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Conjugate Complex numbers

Definition and Conjugate Complex numbers Online Tutoring / Homework Help
DEFINITION OF COMPLEX NUMBER:

If an ordered pair (x, y) of two real numbers x and y is represented by x + iy, where i = √(-1) then the ordered pair (x, y) is known as a complex number or imaginary number.

If the complex number (x, y) is denoted by z then z = (x, y) = x + iy.

When y = 0, then z = (x, 0) = x + i.0 = x i.e., the complex number (x, 0) represents purely real number. Hence, the set of real numbers may be considered as a subset of the set of complex numbers.

Now if x = 0, then z = (0, y) = 0 + i.y = iy that is the complex number z reduces to a purely imaginary number.

If x = 0 and y = 1, then z = (0, 1) = 0 + i.1 = i that is, i represents the unit of a complex quantity. And therefore for this reason, the real number x is called the real part of the complex number z = x + iy and y is called its imaginary part.

Generally two complex numbers (a, b) = a + ib and (c, d) = c + id are defined to be equal if and only if a = c and b = d or in other words when real and imaginary parts of one are equal to the real and imaginary parts of the other respectively.

CONJUGATE COMPLEX NUMBERS:

If two complex numbers are such that they differ only in the sign of their imaginary parts, then one number is said to be the complex conjugate of the other.

The conjugate of a given complex number z is represented by . Hence, if z = x + iy then = x - iy.

For example, conjugate of z1 = 4 + 5i is () = 4 - 5i and conjugate of z1 = 2i is ()= -2i.

Example 1:

If z1 = -3 + 4i and z2 = 12 - 5i, then show that,


Solution:

1) Given that z1 = -3 + 4i and z2 = 12 - 5i

Then z1 + z2 = -3 + 4i + 12 - 5i = 9 - i

Hence = 9 + i                                            (1)

Now ( ) = -3 - 4i and ( ) = 12 + 5i

Hence ( ) + ( ) = -3 - 4i + 12 + 5i = 9 + i                 (2)

Hence from equations (1) and (2) we get,

( )= ( ) + ()                 (Proved)

2) z1 - z2 = -3 + 4i - 12 + 5i = -15 + 9i

Hence () = -15 - 9i                                            (1)

Now ( ) = -3 - 4i and ( ) = 12 + 5i

Hence ( ) - ( ) = -3 - 4i - 12 - 5i = -15 - 9i               (2)

From equations (1) and (2) we get,

() = ( ) - ( )               (Proved)

3) z1 = -3 + 4i and z2 = 12 - 5i

Then z1 z2 = (-3 + 4i)( 12 - 5i)

                  = -36 + 15i + 48i - 20i2

                  = - 36 - 20i2 + 63i

                  = -36 + 20 + 63i (Since i2 = -1)

                  = -16 + 63i

Hence ( ) = -16 - 63i                   (1)

Now ( ) = -3 - 4i and ( ) = 12 + 5i

Hence . = (-3 - 4i)( 12 + 5i)

                           = -36 -15i - 48i - 20i2

                           = -36 -15i - 48i + 20 = -16 - 63i              (2)

From equations (1) and (2) we get,

( ) = () . ( )            (Proved)

4) Given that z1 = -3 + 4i and z2 = 12 - 5i


Now ( ) = -3 - 4i and ( ) = 12 + 5i


From equations (1) and (2) we get,

(Proved)


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