# Addition, subtraction and multiplication of complex numbers

In order to add and subtract complex numbers, you just add and subtract the real and imaginary parts separately. That is \begin{align*} (a+bi)+(c+di) &= (a+c)+(b+d)i \\ (a+bi)-(c+di) &= (a-c) + (b-d)i. \end{align*}

Worked exercise: Add the complex numbers $$3+5i$$ and $$2-3i$$.
Solution: $$(3+5i)+(2-3i) = (3+2)+(5-3)i = 5 +2i.$$

You multiply complex numbers just like you would multiply binomials, remembering that $$i^2 = -1$$.

Worked exercise: Evaluate $$(1-i)(2-3i)$$.
Solution: Expanding like you would with binomials, and applying $$i^2 = -1$$, we have\begin{align*} (1-i)(2-3i) &= 2-3i-2i+3i^2 \\ &= (2-3)-(3+2)i \\ &= -1-5i. \end{align*}

## Problems

1. Add the following complex numbers.
1. $$14 + 6i$$ and $$-1 – i$$
2. $$2+6i$$ and $$1-i$$
3. $$4-i$$ and $$-5-9i$$
4. $$3+2i$$ and $$6i$$
2. Subtract the second complex number from the first complex number.
1. $$5i$$ and $$2-i$$
2. $$1-i$$ and $$9+7i$$
3. $$-7+4i$$ and $$2+5i$$
4. $$i$$ and $$-i$$
3. Evaluate the following.
1. $$(4+i)(-2-3i)$$
2. $$(-2+4i)(8-i)$$
3. $$(8-6i)(-1-2i)$$
4. $$(5+9i)(4-10i)$$