## About "Equality of complex numbers"

Equality of complex numbers :

Two complex numbers are equal when their real parts are equal and their imaginary parts are equal.

a + bi = c + di , a = c and b = d

Let us look into some example problems based on equality of complex numbers.

Example 1 :

Find real numbers x and y such that

(x + yi) (2 - i) = ¡

Solution :

(x + yi) (2 - i) = ¡

By multiplying two complex numbers on the left side, we get

2x - ix + 2iy - i2y = ¡

Here the value of i2 is -1

(2x + y) + i(2y - x)  =  i

By equating the real and imaginary parts

 2x + y  =  0  ----(1)y  =  -2x 2y - x  =  1  ----(2)

Applying the value of y in the second equation

 2(-2x) - x  =  1-4x - x  =  1-5x  =  1x  =  -1/5 y  =  -2(-1/5)y  =  2/5

Example 2 :

Find real numbers x and y such that

(3 + 2i) (x + yi)  =  -i

Solution :

(3 + 2i) (x + yi)  =  -i

3x + 3iy + 2ix - 2yi2  =  -i

3x + 3iy + 2ix - 2y(-1)  =  -i

(3x + 2y) + i(2x + 3y)  =  -i

Equating the real and imaginary parts.

 3x + 2y  =  0 ----(1)2y  =  -3xy  =  -3x/2 2x + 3y  =  -1 ----(2)

Applying the value of y in the second equation

2x + 3(-3x/2)  =  -1

2x - (9x/2)  =  -1

(4x - 9x)/2  =  -1

Multiply both sides by 2

-5x  =  -2

Divide -5 on both sides

x  =  2/5

y =  -3(2/5)/2

=  -(6/5) ⋅ (1/2)

y  =  -3/5

Hence the value of x  =  2/5 and y  =  -3/5.

Example 3 :

Find real numbers x and y such that

(x + 2i) (y - i)  =  -4 - 7i

Solution :

(x + 2i) (y - i)  =  -4 - 7i

xy - ix + i2y - 2i2  =  -4 - 7i

xy - ix + i2y - 2(-1)  =  -4 - 7i

(xy + 2)- i(x - 2y)  =  -4 - 7i

 xy + 2  =  -4 -----(1) - (x - 2y)  =  -7x - 2y  =  7x  =  7 + 2y -----(2)

Applying the value of x in the first equation, we get

(7 + 2y) y +  2  =  -4

7y + 2y2 + 2  =  -4

2y2 + 7y + 2 + 4  =  0

2y+ 7y + 6  =  0

2y+ 4y + 3y + 6  =  0

2y(y + 2) + 3(y + 2)  =  0

(2y + 3) (y + 2)  =  0

 2y + 3  =  0 2y  =  -3y  =  -3/2x  =  7 + 2(-3/2)x  =  7 - 3x  =  4 y + 2  =  0y  =  -2x  =  7 + 2(-2)x  =  7 - 4x  =  3

Hence the solutions are

x  =  4 and x  =  3

y  =  -3/2 and y  = -2

After having gone through the stuff given above, we hope that the students would have understood "Equality of complex numbers".

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