# Complex numbersFind the absolute values of the complex number z = (2-3i)/(-2-4i)

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

To calculate the modulus of the complex number z, we'll re-write the given complex number z.

Since it is not allowed to keep a complex number to denominator, we'll multiply the ratio by the conjugate of the denominator, -2 + 4i:

z = (2 - 3i)(-2 + 4i)/(-2 - 4i)(-2 + 4i)

We've obtained to denominator the difference of squares a^2 - b^2, where a = -2 and b = 4i => a^2 = 4 and b^2 = 16i^2.

But i^2  =-1 => b^2 = -16

a^2 - b^2 = 4 - (-16) = 4+16 = 20

z = (2 - 3i)(-2 + 4i)/(20)

We'll calculate the numerator:

(2 - 3i)(-2 + 4i) = -4 + 8i + 6i + 12

We'll combine real parts and imaginary parts:

(2 - 3i)(-2 + 4i) = 8 + 14i

z = 2(4 + 7i)/20

z = (4 + 7i)/10

The modulus of z is:

|z| = sqrt (x^2 + y^2)

We'll identify x and y from the numbar z:

z = 4/10 + 7i/10

z = 2/5 + 7i/10

x = 2/5 and y = 7/10

|z| = sqrt (16/100 + 49/100)

|z| = sqrt (65/100)

|z| = [sqrt (65)]/10