Complex Numbers: Chapter Review

Vocabulary and notation

Imaginary unit $i$

Defined by the single equation $i^2 = -1$, with every ordinary law of algebra kept. No real number has a negative square, so $i$ lies nowhere on the real line.

Complex number, standard form $a + bi$

Real part $a$ and imaginary part $b$, both real numbers. The imaginary part is the coefficient $b$ with its sign, never $bi$: for $7 - 4i$ it is $-4$.

Pure imaginary number

A number $bi$ with $b \neq 0$, so its real part is $0$. Taking $b = 0$ instead gives a real number: the reals sit inside the complex numbers unchanged.

Conjugate $\overline{z}$

For $z = a + bi$, the number $a - bi$: only the sign of the imaginary part flips. The conjugate of $-3 - 5i$ is $-3 + 5i$, not $3 + 5i$.

Conjugate pair

$p + qi$ and $p - qi$ with $q \neq 0$: equal real parts, opposite imaginary parts, mirror images across the real axis.

Complex plane (Argand diagram)

The plane in which $a + bi$ is the point $(a, b)$. The horizontal real axis holds the numbers with $b = 0$; the vertical imaginary axis holds $0$ and the pure imaginaries.

Discriminant $\Delta = b^2 - 4ac$

For REAL $a \neq 0$, $b$, $c$ it no longer counts roots (there are always two with multiplicity); it locates them: $\Delta > 0$ two real, $\Delta = 0$ one repeated real, $\Delta < 0$ a non-real conjugate pair. Let one coefficient go non-real and none of it holds: $x^2 - 2ix - 1$ has $\Delta = 0$ and the repeated root $i$.

Formulas and theorems

• The definition of $i$, and roots of negatives

  $$i^2 = -1, \qquad \sqrt{-k} = i\sqrt{k}$$

  Use when $k > 0$. The symbol $\sqrt{-k}$ names ONE number, $i\sqrt{k}$, while $x^2 = -k$ has two solutions, $\pm i\sqrt{k}$. Keep the $i$ outside the radical.

• Powers of $i$

  $$\begin{gathered} i^1 = i, \quad i^2 = -1, \quad i^3 = -i, \quad i^4 = 1 \\ i^n = i^r \ \text{ for } \ n = 4q + r \end{gathered}$$

  Use when $n$ a positive integer, $r$ its remainder on division by $4$; remainder $0$ gives $1$. Parity decides nothing: $i^6 = -1$ but $i^8 = 1$. Any four consecutive powers sum to $0$.

• Equality of complex numbers

  $$a + bi = c + di \iff a = c \ \text{ and } \ b = d$$

  Use when $a$, $b$, $c$, $d$ ALL real. One complex equation therefore carries two real ones. The complex numbers have no order, so $<$ and $>$ between them are meaningless.

• Addition and subtraction

  $$(a + bi) \pm (c + di) = (a \pm c) + (b \pm d)\,i$$

  Use when Any two complex numbers. Distribute a minus sign through BOTH terms of the second. Geometrically, adding $c + di$ slides every point $c$ across and $d$ up; $z - w$ is the arrow from $w$ to $z$.

• Multiplication

  $$\begin{gathered} (a + bi)(c + di) \\ = (ac - bd) + (ad + bc)\,i \end{gathered}$$

  Use when Any two complex numbers. Rebuild it by expanding rather than memorizing: each $i^2$ becomes $-1$ and joins the real part with its sign flipped.

• Conjugate product

  $$z\,\overline{z} = (a + bi)(a - bi) = a^2 + b^2 = |z|^2$$

  Use when Any $z = a + bi$. The value is REAL, never negative, and $0$ only for $z = 0$. The $i^2$ turns the difference of squares into a SUM, which is what lets a conjugate clear a denominator.

• Conjugation passes through arithmetic

  $$\begin{gathered} \overline{z + w} = \overline{z} + \overline{w} \\ \overline{zw} = \overline{z}\,\overline{w} \\ \overline{\overline{z}} = z \end{gathered}$$

  Text description

  The point z and the point for its conjugate sit directly above and below each other at equal distances from the real axis, so conjugation reflects the plane across that axis.

  Use when Any complex $z$ and $w$; the product rule repeats into powers. Conjugation fixes exactly the real numbers and reflects the plane across the real axis, so $|\overline{z}| = |-z| = |z|$.

• Division and reciprocals

  $$\begin{gathered} \frac{a + bi}{c + di} = \frac{(a + bi)(c - di)}{c^2 + d^2} \\ \frac{1}{c + di} = \frac{c - di}{c^2 + d^2} \end{gathered}$$

  Use when $c + di \neq 0$, the only forbidden divisor, since $c^2 + d^2 = 0$ only for $c = d = 0$. Multiply NUMERATOR AND DENOMINATOR by the DENOMINATOR's conjugate, which is multiplying by $1$.

  e.g. $\dfrac{1}{i} = \dfrac{-i}{0^2 + 1^2} = -i$, and $i(-i) = 1$.

• Modulus, and the modulus of a product

  $$|a + bi| = \sqrt{a^2 + b^2}, \qquad |zw| = |z|\,|w|$$

  Text description

  The point a plus b i closes a right triangle whose legs along the axes have lengths a and b, so its hypotenuse, the distance from zero, is the modulus.

  Use when Any complex numbers. The modulus is the distance from $0$: always a NONNEGATIVE REAL number, never complex, and $0$ only for $z = 0$. Multiplicativity extends to any number of factors; no such rule holds for a sum.

  e.g. $|(6 + 8i)(4 - 3i)| = 10 \cdot 5 = 50$, with no expansion.

• Distance, and circles

  $$\begin{gathered} |z - w| = \sqrt{(a - c)^2 + (b - d)^2} \\ |z - z_0| = r \end{gathered}$$

  Use when For $z = a + bi$ and $w = c + di$; distance from $0$ is the case $w = 0$. The circle needs $r > 0$ and has center $z_0$, with $|z - z_0| < r$ inside and $> r$ outside.

  e.g. $|(-1 + 2i) - (3 - i)| = |-4 + 3i| = 5$.

• Triangle inequality

  $$\bigl| |z| - |w| \bigr| \;\le\; |z + w| \;\le\; |z| + |w|$$

  Use when Any two complex numbers. The upper bound is an equality only when the arrows point the same way, or one is $0$; never solve with it as an equation.

  e.g. $|z| = 3$ and $|w| = 4$ pin $|z + w|$ into $[1, 7]$.

• Quadratic formula with a negative discriminant

  $$\begin{gathered} x = \frac{-b \pm i\sqrt{|\Delta|}}{2a} \\ \Delta = b^2 - 4ac < 0 \end{gathered}$$

  Use when $a$, $b$, $c$ real and $a \neq 0$. Divide BOTH numerator terms by $2a$. The roots are a conjugate pair $p \pm qi$ with $q \neq 0$, sharing the real part $p = -\tfrac{b}{2a}$, the axis of symmetry.

• Conjugate root theorem

  $$\begin{gathered} az^2 + bz + c = 0 \\ \implies a\,\overline{z}^{\,2} + b\,\overline{z} + c = 0 \end{gathered}$$

  Use when EVERY coefficient must be real, with $a \neq 0$. It says nothing new about a real root, where $\overline{z} = z$; its force is that non-real roots never appear alone, so the inventories are two real, one repeated real, or one conjugate pair.

• Sum and product of the roots, and complete factorization

  $$\begin{gathered} r_1 + r_2 = -\frac{b}{a}, \qquad r_1 r_2 = \frac{c}{a} \\ ax^2 + bx + c = a(x - r_1)(x - r_2) \end{gathered}$$

  Use when Any quadratic with $a \neq 0$; both relations hold whether the roots are real or not. For a non-real pair $p \pm qi$ the product is $p^2 + q^2 = |z|^2 > 0$. Over the complex numbers every quadratic splits into two linear factors.

  e.g. Roots $1 \pm 2i$: $x^2 - 2x + 5 = (x - 1 - 2i)(x - 1 + 2i)$.

Problem types, step by step

Exam traps

• Trap Applying $\sqrt{a}\,\sqrt{b} = \sqrt{ab}$ to two negatives: $\sqrt{-4} \cdot \sqrt{-9} = \sqrt{36} = 6$.

  Fix That rule was proved for $a, b \ge 0$ only. Convert first: $(2i)(3i) = 6i^2 = -6$, the opposite sign.

• Trap Leaving an $i^2$ standing, or reading it as $1$.

  Fix Every $i^2$ becomes $-1$ on sight, so $8 - 10i + 12i - 15i^2$ finishes at $23 + 2i$, not $-7 + 2i$.

• Trap Answering "no solution" when the discriminant is negative.

  Fix No REAL solution, but exactly two complex ones, a conjugate pair. Name the number system with the count.

• Trap Dividing only part of the numerator by $2a$, as $\tfrac{6 \pm 4i}{2} = 6 \pm 2i$ or $3 \pm 4i$.

  Fix The denominator sits under the whole numerator: $\tfrac{6 \pm 4i}{2} = 3 \pm 2i$.

• Trap Using $z^2$ where $z\,\overline{z} = |z|^2$ is meant.

  Fix $(3 + 4i)^2 = -7 + 24i$, while $(3 + 4i)(3 - 4i) = 25$. The two agree only for real $z$.

• Trap Computing the gap between $z$ and $w$ as $|z| - |w|$.

  Fix Subtract first: the distance is $|z - w|$. For $z = 3$ and $w = 3i$ it is $3\sqrt{2}$, while $|z| - |w| = 0$.

• Trap Pairing roots as conjugates without checking that the coefficients are real.

  Fix $x^2 - 2ix - 1 = (x - i)^2$ has the double root $i$, and $-i$ is not a root at all.