**NUMBERS**

**Natural numbers**

The
set of counting numbers 1, 2, 3, 4, …………are called as the natural numbers,
denoted by N.

N
= {1, 2, 3, ...}

**Whole numbers**

The
set of natural numbers together with the number ‘0’ is known as the set of
whole numbers, denoted by W.

W=
{0, 1,2,3, ...)

**Integers**

The
negative whole numbers, the number '0' and the natural numbers together form
the set of Integers, denoted by Z.

Z=
{... -3, -2,– 1, 0, 1.2.3, …..}

**Rational numbers**

Numbers
which can be written in the form p/q where p and q are integers and q ≠0 are
called as rational numbers, denoted by Q.

**Decimal representation of rational numbers**

v If
a fraction (rational number) in its lowest terms has no other prime factors
except 2 and 5. we get a terminating decimal.

**Irrational numbers**

The
numbers, which cannot be expressed as rational numbers is called as an
irrational number.

**Note:**

Every
non-terminating, non-recurring decimal is an irrational number.

**Note :**

A
number may he a rational number or an irrational number, but! it cannot be
both.

**Real numbers**

The
set of all numbers comprising rational numbers and irrational numbers is known
as the set of real numbers, denoted by R.

**Note :**

v On
the number line, there is a point corresponding to every real number and to
every point on the number line, there is a real number. Hence the number line
is called the Real Number line.

v Real
numbers are so called because they can be seen as points representing them on
the number line.

**Complex numbers**

There
is no real number, whose square is a negative number.

E.g.
is not a real number.

Such
numbers are called as imaginary numbers.

The
set of numbers comprising of real numbers and imaginary numbers is known as the
set of complex numbers, denoted by C.

The
ordered pair (a, b) where a and b are real numbers, when expressed in the form
a + ib, is called a complex number.

a is called the real part, and b
is called the imaginary part.

**Note:**

v Every
real number a can be represented as a complex number, (a, 0).

v The
complex numbers a + ib and a - ib are called conjugate complex numbers. Each is
called the conjugate of the other.

v The
sum and product of two conjugate complex numbers are real.

v Every
complex number can be represented as a point in the coordinate plane, by taking
real part on the x-axis and the imaginary part on the y-axis.

Hence, x axis is called real axis

y-axis
called imaginary axis.

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