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 Rational Numbers
 Irrational Numbers
A rational number is a real number
that can be written as a ratio of two
integers.
A rational number written in decimal
...
-8
1.3333…
- 3/4
16
1/2
3.56
An irrational number is a number that
cannot be written as a ratio of two
integers.
Irrational numbers written as decima...
Square roots of
non-perfect
“squares”
Pi
π
√17
 One of the subsets of rational
numbers
Integers are the whole numbers
and their opposites.
Examples of integers are
i. 6
ii. -12
iii. 0
iv. 186
v. -934
Integers are rational numbers because
they can be written as fraction with 1
as the denominator
Whole numbers one of the subsets of
rational numbers
Whole numbers are rational numbers
because they can be written as fraction
with 1 as the denominator
Examples Of whole n...
Whole numbers are rational numbers
because they can be written as
fraction with 1 as the denominator
Whole Numbers
Operation Numbers Remarks
Addition 0+7=7+0=7,
2+3=3+2=5
For any two whole
numbers a and b,
a+b = b+a
Additi...
Integers
Operation Numbers Remarks
Addition -6+5 = (-1)
-7+(-5) = (-12)
Addition is
commutative
Subtraction 7-5 = 2
5-7= ...
 Rational Numbers
 Addition-
i.
3
8
+(
−5
7
)=
21+(−40)
56
=
(−19)
56
ii.
−3
8
+
(−4)
5
=
−15+(−32)
40
=
−47
40
 Subtra...
 Multiplication-
i.
−7
3
×
6
5
= [
−42
15
] =
6
5
× [
−7
3
]
 So, we find that multiplication is commutative.
 Division...
 Whole numbers-
Operation Numbers Remarks
Addition 7+(2+5)=(7+2)+5
LHS= 7+(7)=14
RHS= (9)+5=14
So, we find addition is as...
Integers
Operation Numbers Remarks
Addition (-2)+[3+(-4)]=[(-2)+3]+(-4)
LHS=(-2)+(-1)=(-3)
RHS=1+(-4)=(-3)
Addition is as...
 Rational numbers-
 Addition-

−2
3
+
3
5
+
−5
6
=
−2
3
−7
30
=
−27
30
=
−9
10
 [
−2
3
+
3
5
] +
−5
6
=
−1
15
+ [
−5
6...
 Division-

1
2
÷
−1
3
÷
2
5
=
1
2
÷
−1
2
÷
2
5
 LHS=
1
2
÷
−1
3
÷
2
5
=
1
2
÷
−1
3
×
5
2
=
1
2
÷
−1
3
÷
2
5
=
1
2
÷
−1...
 Multiplication-

−7
3
×
5
4
×
2
9
=
−7
3
×
10
36
=
−70
108
=
−35
54
 (
−3
7
×
5
4
) ×
2
9
=
−30
244
 Multiplication i...
Numbers- Whole, Rational, Integers
Numbers- Whole, Rational, Integers
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Numbers- Whole, Rational, Integers

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Based On class VIII NCERT Syllabus

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Numbers- Whole, Rational, Integers

  1. 1.  Rational Numbers  Irrational Numbers
  2. 2. A rational number is a real number that can be written as a ratio of two integers. A rational number written in decimal form is terminating or repeating.
  3. 3. -8 1.3333… - 3/4 16 1/2 3.56
  4. 4. An irrational number is a number that cannot be written as a ratio of two integers. Irrational numbers written as decimals are non-terminating and non-repeating.
  5. 5. Square roots of non-perfect “squares” Pi π √17
  6. 6.  One of the subsets of rational numbers
  7. 7. Integers are the whole numbers and their opposites. Examples of integers are i. 6 ii. -12 iii. 0 iv. 186 v. -934
  8. 8. Integers are rational numbers because they can be written as fraction with 1 as the denominator
  9. 9. Whole numbers one of the subsets of rational numbers
  10. 10. Whole numbers are rational numbers because they can be written as fraction with 1 as the denominator Examples Of whole numbers: i. 1 ii. 2 iii. 3
  11. 11. Whole numbers are rational numbers because they can be written as fraction with 1 as the denominator
  12. 12. Whole Numbers Operation Numbers Remarks Addition 0+7=7+0=7, 2+3=3+2=5 For any two whole numbers a and b, a+b = b+a Addition is commutative. Subtraction 4-6 ≠ 6-4 Subtraction is not commutative Multiplication 2×6 = 6×2 = 12, 4×5 = 5×4 = 20 Multiplication is commutative Division 7÷3 ≠ 3÷7 Division is not commutative
  13. 13. Integers Operation Numbers Remarks Addition -6+5 = (-1) -7+(-5) = (-12) Addition is commutative Subtraction 7-5 = 2 5-7= (-2) Subtraction is not commutative Division 5÷8 = 5 8 8÷5= 8 5 Division is not commutative Multiplication 5×8 = 40 (-5) ×(-8)= 40 Multiplication is commutative
  14. 14.  Rational Numbers  Addition- i. 3 8 +( −5 7 )= 21+(−40) 56 = (−19) 56 ii. −3 8 + (−4) 5 = −15+(−32) 40 = −47 40  Subtraction- i. −5 7 − 2 3 = −5×3−2×7 21 = −29 21 ii. 2 3 − 5 4 ≠ 5 4 − 2 3  We find that subtraction is not commutative.
  15. 15.  Multiplication- i. −7 3 × 6 5 = [ −42 15 ] = 6 5 × [ −7 3 ]  So, we find that multiplication is commutative.  Division- i. −5 4 ÷ 3 7 = 3 7 ÷ −7 3 ii. −5 4 ÷ 3 7 = −5×7 4×3 = −35 12 = 3 7 ÷ −5 4 3 × 4 7 × −5 = 12 −35  So, we find that division is not commutative.
  16. 16.  Whole numbers- Operation Numbers Remarks Addition 7+(2+5)=(7+2)+5 LHS= 7+(7)=14 RHS= (9)+5=14 So, we find addition is associative with whole numbers. Subtraction 8-(9-11)=11-(9-8) LHS= 8-2=6 RHS= 11-1=10 So, we find subtraction is associative with whole numbers. Division 3÷(2÷9)≠ 9÷(3÷2) LHS= 3÷[ 2 1 × 1 9 ]= 27 2 RHS= 9÷[ 3 1 × 1 2 ]= 18 3 So, we find division is associative with whole numbers. Multiplication 7×(2×5)=(7×2) ×5 LHS= 7×10=70 RHS= 14×5=70 So, we find multiplication is associative with whole numbers.
  17. 17. Integers Operation Numbers Remarks Addition (-2)+[3+(-4)]=[(-2)+3]+(-4) LHS=(-2)+(-1)=(-3) RHS=1+(-4)=(-3) Addition is associative. Subtraction 5-(7-3)= (5-7)-3 LHS= 5-10= (-5) RHS= (-2)-3= (-5) Subtraction is not associative. Division [(-10)÷2]÷(-5)=(-10)÷[2÷(-5)] LHS=(-5)÷(-5)= (-1) RHS= (-10)÷ 2 −5 =(-25) Division is not associative. Multiplication 5×[(-7) ×(-8)]=[5((-7)]×(-8) LHS=5×56=280 RHS=(-35)×(-8)=280 Multiplication is associative.
  18. 18.  Rational numbers-  Addition-  −2 3 + 3 5 + −5 6 = −2 3 −7 30 = −27 30 = −9 10  [ −2 3 + 3 5 ] + −5 6 = −1 15 + [ −5 6 ] = −27 30 = −9 10  Addition is associative with rational numbers.  Subtraction-  −2 3 − −4 5 − 1 2 = 2 3 − −4 5 − 1 2  [ −2 3 − ( −4 5 )] − 1 2 = 22 15 − 1 2 = 29 30  −12 15 − 1 2 = −29 30 ≠ 29 30  Subtraction is associative with rational numbers.
  19. 19.  Division-  1 2 ÷ −1 3 ÷ 2 5 = 1 2 ÷ −1 2 ÷ 2 5  LHS= 1 2 ÷ −1 3 ÷ 2 5 = 1 2 ÷ −1 3 × 5 2 = 1 2 ÷ −1 3 ÷ 2 5 = 1 2 ÷ −1 3 × 5 2 = 1 2 ÷ −5 6 = −6 10  Division is associative with rational number.
  20. 20.  Multiplication-  −7 3 × 5 4 × 2 9 = −7 3 × 10 36 = −70 108 = −35 54  ( −3 7 × 5 4 ) × 2 9 = −30 244  Multiplication is associative with rational numbers.

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