CP 2.6 - Multiplying and dividing in modulus-argument form

Multiply moduli and add directions

Why does multiplying by $1+\mathrm{i}$ and then by $1-\mathrm{i}$ double a complex number? The algebra says $(1+\mathrm{i})(1-\mathrm{i})=2$. The geometry will explain how two changes of direction can cancel while their changes of scale combine.

Start with $\mathrm{i}\times\mathrm{i}=-1$. Both factors have modulus $1$ and argument $\pi/2$; the answer has modulus $1$ and argument $\pi$. The lengths multiplied, while the directions added. This happens for every non-zero pair.

Write $z=r(\cos\alpha+\mathrm{i}\sin\alpha)$ and $w=s(\cos\beta+\mathrm{i}\sin\beta)$. Expansion gives

The two real expressions are exactly the compound-angle formulae. No new trigonometric identity is being smuggled in.

The modulus identity also holds if a factor is zero, but an argument cannot be assigned to that zero factor or to the zero product. The angle rule needs both factors non-zero.

Now change only the angle of $w$, from $\pi/6$ to $5\pi/6$. The sum of the angles leaves the principal range, so it needs one full-turn correction.

When principal arguments are used, the familiar line “arguments add” always comes with this full-turn qualification.

Apply the same correction when the sum is further round the circle.

Divide moduli and subtract directions

Division undoes multiplication: divide the lengths and subtract the direction of the denominator from the direction of the numerator.

For $w=s(\cos\beta+\mathrm{i}\sin\beta)$, its reciprocal is

Multiplying this expression by $w$ gives modulus $1$ and angle zero, so the product is $1$. Multiplying $z$ by this reciprocal establishes the quotient rule. The reciprocal's direction may itself need a full-turn correction at the principal endpoint, but its sine and cosine are unchanged.

The order matters: denominator direction is subtracted from numerator direction. Unlike multiplication, reversing the two numbers changes the result.

The second example changes only the denominator's angle, making it negative. Subtracting that negative direction increases the angle.

A reciprocal both reverses a direction and replaces a scale factor by its reciprocal. These are two separate operations.

Use the same subtraction when the denominator lies below the real axis.

Multiplication rotates and changes scale

Multiplication by a fixed non-zero complex number rotates every position vector through the same angle and multiplies every length by the same positive factor.

Let the fixed multiplier be $k=s(\cos\beta+\mathrm{i}\sin\beta)$. For any non-zero $z$, the product has modulus $s|z|$ and direction $\arg z+\beta$. That is a rotation through $\beta$ about the origin followed by an enlargement with scale factor $s$. You can apply them in either order; the origin itself stays at the origin.

For a concrete case, multiplying $1+\mathrm{i}$ by $2\mathrm{i}$ gives $-2+2\mathrm{i}$. The modulus changes from $\sqrt2$ to $2\sqrt2$, and the angle changes from $\pi/4$ to $3\pi/4$. The picture displays the same calculation as a turn and a stretch.

[DIAGRAM: asset_name: Multiplication by 2i as a rotation and an enlargement; asset_slug: 9fm0-complex-multiplier; caption: Multiplication by 2i turns a vector through a right angle and doubles its length; description: Equal-scale Argand axes show z equals 1 plus i at (1,1) and 2iz equals minus 2 plus 2i at (minus 2,2). Blue and clay position arrows have labelled lengths root 2 and 2root 2. A curved arrow around the origin marks an anticlockwise rotation of pi over 2.]

Division by $k$ reverses both changes: scale factor $1/|k|$ and rotation through $-\arg k$. In particular, division by $\mathrm{i}$ is a clockwise quarter-turn, because $1/\mathrm{i}=-\mathrm{i}$. A multiplier of modulus $1$ rotates without changing any length; a positive real multiplier changes scale without changing direction.

Now return to the opening puzzle. Both $1+\mathrm{i}$ and $1-\mathrm{i}$ have modulus $\sqrt2$, and their arguments are $\pi/4$ and $-\pi/4$. The opposite turns cancel, while the scale factors multiply to $2$. The pair doubles every complex number. Turning back does not undo the stretching!

The following flawed argument mixes up those two jobs.

Quick Recap

• Multiply moduli and add directions for a product.
• Divide moduli and subtract the denominator direction for a quotient.
• Correct directions by whole multiples of $2\pi$ when a principal argument is required; zero has no argument.
• Multiplication by $k\ne0$ enlarges by $|k|$ and rotates by $\arg k$; division reverses both changes.

The opening multipliers have opposite angles but both have scale factor $\sqrt2$. Their turns cancel and their combined scale factor is $2$.