Summary

Summary

• Need for complex numbers. A quadratic $ax^2+bx+c=0$ with negative discriminant has no real root, since no real number squares to a negative number. Adjoining a new symbol $i$ with $i^2=-1$ closes this gap; a complex number is $z=a+ib$ with $a,b\in\mathbb{R}$, where $a=\mathrm{Re}(z)$ and $b=\mathrm{Im}(z)$. Two complex numbers are equal exactly when both parts match; $\mathbb{C}$ cannot be ordered.
• Algebra of complex numbers. Addition/subtraction is componentwise: $(a+ib)\pm(c+id)=(a\pm c)+i(b\pm d)$. Multiplication expands like binomials with $i^2=-1$: $(a+ib)(c+id)=(ac-bd)+i(ad+bc)$. The conjugate $\bar z=a-ib$ satisfies $z\bar z=a^2+b^2$, $z+\bar z=2\,\mathrm{Re}(z)$, $z-\bar z=2i\,\mathrm{Im}(z)$. The reciprocal of a non-zero $z$ is $z^{-1}=\bar z/|z|^2$, and division by $z_2\neq0$ is multiplication by $z_2^{-1}$ (i.e. rationalise using the conjugate of the denominator).
• The Argand plane. $z=x+iy$ is plotted as the point $(x,y)$: the horizontal axis is the real axis, the vertical axis is the imaginary axis. $\bar z$ is the reflection of $z$ in the real axis.
• Modulus and argument. $|z|=\sqrt{a^2+b^2}=OP$, always non-negative. $\arg(z)=\theta$ is the angle $OP$ makes with the positive real axis, satisfying $\cos\theta=a/r,\ \sin\theta=b/r$; the principal value lies in $(-\pi,\pi]$ and must be found by first locating the quadrant, never by trusting $\tan^{-1}(b/a)$ alone.
• Polar representation. Substituting $a=r\cos\theta,\ b=r\sin\theta$ gives $z=r(\cos\theta+i\sin\theta)$, the polar form, with $(r,\theta)$ the polar coordinates of the point representing $z$. …