 If 2

8 = 4 then 2

8 4 = 4 6

Addition Property

Addition Property

Multiplication Property

Multiplication Property

EX.7 = 7 5 = 510 = 104 = 49 = 9

Property of Equality

Reflexive PropertyReflexive Property is A = A when A is Real number.

If 9 = 6+ 3 then 6+ 3 = 9If 6 = 4 + 2 then 4 + 2 = 6 If 7 = 4 + 3 then 4 + 3 = 7 If 12 = 10 + 2 then 10 + 2 = 12

Symmetric Property

Symmetric Property is if A = B, then A = B when A and B is Real number.

If 32 = 9 and 9 = 6 + 3 then 32 = 6 + 3If 52 = 25 and 25 = 21 + 4 then 52 = 21 + 4If 42 = 16 and 16 = 6 + 10 then 62 = 6 + 10If 132 = 169 and 169 = M + 3 then 32 = M + 3

Transitive Property

Transitive Property is if A = B and B = C, then A = C when A ,B and C are Real number.

If 4 3 = 12 and c = 2 then (4 3) + 2 = 12 + 2If 5 2 = 10 and c = 5 then (5 2 ) + 5 = 10 + 5If 7 8 = 56 and c = 9 then (8 7) + 9 = 56 + 9If 8 4 = 32 and c = 3 then (8 4) + 3 = 32 + 3

Addition Property is if A = B, then A + C = B + C when A ,B and C is Real number.

Addition Property

If 4 x 8 = 32 and c = 5 then (4 x 8 )x 5 = 32 x 5If 9 x 6 = 54 and c = 8 then (9 x 6 )x 8 = 54 x 8If 12 x 8 = 96 and c = 9 then (12 x 8 )x 9 = 96 x 9

Multiplication Property

Multiplication Property is if A = B, then A × C = B ×C when A ,B and C is Real number.

ADDITIONALPROPERTY

Closure Property

Commutative Property

Associativity

Additive identityInverse property

Closure PropertyIf 4, -3 R then 4 + (-3) R

If 20, 34 R then 20 + 34 R If -45, -9 R then 45 + (-9) R

Closure Property is if a, b R Then a + b R Ex. 3, 6 R then 3 + 6 R

Commutative Property of Addition

6 + 3 = 3 + 6 (-56) + 4 = 4 + (-56)

(-56) + (-99) = (-99) + (-56)

Commutative Property of Addition is a + b = b + a ex. 2 + 4 = 4 + 2

Associativity10 + (2 + 8) = (10 + 2) + 8

26 + (13 + 34) = (26+ 13) + 34 (-99) + ((-2) + 8) = ((-99)+ (-2)) + 8

Associativity a + (b + c) = (a + b) + c ex. 6 + (7 + 2) = (6 + 7) + 2

Additive identity0 + 8 = 8 = 8 + 0

0 + 10 = 10 = 10 + 00 + (-9) = -9 = (-9) + 0

Additive identity is 0 + a = a = a + 0

inverse property of addition

(-15) + 15 = 0 = 15 + (-15)(-47) + 47 = 0 = 47 + (-47)(-99) + 99 = 0 = 99 + (-99) (-65) + 65 = 0 = 65 + (-65)

Inverse property of addition For real number a will have -a is (-a) + a = 0 = a + (-a) ex. (-5) + 5 = 0 = 5 + (-5)

1. When the square of a certain number is diminished by 9 times the number the result is 36. Find the number.

square of a certain number = 2x

9 times the number = x9

3692 xx

03692 xx0)3)(12( xx

312 orx

2. A certain number added to its square is 30. Find the number.

A certain number = xits square =

2x

302 xx

0302 xx0)5)(6( xx

5)6( orx

3. The square of a number exceeds the number by 72. Find the number.

a number = xsquare of a number =

2x

722 xx0722 xx0)8)(9( xx

89 orx

4. Find two positive numbers whose ratio is 2:3 and whose product is 600.

numbers whose ratio is 2:3 = 2x:3xFirst numbers =x

xx2

3:

product is 600 = 600)2

3( xx

6002

3 2

x

12003 2 x

3

12002 x

4002 x

04002 x

02022 x

0)20)(20( xx

2020 orx

20xpositive numbers

two positive numbers 30:20)20(2

3:20

2

3: xx

5. The product of two consecutive odd integers is 99. Find the integers.

odd integers = xconsecutive odd integers = 2x

99)2( xx9922 xx09922 xx0)11)(9( xx

119 orx

If x = 9 consecutive odd integers= 9+2=11

If x = -11 consecutive odd integers= -11+2=-9

6. Find two consecutive positive integers such that the square of the first is decreased by 17 equals 4 times the second.First number = xSecond number = 1x

)1(4172 xx0)1(4172 xx044172 xx02142 xx0)3)(7( xx

37 orx7x

First number = 7 Second number = 7+1=8

7. The ages of three family children can be expressed as consecutive integers. The square of the age of the youngest child is 4 more than eight times the age of the oldest child. Find the ages of the three children.

youngest child = xoldest child = 2x

)2(842 xx16842 xx

02082 xx0)10)(2( xx

102orx 10x

The ages of three family children = 10,11,12 year old

8. Find three consecutive odd integers such that the square of the first increased the product of the other two is 224.

224)4)(2(2 xxx2248622 xxx021662 2 xx010832 xx0)9)(12( xx

912orx 9x

three consecutive odd integers = 9,10,11

9. The sum of the squares of two consecutive integers is 113. Find the integers.

113)1( 22 xx1131222 xxx011222 2 xx0562 xx0)8)(7( xx

87 orx

If x= 7 two consecutive integers is 7 and 8 If x= -8 two consecutive integers is -8 and -7

10. The sum of the squares of three consecutive integers is 302. Find the integers.

302)2()1( 222 xxx302)44()12( 222 xxxxx

029763 2 xx09922 xx0)9)(11( xx

911orx

If x= -11 three consecutive integers is -11,-10,-9If x= 9 three consecutive integers is 9,10,11

11. Two integers differ by 8 and the sum of their squares is 130.

130)8( 22 xx130641622 xxx066162 2 xx03382 xx0)11)(3( xx

113 orx If integers is -3 two integers is -3 and -11If integers is 11 two integers is 11 and 3

12. rectangle is 4 m longer than it is wide, its area is 192 m2. Find the dimensions of the rectangle.

192)4( xx19242 xx019242 xx0)12)(16( xx

1216 orx 12x

12

412

Dimensions of the rectangle= 12 x 16 = 192