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The
number system consists of natural numbers, whole numbers, integers, rational
numbers and irrational numbers. Real
and imaginary numbers also form part of the entire set of numbers.
What
we will study in this chapter :
1.
Natural Numbers
2. Whole Numbers
3. Integers
4. Rational and
Irrational Numbers
5. Real Numbers
1.
Natural
Numbers
The numbers 1,2,3,4,5Ö.etc. are called Natural
numbers.. Each number is equidistant from each other by the unit number 1.
The set of natural numbers can be represented by a set {1,2,3,4Ö.} or by a
number line.
The
results of all operations done on the natural numbers will again be a
natural number. The number line extends to infinity, which is denoted as  .
2.
Whole
Numbers
If we include
ì0î in the set of natural numbers, then the set of numbers is called
whole numbers. The set is shown as {0,1,2,3,4Ö..}. Whole numbers can also
be shown on the number line, with the starting number as zero. All
operations involving whole numbers will again give rise to another whole
number. With the introduction of ì0î, one has to remember that
multiplication of any number with ì0î is always ì0î, division with
ì0î is infinity ().
Division of ì0î with ì0î is an indeterminate quantity.
3.
Integers
The set of integers
consists of all natural numbers, zero and the negative of all natural
numbers. The set of integers can be represented as a set { ÖÖ-4, -3, -2,
-1, 0, 1, 2, 3,,4Ö.} On the number line, the integers can be shown as
given below :
The
number line for the integers extends from -
to + .
Operations of all integers are also integers. With negative integers, one
has to be a bit careful as ñ1 > ñ2,
but with natural numbers 1 < 2.
4. Rational and Irrational Numbers
Rational numbers
form the next higher step for the system of numbers. All integers separate
from each other by a unit number 1. But
there could be numbers less than 1 in between two numbers on the number
line. Such numbers are called fractions.
For
example 1/2 will fall at the exact mid point between 0 and 1 on the number
line.
Similarly
a number ñ 3/2 will be at the exact mid point between ñ1 and ñ2.
Thus
the set of rational numbers includes all integers and all fractions. It has
to be borne in mind that fractions of numbers can have recurring digits, For example 8/3
= 2.6666, which is written as 2.6. Even these numbers can be represented on
the number line.
But
there are other numbers such as  = 22/7,
 2,
3,
5,
7 , etc. which cannot be represented as recurring decimal integers. These
are called irrational numbers. The more significant decimal places that can
be accommodated, the more closer to value of the fraction one is able to
obtain. Since it is practically
not feasible to write all the decimals, the number is truncated to the
required significant decimal place that is needed for calculations.
Irrational numbers are also represented on the number line. All
operations with irrational numbers again belong to the set of rational or
irrational numbers and the number line.
5.
Real Numbers
Real numbers consist of
all rational and irrational numbers. Any
appropriate point on the number line and vice versa gives real numbers. The
set of real numbers thus consists of each and every point on the number
line.
In addition to real
numbers there are imaginary numbers like the square root of negative numbers.
Imaginary numbers also form an entire set, but this set is
represented by a plane, where the X-axis gives the real part of the number
and the Y-axis gives the imaginary part of the number.
Further explanation of imaginary numbers is beyond the scope of this
syllabus.
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