Complex Numbers and Their Algebra: Quick Revision

A brief note on Complex Numbers and Their Algebra. Read it before a test. Try each question at the end before you open its answer.

The number i

• No real number squares to a negative number. So we bring in $i$, with $i^2 = -1$.
• For $a > 0$, $\sqrt{-a} = \sqrt{a}\, i$. So $\sqrt{-25} = 5i$.
• A complex number is $z = a + ib$: the real part is $a$, the imaginary part is $b$. Both are real numbers.
• $a + ib = c + id$ means $a = c$ and $b = d$: one equation gives two.
• Complex numbers cannot be put in order. In Python, $i$ is written 1j.

Adding, subtracting and multiplying

• Add and take away part by part:
  • $(a + ib) \pm (c + id) = (a \pm c) + i(b \pm d)$
• Multiply out and put $-1$ for $i^2$:
  • $(a + ib)(c + id) = (ac - bd) + i(ad + bc)$
• The closure, commutative, associative and distributive laws hold. The identities are $0$ and $1$.
• Every algebraic identity still works, and $x^2 + 1 = (x + i)(x - i)$.

The inverse and division

• For $a + ib \neq 0$:
  • $\dfrac{1}{a + ib} = \dfrac{a - ib}{a^2 + b^2}$
• To divide, multiply the top and the bottom by the bottom with the sign of $i$ changed.

The powers of i

• $i^1 = i$, $i^2 = -1$, $i^3 = -i$, $i^4 = 1$, and the cycle repeats.
• $i^n = i^r$, where $r$ is the remainder of $n$ on division by $4$. This works for negative $n$ too.
• Four powers in a row add up to $0$.
• $\sqrt{a}\sqrt{b} = \sqrt{ab}$ fails when $a$ and $b$ are both negative: write each root with $i$ first.

A worked example

Find the real numbers $x$ and $y$ for which $(x + iy)(1 + 2i) = 7 + 4i$.

1. Divide. $x + iy = \dfrac{7 + 4i}{1 + 2i}$.
2. Multiply the top and the bottom by $1 - 2i$. The bottom is $1 + 4 = 5$. The top is $7 - 14i + 4i - 8i^2 = 15 - 10i$.
3. Simplify. $x + iy = \dfrac{15 - 10i}{5} = 3 - 2i$, so $x = 3$ and $y = -2$.

Check: $(3 - 2i)(1 + 2i) = 3 + 6i - 2i - 4i^2 = 7 + 4i$.

Practice with new numbers

A program to try

The program checks the rules of this note against Python's own arithmetic: the rule for a product, the rule for a quotient, and the cycle of the powers of $i$.

Mistakes to avoid

• Keeping $i$ in the imaginary part. In $2 - 9i$ the imaginary part is $-9$.
• Leaving $i^2$ in an answer. Always replace $i^2$ with $-1$.
• Dividing part by part. Multiply the top and the bottom by the bottom with the sign of $i$ changed.
• Two negative roots under one sign. $\sqrt{-5}\sqrt{-20} = -10$, not $10$.
• Writing i in Python. Python needs j, as in 2 + 3j.

Check yourself

1. Write $\sqrt{-50}$ as a multiple of $i$, in its simplest form.

2. Find $(3 - i)(-2 + 5i)$.

3. Find $i^{82} + i^{83}$.

4. Find the inverse of $1 + 3i$.

5. Is $(1 + i)^2$ real, purely imaginary, or neither?