Question

The graph represents the complex numbers z1 and z2. what are their conjugates? what is the conjugate of their product? the conjugate of z1 is ___. the conjugate of z2 is ___the conjugate of the product of z1 and z2 is _____.

Answer 1

Explanation:

Complex numbers in rectangular form are written as z = a + b i

a is the real part and b is the imaginary part.

To graph a complex number we have to draw a vector that starts in the origin and ends at point (a, b).

For z1:

• a1 = 2

,

• b1 = 3

For z2:

• a2 = -5

,

• b2 = -4

These complex numbers are:

• z1 = 2 + 3i

,

• z2 = -5 - 4i

Their conjugates are the same number with opposite sign in the imaginary part

To find the conjugate of the product of z1 and z2 we could multiply the numbers and then find the conjugate or find the conjugate of both numbers and then multiply them. This is because of the product of complex conjugates theorem:

$\bar{(z_1\cdot z_2)}=\bar{z_1}\cdot\bar{z_2}$

Since we've already got the conjugate of each number let's use this theorem:

$\begin{gathered} \bar{(z_1\cdot z_2)}=\bar{z_1}\cdot\bar{z_2}=(2-3i)(-5+4i)=2\cdot(-5)+2\cdot4i-3i\cdot(-5)-3i\cdot4i \\ \bar{(z_1\cdot z_2)}=-10+8i+15i-12i^{2} \\ \text{ since i²=-1} \\ \bar{(z_1\cdot z_2)}=-10+(8+15)i+12 \\ \bar{(z_1\cdot z_2)}=2+23i \end{gathered}$

Answer:

The conjugates of these complex numbers are:

• z1* = 2 - 3i

,

• z2* = -5 + 4i

The conjugate of the product of z1 and z2 is:

• (z1 z2)* = z1* z2* = 2 + 23i