Review of Number Systems and Real Numbers
In earlier grades, you learned about the extension of numbers up to R, the set of real numbers.
You are familiar with counting numbers such as 1, 2, 3, …, etc. This set of numbers is called natural numbers. It is
denoted by N = { 1, 2, 3, …}.
However, since the natural number 1 - 1 = 0 does not exist, the natural number system was extended to include
the number '0'. This new set is called the set of whole numbers. It is denoted by W = { 0, 1, 2, 3, …}.
Next, the set of integers was introduced to make up for the deficiency of negative numbers such as 5 - 8 = -3. This
set is denoted by Z = {…-3, -2, -1, 0, 1, 2, 3, …}.
We found, however, that there is no integer by which you can multiply 3 to get 2. This necessitated the introduction
of new numbers in the form p/q, so that 3 ∗ 2 / 3 = 2. Such numbers are called rational numbers. This set
is denoted by Q
Q = {: p, qɛZ and q ≠0 }
Rational numbers can be represented as decimals.
Rational numbers of the form
7/5= 1.4
6/8 = 0.75
are called terminating decimals.
Rational numbers of the form
8/2 = 0.222..., = 
3/11 = 0.2727…, = 
are called non-terminating repeating decimals.
There is a set of non-terminating, non-repeating decimals of the form
1.232233222333…
12.11123457198….
These cannot be written in the form p/q, and are called irrational numbers.
Also, there is no rational number whose square is 2. So √2 is an irrational number.
The set of irrational numbers is denoted by I.
Rational numbers and irrational numbers taken together form the set of real numbers, or R = Q U I
The set of real numbers R has the following properties.
Closure properties of addition and multiplication
If a, b ɛR then a + b ɛ R
If a, b ɛR then a ∗ b ɛ R
Example
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5 + (-6) = -1 ɛR where 5, -6 ɛR
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3 ∗√7= 3√7 ɛR where 3, √7ɛR
Associative property of addition and multiplication
If a, b, c ɛR then a+(b+c) = (a+b)+c
If a, b, c ɛR then a∗(b∗c) = (a∗b)∗c
Example
1, 2, -5 ɛR
1+(2+(-5)) = (1+2)+(-5)
1+(2-5) = 3+(-5)
1-3 = 3-5
-2 = -2
4, 3, -6 ɛR
4∗(3∗(-6)) = (4∗3)∗(-6)
4∗(-18) = 12∗(-6)
-72 = -72
Commutative property of addition and multiplication
If a, b ɛR then a+b = b+a
If a, b ɛR then a∗b = b∗a
Example:
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3, 10 ɛR
3+10 = 10+3
13 = 13
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-4, 2 ɛR
(-4)∗2 = 2∗(-4)
-8 = -8
Additive Identity
For every a ɛ R there exists a b ɛ R such that
a+b = b+a = a
In this case b = 0.
Example:
13 ɛR
13+0 = 0+13 = 13
Therefore '0' is the additive identity of R.
Additive Inverse
For every a ɛR there exists an a'ɛ R such that
a+a' = a'+a = 0.
We know that a' = -a.
-a is called the additive inverse of a.
Example:
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6/5 ɛR.
The additive inverse is - 6/5.
6/5+ (6/-5) = (6/-5) + 6/5= 0.
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-2 ɛR its additive inverse is 2.
So -2 + 2 = 2 + (-2) = 0.
Multiplicative Identity
For every a ɛR there exists an a1 ɛR such that
a∗a1 = a1∗a = a
Obviously, a1 = 1.
Therefore, 1 is called the multiplicative identity of R.
Multiplicative Inverse
For every a ɛR where a ≠ 0, there exists an a' ɛR
such that a ∗ a' = a' ∗ a = 1.
Obviously a ' = 1/a = (a)-1
So, for every a ɛR, a ≠ 0, 1/a ɛR such that
a ∗ 1/a= 1/a ∗ a = 1.
Therefore 1/a is called the multiplicative inverse of a.
Example
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1/-5ɛR. Its multiplicative inverse is -5 as
1/-5 ∗ (-5) = (-5) ∗ 1/-5= 1.
2 ∗;R then 1/2 ɛR.
2 ∗ 1/2 = 1/2 ∗ 2 = 1
1/2 is the multiplicative inverse of 2.
Distributive property
The two binary operations '+' and '∗' are defined in R as, a, b, c ɛR, then a ∗(b+c) = a ∗b + a ∗c.
Then multiplication is distributive over addition in the set of real numbers.
Example:
14, 6, 9 ɛR
14 ∗(6+9) = 14 ∗6 + 14 ∗9
14 ∗15 = 84 + 126
210 = 210
Additional properties for real numbers
In addition to these properties, the set of real numbers has the following properties:
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If a, b ɛR only one of the following is true.
a) a < b b) a = b c) a > b
This is called the Law of Trichotomy.
e.g. Consider 5, 6 ɛR.
5 < 6 or 6 > 5.
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If a, b, c ɛR and if a > b, b > c then a > c.
This is called Transitive Property.
e.g. Consider 7, 3, 1 ɛR.
7 > 3, 3 > 1 then 7 > 1.
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If x, y ɛR and z ɛR; z ≠0
such that x < y then x + z < y + z.
e.g.
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10, 12 ɛR and 5 ɛR.
10 < 12
10 + 5 < 12 + 5
15 < 17
4, 6 and –13 ɛR.
4 < 6
4 - 13 < 6 - 13
-9 < -7
If x, y, z ɛR, z is a positive real number such that x y
Then x ∗ z < y ∗ z
e.g. 1, 9, 6 ɛR.
1 < 9
1 ∗ 6 < 9 ∗ 6
6 < 54
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If however x, y, z ɛR and z is a negative number then
If x < y
x ∗ z > y ∗ z
e.g. 2,11 ∗;R, -6 ɛR.
2 < 11
2 ∗ (-6) > 11 ∗ (-6)
-12 > -66
Absolute values of real numbers
Recall that in grade-8 you learned about absolute values. The absolute value of a number x is given by
| a | = a if a
The absolute value is always non-negative.
Example 1
|-15|= -(-15) -15 < 0
Example 2
|12/7| = 12/7 12/7 > 0
Try these questions
Name the property used in the statement to make it true.
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5+3 = 3+5
Answer: Commutative property
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7 ∗(4 ∗6) = (7∗4) ∗6
Answer: Associative property of multiplication
5+0 = 0+5
Answer: Additive identity
-
7+9 = 16
Answer: Closure property of addition
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1 ∗(-18) = (-18) ∗1= -18
Answer: Multiplicative identity
.
4/3+ (4/-3) = (4/-3) + 4/3= 0
Answer: Additive inverse
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14 ∗9 = 126
Answer: Closure property of multiplication
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6/17∗17/6= 17/6 ∗6/17=1
Answer: Multiplicative inverse
(12+9) + 16 = 12 + (9+16)
Answer: Associative property of multiplication
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14/6∗ 8/9 = 8/9 ∗ 14/7
Answer: Commutative property of multiplication