Polar form, multiplication and division | CAIE 9709 Pure Mathematics 3 Notes

Pure Mathematics 3

Mathematics9709

Multiply lengths and add angles

For non-zero complex numbers, let $r_1$ and $r_2$ be their positive moduli. A divisor cannot have modulus 0.

r_1e^{i\theta_1}\,r_2e^{i\theta_2}

=r_1r_2e^{i(\theta_1+\theta_2)}

\frac{r_1e^{i\theta_1}}{r_2e^{i\theta_2}}

=\frac{r_1}{r_2}e^{i(\theta_1-\theta_2)}

Multiplication: multiply moduli and add arguments.
Division: divide moduli and subtract the denominator’s argument.
Reduce the final argument into the interval the question requests.

The algebra is a scale and rotation

Complex multiplication is scale + rotation

before

modulus r
argument θ

× 2e^(iπ/3)
→

after

modulus 2r
argument θ + π/3

length × 2

rotate +60°

Multiplication multiplies moduli and adds arguments.

Adding a complex number translates a point by that vector; subtracting it translates by the opposite vector.
Conjugating reflects it in the real axis.
Multiplying by $re^{i\alpha}$ scales lengths by r and rotates anticlockwise through α.
Dividing by the same number scales by $1/r$ and rotates through $-\alpha$ .

Key idea

These effects explain the modulus and argument laws; they are not separate facts to memorise.

Operation

Real part of z2

Imaginary part of z1

Real part of w1

Imaginary part of w2

z = 2 + 1i

w = 1 + 2i

zw = 0 + 5i

Scale every length by 2.24 and rotate by 63.43°.

The algebra and the geometry describe the same transformation. Change one value at a time and predict the result before reading it.

Current-paper arguments without expansion

Worked example

Use argument laws directly

A current paper gives a long quotient. Rename its two unit complex numbers first:

a=\cos(\pi/7)+i\sin(\pi/7)

b=\cos(\pi/7)-i\sin(\pi/7)

u=\frac{a^4}{b}

The numerator has argument $4\pi/7$ . The denominator has argument $-\pi/7$ .

\arg u=\frac{4\pi}{7}-\left(-\frac{\pi}{7}\right)

Answer

\arg u=\frac{5\pi}{7}

Conjugation is reflection in the real axis, so it reverses the argument.

Answer

\arg u^*=-\frac{5\pi}{7}

(9709/32/O/N/24 Q5(a,b))

Your turnIndependent polar-operation check [4 marks]

Given $z=2e^{i\pi/6}$ and $w=3e^{-i\pi/4}$ , find zw and z/w in exponential form.

Show worked answer

Answer

zw=6e^{-i\pi/12}

\dfrac{z}{w}=\dfrac23e^{5\pi i/12}

Multiply lengths and add angles

For non-zero complex numbers, let $r_1$ and $r_2$ be their positive moduli. A divisor cannot have modulus 0.

r_1e^{i\theta_1}\,r_2e^{i\theta_2}

=r_1r_2e^{i(\theta_1+\theta_2)}

\frac{r_1e^{i\theta_1}}{r_2e^{i\theta_2}}

=\frac{r_1}{r_2}e^{i(\theta_1-\theta_2)}

Multiplication: multiply moduli and add arguments.
Division: divide moduli and subtract the denominator’s argument.
Reduce the final argument into the interval the question requests.

The algebra is a scale and rotation

Complex multiplication is scale + rotation

before

modulus r
argument θ

× 2e^(iπ/3)
→

after

modulus 2r
argument θ + π/3

length × 2

rotate +60°

Multiplication multiplies moduli and adds arguments.

Adding a complex number translates a point by that vector; subtracting it translates by the opposite vector.
Conjugating reflects it in the real axis.
Multiplying by $re^{i\alpha}$ scales lengths by r and rotates anticlockwise through α.
Dividing by the same number scales by $1/r$ and rotates through $-\alpha$ .

Key idea

These effects explain the modulus and argument laws; they are not separate facts to memorise.

Operation

Real part of z2

Imaginary part of z1

Real part of w1

Imaginary part of w2

z = 2 + 1i

w = 1 + 2i

zw = 0 + 5i

Scale every length by 2.24 and rotate by 63.43°.

The algebra and the geometry describe the same transformation. Change one value at a time and predict the result before reading it.

Current-paper arguments without expansion

Worked example

Use argument laws directly

A current paper gives a long quotient. Rename its two unit complex numbers first:

a=\cos(\pi/7)+i\sin(\pi/7)

b=\cos(\pi/7)-i\sin(\pi/7)

u=\frac{a^4}{b}

The numerator has argument $4\pi/7$ . The denominator has argument $-\pi/7$ .

\arg u=\frac{4\pi}{7}-\left(-\frac{\pi}{7}\right)

Answer

\arg u=\frac{5\pi}{7}

Conjugation is reflection in the real axis, so it reverses the argument.

Answer

\arg u^*=-\frac{5\pi}{7}

(9709/32/O/N/24 Q5(a,b))

Your turnIndependent polar-operation check [4 marks]

Given $z=2e^{i\pi/6}$ and $w=3e^{-i\pi/4}$ , find zw and z/w in exponential form.

Show worked answer

Answer

zw=6e^{-i\pi/12}

\dfrac{z}{w}=\dfrac23e^{5\pi i/12}