T.So
1, 2, 3, …
Whole Number
Natural Number
Real
Numbers
Irrational
Number
0, 1, 2, 3, …
0.5254578
Rational Number
Integer
Negative (less than 0)
Zero
Divide all number is no
answer
Positive (more than 0)
Number divide 0 is 0
Multiply all number is 0
Plus all number is all number
Minus all number is all number
No.
Property of equality
Number
1
8 = 8
Reflexive Property
2
9 = 9
Reflexive Property
3
100 = 100
Reflexive Property
4
If 6 = 5 + 1 then 5 + 1 = 6
Symmetric Property
5
If 8 = 3 + 5 then 3 + 5 = 8
Symmetric Property
6
If 10 = 7 + 3 then 7 + 3 = 10
Symmetric Property
7
If 42 = 16 then 16 = 7 + 9
9
8
If 52 = 25 then 25 = 20 + 5
20 + 5
52 =
9
If 72 = 49 then 49 = 40 + 9
40 + 9
72 =
42 = 7 +
Transitive 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
Addition Property
12 If 3  2 = 6 then (3  2) + 5 = 6 + 5
Addition Property
13 If (4  3) = 12 then (4  3)  5 = 12 
5
Multiplication Property
14 If (5  2) = 10 then (5  2)  6 = 10  6
Multiplication Property
15 If 8 = 4 then
2

4
=
4

6
2
8
Multiplication Property
Property of Equality
EX.
7 =7
5 =5
10 = 10
4 =4
9 = 9
Reflexive Property
Reflexive Property is A = A
when A is Real number.
Symmetric Property
If 9 = 6+ 3 then 6+ 3 = 9
If 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 is if A = B, then
A = B when A and B is Real number.
Transitive Property
If
If
If
If
32 = 9 and 9 = 6 + 3 then 32 = 6 + 3
52 = 25 and 25 = 21 + 4 then 52 = 21 + 4
42 = 16 and 16 = 6 + 10 then 62 = 6 + 10
132 = 169 and 169 = M + 3 then 32 = M + 3
Transitive Property is if A = B and B = C, then A = C when
A ,B and C are Real number.
Addition Property
If
If
If
If
43
52
78
84
=
=
=
=
12
10
56
32
and
and
and
and
c
c
c
c
=
=
=
=
2
5
9
3
then
then
then
then
(4  3) + 2 = 12 + 2
(5  2 ) + 5 = 10 + 5
(8  7) + 9 = 56 + 9
(8  4) + 3 = 32 + 3
Addition Property is if A = B, then A + C = B + C when
A ,B and C is Real number.
Multiplication Property
If 4 x 8 = 32 and c = 5 then (4 x 8 )x 5 = 32 x 5
If 9 x 6 = 54 and c = 8 then (9 x 6 )x 8 = 54 x 8
If 12 x 8 = 96 and c = 9 then (12 x 8 )x 9 = 96 x 9
Multiplication Property is if A = B, then A × C = B ×C
when A ,B and C is Real number.
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 = x
2
9 times the number = 9 x
x  9 x  36
2
x  9 x  36  0
2
( x  12 )( x  3 )  0
x  12
or  3
2. A certain number added to its square is 30. Find the
number.
A certain number =
2
its square = x
x
x  x  30
2
x  x  30  0
( x  6 )( x  5 )  0
2
x  (  6 ) or 5
3. The square of a number exceeds the number by 72.
Find the number.
a number =
x
square of a number =
x
2
x  x  72
2
x  x  72  0
( x  9 )( x  8 )  0
2
x   9 or
8
4. Find two positive numbers whose ratio is 2:3 and whose
product is 600.
First numbers = x
numbers whose ratio is 2:3 = 2x:3x
x:
product is 600 = x (
3x
3
3
x
2
x )  600
2
2
x  400
2
 600
2
x  400  0
2
3 x  1200
2
x  20
2
x 
2
1200
3
2
0
( x  20 )( x  20 )  0
x  20
positive numbers
or  20
x  20
two positive numbers x :
3
2
x  20 :
3
2
( 20 )  20 : 30
5. The product of two consecutive odd integers is 99. Find
the integers.
odd integers =
x
consecutive odd integers =
x ( x  2 )  99
x  2 x  99
2
x  2 x  99  0
2
( x  9 )( x  11 )  0
x  9 or  11
x2
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 = x
Second number =
x 1
x  17  4 ( x  1)
2
x  17  4 ( x  1)  0
2
x  17  4 x  4  0
2
x  4 x  21  0
First number = 7
( x  7 )( x  3 )  0 Second number = 7+1=8
x  7 or  3
x  7
2
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 = x
oldest child = x  2
x  4  8( x  2 )
2
x  4  8 x  16
2
x  8 x  20  0
( x  2 )( x  10 )  0
x   2 or 10
x  10
2
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.
x  ( x  2 )( x  4 )  224
2
2
x  x  6 x  8  224
2
2 x  6 x  216  0
2
x  3 x  108  0
( x  12 )( x  9 )  0
x   12 or 9
x 9
2
three consecutive odd integers = 9,10,11
9. The sum of the squares of two consecutive integers is 113.
Find the integers.
x  ( x  1)  113
2
2
x  x  2 x  1  113
2
2 x  2 x  112  0
2
x  x  56  0
( x  7 )( x  8 )  0
x  7 or  8
2
2
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.
x  ( x  1)  ( x  2 )  302
2
2
2
x  ( x  2 x  1)  ( x  4 x  4 )  302
2
2
2
3 x  6 x  297  0
2
x  2 x  99  0
( x  11 )( x  9 )  0
x   11 or 9
2
If x= -11 three consecutive integers is -11,-10,-9
If x= 9 three consecutive integers is 9,10,11
11. Two integers differ by 8 and the sum of their squares
is 130.
x  ( x  8 )  130
2
2
x  x  16 x  64  130
2
2 x  16 x  66  0
2
x  8 x  33  0
( x  3 )( x  11 )  0
x   3 or 11
2
2
If integers is -3 two integers is -3 and -11
If 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.
x ( x  4 )  192
2
x  4 x  192
2
x  4 x  192  0
( x  16 )( x  12 )  0
x   16
or 12
x  12
Dimensions of the rectangle
= 12 x 16 = 192
12
12  4