CHAP.TWO: SETS OF NUMBERS
II.2.1. SETS OF REAL NATURAL NUMBERS (N)
The set of real natural numbers N is the set of counting numbers which do not
include zero. They also form and infinite set,
extending in the positive direction.
N  1,2,3,4...orN  1,2,3,...  
OPERATIONS IN THE SET OF NATURAL NEMBERS
I.
Addition:
x, y  N
• X 1  X 
• X  Y   ( X  Y )
Laws over addition:
X , Y , Z  N ,
•(X+Y) +Z=X+(Y+Z) Addition of any three natural numbers is associative i.e order
of addition does not matter
•X+Y=Y+X Addition of any two natural numbers is commutative
II.
Multiplication
X , Y , Z  N ,
• (XY) Z = X (YZ) Multiplication of any three natural numbers is associative
•x y=y x Multiplication of any two natural numbers is commutative.
Note:
i.
Addition and Multiplication are connected by the distribution property
as follows:
X , Y , Z  N ,
Z (X+Y) = ZX+ZY, The multiplication is distributive over addition.
ii.
Order relation is compatible with Addition and Multiplication as follows:
a, b, c  N ,
.a  b  a  c  b  c,
.a  b  ac  bc
Note:
i.
ii.
The substraction of a-b =x is possible if a ≥ b, and the result x is called
the difference between a and b. If a<b, the substraction in the set of
natural numbers is impossible.
The division of a and b is an operation between natural numbers a and b
and associated with natural numbers q and r such that:
A= bq+r, where r < b if r=0, a= bq and b is a multiple of a or is a divisor of a.
II.2.2.THE SET THE PRIME INTEGERS (Z)
This is a set of positive and negative whole numbers. This set was introduced so
as to solve equations like x+1=0, x=-1. This is because the equation x+1=0 cannot
be solved in the set of the real natural numbers N. Therefore,
Z  ...,2,1,0,1,2,3,...
Operation in the set of Z
1. Addition in Z
x, y, z  Z ,
•X+YϵZ Addition of prime integers is closed.
•0+x=x+0, 0 is the identity element under addition i.e does not change the result
•X+Y=Y+X=0, X and Y are the opposite numbers. All the prime integers have their
opposite, what was not found in the set of natural numbers.
Laws under addition of prime integers.
X+Y=Y+X Addition of any two prime integers is commutative
X+ (Y+Z) = (X+Y) Z Addition of any three prime integers is associative.
2. Multiplication in the set of Z
x, y, z  Z
•XY=YZ Multiplication of any two prime integers is commutative
•X (YZ) = (XY) Z Multiplication of any three prime integers is associative
•1x=x1=x, 1 is the identity element under multiplication.
The set of integers is ordered under the relation “≤”, and the sets N and Z are
isomorph.
Note: All the properties of Natural numbers holds for the prime integers, but for
the prime numbers, it has an important one which is the fact that all integers
have opposite integers; which give the limitation of substraction of natural
numbers.
II.2.3. THE SET OF RATIONAL NUMBERS (Q)
This is the set of numbers in the form of
a
, a, b  Z , b  0 .The division of two
b
integers is possible only if b is a multiple of a, in other cases there is no integer
a
b
which can verify the equation a  bx, x  (b  0)
e.g: 3x+5=0 has no root in the set of integers, but 3x=6 has 2 as a root of the
a
b
equation. From the equation orab 1 . All the integers are also rational numbers as
they can be written in the form of
a
b
4
1
e.g: 4= 
8 16
 , etc the set of Rational numbers Q has the same properties of
2 4
Addition, Multiplication as the integers (Z). These operations again have the same
properties of commutativity, associativity and distributivity of Multiplication over
Addition as the integers.
INVERSE OF RATIONAL NUMBERS
X  Q, and , X  O, X 1 , Is the inverse of X.
1
3
1
3
e.g: The inverse of 3 is , as 3x =1 and 1 is the identity element under
Multiplication of rational numbers.
Note: Every rational number can be described as either terminating or nonterminating but repeating (or recurring decimals).
a. Terminating decimals.
1
 0.1  10%
10
1
 0.25  25%
4
1
 0.125  12.5%
8
b. Repeating or recurring decimals
e.g: 0.48484848…
5
 0.454545...  0.45, The period is 45 repeats. Algebra can be used to find
11
rational number in fractional form
e.g: Convert the following recurring decimal number in the form of fraction:
1) 0.48484848…
Solution:
Let x =0.484848… (eqn 1)
100x =48.484848… (eqn 2)
(Eqn 2)- (Eqn1), we get
99x=48
X=
48 16
or
99 33
2) 0.16  0.16666...
Let x=0.1666…
10x =1.666… (Isolate the non-recurring decimal number)… (eqn1)
100x =16.666… (eqn2)
(eqn2) – eqn1), we get
90x =15
X=
15
1
or
90 16
EXERCISES.
Express each of the following as a rational number or in the fractional form:
1.
2.
3.
4.
5.
0.363
0.04
 2.39
0.3 6
 1.046
II.2.4. THE SET OF IRRATIONAL NUMBERS (I)
This is the set of numbers which cannot be expressed in the form of
a
, b  0, a, b  Z or the numbers which are not terminating nor repeating.
b
e.g:
2 , 3 , 5 ,…
Operations with radicals
1. Addition or Substraction of radicals.
To Add or Substract like radicals, we base the operation on the distributive
property of numbers
e.g:1) 3 2 +5 2 = (3+5) 2
=8 2
2) 3 3 -2 3 +4 3 = 3 2 +4 2 +5 3 -2 3
=7 2 +3 3
2. Multiplication of radicals
If a, b ≥ 0, then a x b = ab or ab = a x b
e.g; 18 = 9x2
= 9x 2
=3 2
EXERCISSES
1.
2.
3.
4.
5
5
7
2
8 - 27 + 50 -2 12
3 -2 7 +3 3 -4 3
3 -6 75
3 (3 6 -2 5 )
3. Dividing radicals
In order to simplify radical expression, very often, it is useful to have no radicals in
the denominator, that process of obtaining a non-radical denominator is called
rationalizing the denominator.
e.g: 2
√2+3√3
3√6
=
(2√2+3√3)×√6
3√6×√6
1
2
9
= +√3 + √2
2
If the denominator is the sum or the difference of two terms (two radicals or a
radical and any non-radical) we multiply both numerator and the denominator by
the conjugate both the denominator
2−3√3
e.g:
√3+√2
=
=
(2−3√2)(√33−√2)
(√3+√2)(√3−√2)
(2−3√2)(√3−√2)
1
Laws of addition, substraction, multiplication and division of rationals are also
applied to Irrationals, with particular cases of irrationals where ab =√𝑎 × √𝑏
√𝑎
and
√𝑏
𝑎
= √ , b > 0.
𝑏
II.2.5.THE SET OF REAL NUMBERS (R)
The set of the real numbers (R) is the union of rationals (Q) and Irrationals (I), i.e.
the real numbers (R) consist of all rational numbers and all the Irrational
numbers(I).   Q  I
Therefore, from the diagram above the real numbers (R) consist of all rational
numbers and all the Irrational numbers (I).