Adding, subtracting and multiplying

What you will be able to do

• add and take away complex numbers;
• multiply complex numbers, using $i^2 = -1$;
• use the properties of the operations, and the algebraic identities, with complex numbers.

Adding and taking away

Treat $i$ like a letter in algebra. Collect the real parts, and collect the imaginary parts:

Example. For $z_1 = 5 + 2i$ and $z_2 = 1 - 6i$:

• $z_1 + z_2 = (5 + 1) + (2 - 6)i = 6 - 4i$;
• $z_1 - z_2 = (5 - 1) + (2 + 6)i = 4 + 8i$.

The sum of two complex numbers is again a complex number. This is the closure law. Addition is also commutative, $z_1 + z_2 = z_2 + z_1$, and associative, $(z_1 + z_2) + z_3 = z_1 + (z_2 + z_3)$. The number $0 = 0 + 0i$ is the additive identity. The additive inverse of $a + ib$ is $-a - ib$, because the two add up to $0$.

Multiplying

Multiply out the brackets as you would in algebra. Then replace $i^2$ with $-1$:

Example. $(3 + 4i)(2 - 5i) = 6 - 15i + 8i - 20i^2 = 6 - 7i + 20 = 26 - 7i$.

Watch the last step. The term $-20i^2$ becomes $+20$, and it joins the real part.

Multiplication has the same good properties. It is closed, commutative and associative. The number $1 = 1 + 0i$ is the multiplicative identity. The distributive law holds: $z_1(z_2 + z_3) = z_1z_2 + z_1z_3$.

The identities still hold

Because the laws above hold, every algebraic identity you know still works for complex numbers. For example:

• $(z_1 + z_2)^2 = z_1^2 + 2z_1z_2 + z_2^2$
• $(z_1 + z_2)(z_1 - z_2) = z_1^2 - z_2^2$

Example. $(3 + 2i)^2 = 3^2 + 2 \times 3 \times 2i + (2i)^2 = 9 + 12i - 4 = 5 + 12i$.

One identity is new and very useful. For any real $x$:

So $x^2 + 1$, which cannot be factorised with real numbers, does factorise with complex numbers.

A worked example

Simplify $(1 + i)^3$.

1. Square first. $(1 + i)^2 = 1 + 2i + i^2 = 1 + 2i - 1 = 2i$.
2. Multiply by $1 + i$ once more. $(1 + i)^3 = 2i(1 + i) = 2i + 2i^2 = 2i - 2$.
3. Write it as $a + ib$. $(1 + i)^3 = -2 + 2i$.

Check with Python: (1 + 1j) ** 3 gives (-2+2j).

Practise with new numbers

Try it in Python

Python knows the rules for complex numbers, so it can check hand work. The program adds, takes away and multiplies two numbers. Then it uses the rule for a product, with the real and imaginary parts taken out by .real and .imag.

Common mistakes

• Forgetting that $i^2 = -1$. $(2i)(3i) = 6i^2 = -6$. Do not leave $i^2$ in an answer, and do not write $+6$.
• Adding real and imaginary parts together. $3 + 4i$ is not $7i$ and not $7$. The two parts never mix in a sum.
• Losing a sign in a difference. $(2 + i) - (5 - 3i) = -3 + 4i$. The minus sign changes $-3i$ into $+3i$.
• Writing i in Python. Python reads i as a variable name, so 3 + 4i stops with SyntaxError. Write 3 + 4j.

Remember these

• Add and take away part by part.
• $(a + ib)(c + id) = (ac - bd) + i(ad + bc)$.
• Addition and multiplication are closed, commutative and associative, with identities $0$ and $1$. The distributive law holds.
• All the algebraic identities hold, and $x^2 + 1 = (x + i)(x - i)$.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. Find $(7 - 2i) - (3 + 5i) + (-1 + 4i)$.

2. Multiply $(4 + i)(2 - 3i)$.

3. What is the additive inverse of $-6 + 9i$?

4. Use an identity to find $(5 + 2i)(5 - 2i)$.

5. Factorise $x^2 + 16$ with complex numbers.