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Home » Math Homework Help » Algebra Homework Help » Complex Numbers
Complex Numbers
A complex number is a number which can be written in the form a + ib, where a and b are real numbers and i = . It is generally denoted by a single letter z. The real number a  is called the real part of the complex number z = a + ib and the real number b is called the imaginary part of z. Thus +2i, 2 – i, 10 + 0i, 0 + 5i, 1/5 + , 7 + (-1)1/2, + 5 are all complex numbers.

Alternatively, a complex number is also defined as ordered pair (a, b) of real numbers. Thus if z is a complex number, then

z = a + ib = (a, b), for some a, b R.

Set of complex numbers: The set of complex numbers is denoted by C, where

C = {z | z = a + ib, ∀ a, b R}

Or, C = {z | z = (a, b), ∀ a, b R}

In other words, the cartesian product R × R consisting of the ordered pairs of real numbers is called the set of complex numbers.

Zero complex numbers: The complex number z = a + ib, a, b R is said to be zero complex number or zero of C if and only if a = 0 and b = 0.

Negative of a complex number: The complex number,

– z =a – ib

is called the negative of the complex number z = a + ib and vice versa.

Equality of two complex numbers: Two complex numbers are said to be equal of an only if there real and imaginary parts are separately equal, i.e.

a + ib = c + id a = c and b = d.

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