Real numbers are either rational or
irrational. A rational number is a ratio or quotient of two integers. Rational numbers can be
represented as integers, fractions, terminating decimals and recurring or repeating decimals.
Irrational numbers are the numbers that
cannot be expressed as the ratio of two integers. An irrational number expressed as a decimal never repeat or terminate.
The
real number system is the set of all decimal
numbers. All rational and all irrational numbers
are real numbers. The real numbers are all
numbers on the number line.
R = Q U I,
R
is the set of real numbers.
:: The absolute value of a real number a is its distance from the
origin
Properties of absolute value
Examples
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
:: Properties
of real numbers
Properties
Examples
1
a + b
= b + a
1
−1 +
4 = 4 +
(−1)
= 3
2
(a + b)
+ c
= a + (b + c)
2
(−3
+ 2) + 7 =
−3 + (2 + 7) =
6
3
a +
0 = a
3
−
8
+ 0 =
− 8
4
a + (−
a) =
0
4
9
+ (- 9)
=
0
5
a
· (b + c) =
a ·
b + a
· c
5
−3
·
(−1 + 7) =
−3 ·
(−1) + (−3) ·
7 = −
18
6
a
· b =
b · a
6
(−
4)
·
2
=
2 ·
(−
4)
=
− 8
7
a + (−
b) =
a −
b
7
9
+ (- 4)
=
9 -
4 =
5
8
−
(a + b) =
− a
− b
8
−
(−3 +
2) =
− (−3) − 2 =
3 − 2 =
1
9
b
− a =
− (a
−
b)
9
5
- 7
= - (7
-
5) =
- 2
10
a
· (−
b) =
− a· b
10
3
·
(-
5) =
-3
·
5 =
-15
11
(−
a)· (−
b) =
a ·
b
11
(-
3) ·
(-
6) = 3 ·
6 =
18
12
−
(−
a) =
a
12
-
(-
7) =
7
13
a
· 0 =
0
13
(-11)
·
0 =
0
14
14
15
a
· 1 =
1· a
= a
15
-
5 ·
1 = 1 ·
(-
5) =
-
5
16
(−1)
· a
= −
a
16
(-1) · 4 = -
4
17
17
18
18
19
19
20
20
21
21
22
22
23
23
24
24
25
25
26
26
27
27
28
if
28
if
29
if
29
if
:: Decimal fractions
are vulgar fractions whose denominator is a power
of ten.
Examples:
:: Recurring
decimal to fraction
conversion
Rational number which cannot be expressed as a
decimal fraction converts to recurring
decimal.
The vulgar fractions in lowest terms whose prime factors in the
denominator are other than 2 or 5 convert to the purely recurring
decimals, while those fractions whose denominator is a product of
2's and/or 5's besides the prime numbers convert to the mixed
recurring decimals.
Examples:
Number of 9’s in the denominator equals the number of digits of the period
while 0’s equals to the number of digits in the non-recurring pattern.
