Let Z and W be complex numbers with Z=1-2i and W=3+i

Question

a) find the Value of 2z -3w?

b) write in the form a+bi where a and b are real numbers, i(w-z)/w*?

(w* is the complex conjugate of w)

Answer 1

Hi Ali!

Complex numbers are difficult to wrap your head around, but they often work just like the real numbers and variables that you're used to!

For part a) we'll substitute the values they gave us for z and w into the expression: 2z - 3w

2z - 3w = 2(1 - 2i ) - 3(3 + i )

Then we'll distribute the 2 and 3 outside the parenthesis and combine like-terms:

2(1 - 2i ) - 3(3 + i ) = 2 - 4i - 9 - 3i = -7 - 7i

And there's your answer! 2z - 3w = -7 - 7i

b) i(w-z)/(w*)

We'll start by substituting like we did before, keeping in mind that " *" denotes the complex conjugate, which simply means that the sign of the "imaginary part" is flipped (negative to positive, or vice versa). So in this case: w* = 3 - i

Substituting, we get:

i (3 + i - (1 - 2i )) / (3 - i ) , which looks like (and is) a mess - but let's start by simplifying what's in the parentheses.

Distributing the negative sign and combining like terms in the numerator:

= i (3 + i - (1 - 2i )) / (3 - i )

= i (3 + i - 1 + 2i )) / (3 - i )

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

Then, we'll distribute the i in the numerator. This is a good time to remember that i is just the sqrt( -1), so

i * i = -1

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

= (2i + 3(i *i )) / (3 - i )

= (2i + 3(i *i )) / (3 - i )

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

Where I rearranged the numerator so it would be in the custom (a + bi) form.

Now in order to simplify the expression so that we don't have a complex denominator, we'll multiply by a special "1" : The complex conjugate of the denominator, over itself.

You'll notice that when a complex number is multiplied by its complex conjugate, a real number is returned:

ww* = (3 - i )(3 + i ) = 9 + 3i - 3i - (i ²)** = 9 - (-1) = 10

**In order to avoid mistakes with all of the negative signs that come with multiplying i 's, I separate the step of simplifying i ² from distributing any negative signs nearby.

This method works because the middle terms will always cancel out due to the opposite signs. Now, in order to not change the value of the expression, we have to multiply the numerator by the same complex number we used to simplify the denominator, so that overall it looks like this:

= [( -3 + 2i ) / (3 - i )] * [(3 + i ) / (3 + i )]

= [( -3 + 2i )(3 + i )] / [(3 - i )(3 + i )]

= [( -3 + 2i )(3 + i )] / 10 *The 10 coming from the step shown above*

= (1/10)(-9 - 3i + 6i + 2 i ²) *Here I moved the 10 out of the denominator to simplify*

= (1/10)(-9 + 3i + 2( -1))

=(1/10)(-11 + 3i )

= -1.1 + 0.3i *And that's our answer!*