Number System
There are innumerable
number of numbers
between each whole number
Number Line:
-3 -2 -1 0 1 2 3
Natural Numbers : The numbers 1,2 ,3, 4,…. Used for counting are natural
numbers. 0, -7, 13.2, ⅞ are not.
Peano Postulates: PI. 1 N => 1 is a natural number
PII. For any n  N, there is n* (i.e., a number n +1)
PIII. If n, m N, and m* = n*, then m = n
PIV. Any subset S of N is equal to N, if 1 N and mS
Basic laws: (symbol:  = for all or every)
Closure Law:
for m, n N, m + n N
Cumulative Law:
m + n = n + m  m, nN
Associate Law:
m + (n + p) = (m + n) + p m, n, p N
Cancellation Law:
m + p = n + p => m = n m, n, p N
Trichotomy Law: one and only one of the following is possible:
(a) m = n;
(b) m > n; or (c) m < n
Transitive Law: m >n and n >p => m >p
Anti-Symmetric Law: m >n and m >n => m = n
Monotone property Law: m >n => m +p > n+ p and m > n => mp >np for
any m, n, p belonging to N
Number System
Integers: Whole numbers positive, negative or zero. +14, –29, 0
are integers; √7, 0.564, ¾ are not integers.
Odd numbers: 3, 5, 29; Even numbers: 4, 80, 38 etc
Prime Numbers: an integer other than 0 and 1 is a prime if and
only if its divisors (without reminder) are 1 and the number itself.
Ex: 3, 59, 83 etc.
Rational number: a number that can be expressed in the form p/q,
where p and q are any integer and q ≠ 0. Ex: ⅞, 3, 2.5,–¾ etc.
Rational numbers can be expressed in decimal form, both
terminating and non-terminating. Ex: 4/5 (terminating) or, 10/3,
29/ (both non-terminating, either recurring or non-recurring)
7
Irrational numbers: cannot be expressed in the form p/q as the
rational numbers can be. Ex: √3, (2 + √5) etc. √8 is irrational but 3 8
Is not.
All the types of numbers stated above are real numbers. There are
also imaginary numbers. Standard imaginary number is 1 = i and
i2 = –1, i3 = –i i4 =1, i5 = i Explain the concepts of complex
numbers (a + ib) and complex conjugate numbers.
Number System
Illustrations and Problems
ab
ab
For any two real number a and b, there is 2 such that a <
<b
2
3  2i
Solve:
5  3i
3  2i (3  2i )(5  3i )
Solution:
=
5  3i (5  3i )(5  3i )
find the answer, explain
the idea of rationalization.
Problem: Find the square root of 6 + 8 1 .
Solution: 6 + 8 1 = 6 + 8i and let the square root of 6 + 8i = a + ib i.e.,
6  8i = a + ib => 6 + 8i = (a2 – b2) + 2iab, from where,
16
a2 – b2= 6 …(1) and 8i = 2iab or, ab = 4 => b2= 2 ; putting the value of b2
a
16
in (1), a2 – 2 = 6 or, a4 – 6a2 – 16 = 0, solving which we get a2 = 8 and
a
using this in (i), b2 = 2 making a = √8 and b = √2
Therefore, 6  8i =  (√8 + i√2)
Number System: Problems
Which of the following statements are true and which ones are false?
Ans a. F; b. T; c. T; d. F
a. Every real number is a rational number
e. T; f. F; g. T; h. F
b. Every irrational number is a real number
i. T; j. F; k. F; l. F
c. A real number is either rational or irrational
m. T n. T
d. There can be a real number which is both rational and irrational
o. T p. T
e. The product of two rational numbers is rational
f. The sum of two irrational numbers is irrational [(p + √q) + p – √q)]
g. The product of two odd integers is an odd integer
h. For any real number x there is a real number y such that xy = 1
[y =0, xy ≠ 1]
i.
j.
k.
l.
m.
n.
o.
p.
If x is rational and y is irrational, then xy is irrational
If x < y, then x2 < y2
[– 1< 1 but (– 1)2 > 1]
If x > 0, then x2 > x
[x = ½ >0 but ½2 > ½]
Quotient of natural numbers is natural
If a > b and c > o, then ac > bc
If a ≤ b and b ≤ a, then a = b
If a > b then a = b + c, where c is any possible number
If a > 0 and b > 0, then a2 > b2 in all cases. [True if a > b]
Number System: Problems
Which of the following statements are true and which ones are false?
a. If a < b then a  a  b  b
Ans:
b. a  c  ad  bc 2
a. T b. T c. 41 + 13i
b d
c. Multiply 4 – 3i by 3 + 7i
d. −8/29 e. 3 + i
3  2i 3  2i

d. Simplify
2  3i 2  3i
e. Simplify:
9  7i
2  3i