Question

the complex numbers z and w are graphed. which expression represents z + w?

• $(2 + i) + (3 + i)$
• $(2 - i) + (3 + i)$
• $(2 - i) + (3 + 2i)$
• $(2 + i) + (3 + 2i)$

Explanation:

Step1: Determine coordinates of z and w

In the complex plane, the real part is the x - coordinate and the imaginary part is the y - coordinate (with the imaginary unit \(i\)). For complex number \(z\), looking at the graph, the real part (x - coordinate) is \(2\) and the imaginary part (y - coordinate) is \(- 1\) (since it's below the real axis), so \(z=2 - i\). For complex number \(w\), the real part is \(3\) and the imaginary part is \(2\) (above the real axis), so \(w = 3+2i\).

Step2: Find the expression for \(z + w\)

Substitute the values of \(z\) and \(w\) into the addition of complex numbers. So \(z + w=(2 - i)+(3 + 2i)\).

Answer:

\((2 - i)+(3 + 2i)\) (corresponding to the third option: \((2 - i)+(3 + 2i)\))