Complex Numbers

What you need to know

• Powers of $i$. The number $i$ has $i^2=-1$. Its powers $i^1,i^2,i^3,i^4,i^5,\dots$ are $i,-1,-i,1,i,\dots$, repeating every $4$ steps. Example: $2026$ leaves remainder $2$ when divided by $4$, so $i^{2026}=i^2=-1$.
• Parts of a complex number. Write $z=a+bi$ with $a$, $b$ real. The conjugate is $\bar z=a-bi$. The modulus $|z|=\sqrt{a^2+b^2}$ is the distance from $0$ to the point $(a,b)$. Also $z\bar z=a^2+b^2=|z|^2$. Example: for $z=3+4i$, $\bar z=3-4i$, $|z|=5$, and $(3+4i)(3-4i)=25$.
• Dividing. Multiply top and bottom by the conjugate of the bottom; the bottom becomes real. Example: $\dfrac5{2-i}=\dfrac{5(2+i)}{(2-i)(2+i)}=\dfrac{5(2+i)}5=2+i$.
• Moduli multiply. $|zw|=|z|\cdot|w|$ and $|z^n|=|z|^n$. Example: $|(3+4i)^3|=5^3=125$.
• Matching parts. Two complex numbers are equal exactly when their real parts match and their imaginary parts match. So one complex equation is two ordinary equations. Example: $(a+bi)(1+i)=(a-b)+(a+b)i$, so if it equals $3+i$, then $a-b=3$ and $a+b=1$.
• Polar form. A complex number at distance $r$ from $0$, at angle $\theta$ from the positive real axis, is $z=r(\cos\theta+i\sin\theta)$. Multiplying multiplies the distances and adds the angles, so $z^n=r^n(\cos n\theta+i\sin n\theta)$. Example: $1+i$ has $r=\sqrt2$ and angle $45^\circ$, so $(1+i)^2$ has $r=2$ and angle $90^\circ$: $(1+i)^2=2i$.
• Roots of unity. The $n$th roots of unity are the $n$ solutions of $z^n=1$. They are the corners of a regular $n$-gon on the circle of radius $1$ around $0$, and for $n\ge2$ they add up to $0$. Example: the $4$th roots are $1,i,-1,-i$, with sum $0$. The cube roots are $1,\omega,\omega^2$, where $\omega=\frac{-1+i\sqrt3}2$ has $\omega^3=1$ and $\omega^2+\omega+1=0$.
• Conjugate pairs. If a polynomial with real coefficients has a root $z=a+bi$, then $\bar z$ is a root too. Together they give the real factor $(x-z)(x-\bar z)=x^2-2ax+|z|^2$. Example: if $3+2i$ is a root, so is $3-2i$, and $x^2-6x+13$ is a factor.

How AMC 10 tests it

• Problems 5–12: find $(1+i)^{20}$ or $i^{2025}+i^{2026}$ using the cycle of $i$ or $(1+i)^2=2i$.
• Problems 8–15: find real $a,b$ with $(a+bi)^2=3+4i$, or make $\frac{a+bi}{2-i}$ real, by matching parts.
• Problems 12–18: "how many complex $z$ satisfy $z^2=\bar z$?" Set $z=a+bi$ or use polar form.
• Problems 15–22: geometry. $|z-1|=|z+1|$ is a line and $|z-2i|=3$ is a circle; the question asks for an area, a distance, or a count.
• Now and then, a quadratic with roots that are not real. Vieta's Formulas still work.

Standard approaches

1. For powers of $i$, use the remainder of the exponent when divided by $4$. For powers of $1\pm i$, square first: $(1+i)^2=2i$, $(1-i)^2=-2i$. Example: $(1+i)^{20}=(2i)^{10}=2^{10}i^{10}=-1024$.
2. For equations, write $z=a+bi$, multiply out, and set real parts equal and imaginary parts equal.
3. For conditions on the modulus, use $|z|^2=a^2+b^2$ to avoid square roots. Read $|z-w|$ as the distance between the points $z$ and $w$. Example: $|z-2i|=3$ is the circle of radius $3$ around $(0,2)$.
4. For $z^n=1$, or for multiplying by the same number again and again, switch to polar form and follow the angle.
5. Check your final answer with $|z^n|=|z|^n$. It catches most arithmetic slips.

Worked example

Let $z$ be a complex number with $|z|=2$ whose real part is $1$. What is $z^6$?

(A) $-64$ (B) $-8$ (C) $8$ (D) $64$ (E) $128$

Write $z=1+bi$ with $b$ real. Then $|z|^2=1+b^2=4$, so $b^2=3$ (and $b=\pm\sqrt3$). Keep $b$ as a letter, using $b^2=3$ and $i^2=-1$:
$$z^2=1+2bi+b^2i^2=1-3+2bi=-2+2bi,$$
$$z^3=(1+bi)(-2+2bi)=-2+2bi-2bi+2b^2i^2=-2-6=-8 .$$
So $z^6=(z^3)^2=(-8)^2=64$. The modulus check agrees: $|z^6|=|z|^6=2^6=64$.

In polar form, $z=2(\cos60^\circ\pm i\sin60^\circ)$ (the real part is $2\cos60^\circ=1$), so $z^6$ has modulus $2^6$ and angle $\pm360^\circ$, which points along the positive real axis: $z^6=64$.

The answer is $\boxed{\textbf{(D)}\ 64}$.

Pitfalls

• Treating $i$ as an ordinary letter and forgetting $i^2=-1$ when multiplying out. Example: $(2+i)^2=4+4i+i^2=3+4i$, not $5+4i$.
• Computing $|a+bi|$ as $a^2+b^2$ (no square root) or as $\sqrt{a^2-b^2}$. Example: $|3+4i|=5$, not $25$.
• Dividing without the conjugate, or multiplying only the top by it.
• Forgetting $z=0$ when counting solutions of equations like $z^n=\bar z$. Example: $z^2=\bar z$ has $4$ solutions: $0$ and the three cube roots of unity.