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## 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
1. 5 + (-6) = -1 ɛR where 5, -6 ɛR

2. 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:

1. 3, 10 ɛR

3+10 = 10+3

13 = 13

2. -4, 2 ɛR

(-4)∗2 = 2∗(-4)

-8 = -8

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.

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:

1. 6/5 ɛR.

The additive inverse is - 6/5.

6/5+ (6/-5) = (6/-5) + 6/5= 0.

2. -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
1. 1/-5ɛR. Its multiplicative inverse is -5 as

1/-5 ∗ (-5) = (-5) ∗ 1/-5= 1.

2. 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:

1. 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.

2. 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.

3. If x, y ɛR and z ɛR; z ≠0

such that x < y then x + z < y + z.

e.g.

1. 10, 12 ɛR and 5 ɛR.
10 < 12
10 + 5 < 12 + 5
15 < 17

2. 4, 6 and –13 ɛR.
4 < 6
4 - 13 < 6 - 13
-9 < -7

1. 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

2. 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.

1. 5+3 = 3+5
2. Answer: Commutative property

3. 7 ∗(4 ∗6) = (7∗4) ∗6
4. Answer: Associative property of multiplication

5. 5+0 = 0+5

7. 7+9 = 16

9. 1 ∗(-18) = (-18) ∗1= -18
10. Answer: Multiplicative identity

11. 4/3+ (4/-3) = (4/-3) + 4/3= 0

13. 14 ∗9 = 126
14. Answer: Closure property of multiplication

15. 6/17∗17/6= 17/6 ∗6/17=1
16. Answer: Multiplicative inverse

17. (12+9) + 16 = 12 + (9+16)
18. Answer: Associative property of multiplication

19. 14/6∗ 8/9 = 8/9 ∗ 14/7
20. Answer: Commutative property of multiplication