How Can Complex Numbers Represent Addition and Rotation Geometrically?

Complex numbers are represented as vectors on an Argand diagram. Adding complex numbers performs vector addition, while multiplying by a complex number in polar or exponential form rotates and scales the original vector about the origin.

This is HL-only content from AHL 1.13, Complex numbers continued.

A complex number $z=a+bi$ corresponds to the point ((a,b)), or equivalently the vector from the origin to ((a,b)). Its real part gives the horizontal component, while its imaginary part gives the vertical component.

Geometric Addition

For $z_1=a+bi$ and $z_2=c+di$,

$$z_1+z_2=(a+c)+(b+d)i.$$

This adds corresponding vector components. For example,

$$(2+i)+(-1+3i)=1+4i.$$

Geometrically, the vectors $(2,1)$ and $(-1,3)$ combine to produce the resultant vector $(1,4)$, using the parallelogram or head-to-tail rule.

Geometric Rotation

Write the multiplier in polar form as

$$w=r(\cos\theta+i\sin\theta)=re^{i\theta}.$$

If $z=\rho e^{i\phi}$, then

$$zw=r\rho e^{i(\phi+\theta)}.$$

The new modulus is $r\rho$, and the new argument is $\phi+\theta$. Therefore, $r$ controls scaling, while $\theta$ controls rotation. A negative angle gives a clockwise rotation.

For example, $i=e^{i\pi/2}$, so

$$i(1+i)=-1+i.$$

Thus, $(1,1)$ rotates anticlockwise through $\frac{\pi}{2}$ to $(-1,1)$, with unchanged length.

For another example, multiplying $z=2-2i$ by $2i$ gives $4+4i$. The point $(2,-2)$ moves to : its modulus doubles, and its direction rotates anticlockwise through . Multiplication is centred at the origin; it does not translate the point.

This can also be verified using modulus and argument. For $z=2-2i$, $|z|=2\sqrt{2}$ and . For , the modulus is and the argument is . Their product therefore has modulus and argument , matching .

A common misconception is that addition causes rotation. It does not: addition combines vectors; multiplication produces rotation and scaling.

In an IB Mathematics AI HL response, state both the algebraic result and its geometric meaning. Check the new modulus and argument to verify the scale factor and rotation angle.