How do you divide #10/(10+5i)#?

Question

Sadie Alexander · Accepted Answer

#= 4/5 -2/5 i#

you gotta multiply the denominator by its conjugate to make it real...and of course do the same to the denominator, with the result that the numerator will have the real and imaginary parts

BACKGROUND:

the conjugate of #z = x + i y# is #tilde z = x - i y#

and #z * tilde z = x^2 + y^2 = abs(z)^2#

so here we have

#10/(10+5i) * (10-5i)/(10-5i)#

#= 2/(2+i) * (2-i)/(2-i)# .....to make numbers a bit lighter

#= (2 (2-i))/(2^2+ 1^2)#

#= 2/5* (2-i)#

#= 4/5 -2/5 i#

Serenity Johnson · Answer

undefined To divide 10 by (10 + 5i), you first need to rationalize the denominator. To do this, multiply both the numerator and the denominator by the conjugate of the denominator, which is 10 - 5i. This will eliminate the imaginary component from the denominator.

10/(10 + 5i) * (10 - 5i) / (10 - 5i) = (10 * (10 - 5i)) / (10 * (10 - 5i) + 5i * (10 - 5i))

Now, multiply out the numerators and denominators:

Numerator: 10 * (10 - 5i) = 100 - 50i
Denominator: (10 * (10 - 5i)) + (5i * (10 - 5i)) = 100 - 50i + 50i + 25 = 125

So, the final result is:

(100 - 50i) / 125

Which simplifies to:

(20 - 10i) / 25

Further simplification yields:

20/25 - (10i)/25

So, the division of 10 by (10 + 5i) is:

20/25 - (10/25)i

Which can be simplified to:

4/5 - (2/5)i