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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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