Complex Numbers — NDA Mathematics

Formulas (4)

• What a complex number is · Fundamentals of a complex number
  $$i^2=-1 \qquad a+ib=c+id \iff a=c,\ b=d \qquad (a+ib)(c+id)=(ac-bd)+i(ad+bc)$$
• Conjugate; purely real / purely imaginary · Conjugate identities
  $$\overline{a+ib}=a-ib \qquad z\bar z=|z|^2 \qquad \operatorname{Re}(z)=\dfrac{z+\bar z}{2} \qquad \operatorname{Im}(z)=\dfrac{z-\bar z}{2i} \qquad \overline{z_1z_2}=\bar z_1\,\bar z_2$$
• Modulus and the triangle inequality · Modulus properties
  $$|z|=\sqrt{a^2+b^2} \qquad |z_1z_2|=|z_1|\,|z_2| \qquad \left|\dfrac{z_1}{z_2}\right|=\dfrac{|z_1|}{|z_2|} \qquad |z|^2=z\bar z \qquad |z_1+z_2|\le|z_1|+|z_2|$$
• Argument and polar form · Polar form and argument
  $$z=r(\cos\theta+i\sin\theta)=re^{i\theta} \qquad \arg(z_1z_2)=\arg z_1+\arg z_2 \qquad \arg\!\left(\dfrac{z_1}{z_2}\right)=\arg z_1-\arg z_2$$

Watch out for (4)

• Purely imaginary is the real-part-zero condition, not the imaginary-part-zero one
  → Conjugate; purely real / purely imaginary
• $z\bar z=|z|^2$, not $|z|$
  → Modulus and the triangle inequality
• Modulus does not distribute over a sum
  → Modulus and the triangle inequality
• The principal argument depends on the quadrant, not just $\tan^{-1}(b/a)$
  → Argument and polar form

Formulas (2)

• Powers of i (the period-4 cycle) · Powers of i
  $$i^2=-1 \qquad i^3=-i \qquad i^4=1 \qquad i^{4k+r}=i^r \qquad i^k+i^{k+1}+i^{k+2}+i^{k+3}=0$$
• De Moivre's theorem and roots · De Moivre's theorem and nth roots
  $$(\cos\theta+i\sin\theta)^n=\cos n\theta+i\sin n\theta \qquad z^n=r^n e^{in\theta} \qquad z^{1/n}=r^{1/n}e^{i(\theta+2k\pi)/n},\ k=0,\ldots,n-1$$

Watch out for (1)

• Reduce the exponent of $i$ mod 4 — and watch the remainder
  → Powers of i (the period-4 cycle)

Formulas (1)

• 1, ω, ω² and their identities · Cube roots of unity identities
  $$\omega^3=1 \qquad 1+\omega+\omega^2=0 \qquad \bar\omega=\omega^2 \qquad \omega^n=\omega^{\,n\bmod 3}$$

Watch out for (1)

• $\omega^2=\bar\omega$, but $\omega^2\ne-\omega$
  → 1, ω, ω² and their identities