Chapter P.6, Problem 4E

Solution

To write the expression $\left(2+i\right)-\left(9i-3\right)$ in the standard form $a+bi$ .

  $5-8i$

Given:

The expression, $\left(2+i\right)-\left(9i-3\right)$ .

Concept used:

• To add or subtract two complex numbers, simply add/subtract the real parts together and the imaginary parts together. That is,

  $\left(a+bi\right)+\left(c+di\right)=\left(a+c\right)+\left(b+d\right)i$

• The distributive property: $a\left(b+c\right)=a\cdot b+a\cdot c$
• Distributing negative sign inside the parenthesis will change the sign of all terms inside the parenthesis.

Calculation:

In order to add the given complex numbers, first remove the parentheses and distribute the negative sign inside the second parenthesis. Distributing negative sign inside the parenthesis will change the sign of all terms inside the parenthesis.

Then combine the real parts and the imaginary parts together to simplify further using basic algebraic operations as shown below,

  $\begin{array}{c}\left(2+i\right)-\left(9i-3\right)=2+i-9i+3\\ =\left(2+3\right)+\left(i-9i\right)\\ =\left(5\right)+\left(-8i\right)\\ =5-8i\end{array}$

Thus, the given expression can be written in the form of $a+bi$ as $5-8i$ .