Mod-arg & Polar forms - Exercises | IB Math AAHL | Perplex

Mod-arg & Polar forms

rcisθ and eiθ

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Exercises

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Practice exam-style mod-arg & polar forms problems

Key Skills

Complex Argument

AHL HL 1.12

The argument of a complex number is the angle that it forms with the real (x) axis on the complex plane:

By noticing a right angled triangle, we can say that

tan(argz)=ab🚫

When a>0:

argz=arctan(ab)🚫

Note on convention: By convention, the argument is usually given in radians in the range [−π,π]. It will be made clear by the IB which range is preferred in a given question.

Complex argument when a<0

AHL HL 1.12

If a<0, then argz is in the second or third quadrant, which are not in the range of arctan. We therefore need to add or subtract π to get the correct argument:

When a<0:

arg(z)=arctan(ab)±π🚫

When z is in the second quadrant, we add π; when z is in the third quadrant, we subtract π.

Note on convention: By convention, the argument is usually given in radians in the range [−π,π]. It will be made clear by the IB which range is preferred in a given question.

Complex Argument when a=0 (purely imaginary)

AHL HL 1.12

If a=0, then tan(argz)=ab is undefined. tanθ is also undefined for θ=2π,23π… So when a=0 we have

arg(bi)=⎩⎪⎪⎪⎨⎪⎪⎪⎧2πb>0 −2πb<0🚫

This can be seen on the complex diagram by remembering that bi lies on the yi axis:

Note on convention: By convention, the argument is usually given in radians in the range [−π,π]. It will be made clear by the IB which range is preferred in a given question.

Polar form of complex numbers

AHL AA 1.13

The modulus ∣z∣ and argument argz uniquely define the complex number z. That means we can represent any complex number using its modulus and argument instead of a+bi:

It is conventional to call r=∣z∣ and θ=argz. Using trigonometry, we deduce that

z=r(cosθ+isinθ)📖

And we use the shorthand cisθ=cosθ+isinθ:

z=rcisθ📖

Euler's form

AHL AA 1.13

There is one more way to express complex numbers:

z=rcisθ=reiθ📖

For example, we can write z=−1=1⋅cis(π)=eiπ. This leads to the classic result

eiπ+1=0🚫

We call this Euler's form (or sometimes exponential form) because of the presence of Euler's number, e.

Multiplying in polar / Euler form

AHL AA 1.13

The main advantage of Euler's form is that it makes multiplying complex numbers much easier:

r1eiα⋅r2eiβ=r1r2ei(α+β)🚫

In words, when we multiply two complex numbers the arguments add and the moduli multiply.

Similarly for division:

r2eiβr1eiα=r2r1ei(α−β)🚫

In words, when we divide one complex number from another, we subtract the arguments and divide the moduli.

In polar form this becomes:

r1cis(α)⋅r2cis(β)=r1r2cis(α+β)🚫

Complex conjugate in polar form

AHL AA 1.13

If z=rcisθ=reiθ, then the conjugate z∗ is

z∗=rcis(−θ)=re−iθ🚫

Complex Argument

AHL HL 1.12

The argument of a complex number is the angle that it forms with the real (x) axis on the complex plane:

By noticing a right angled triangle, we can say that

tan(argz)=ab🚫

When a>0:

argz=arctan(ab)🚫

Note on convention: By convention, the argument is usually given in radians in the range [−π,π]. It will be made clear by the IB which range is preferred in a given question.

Complex argument when a<0

AHL HL 1.12

If a<0, then argz is in the second or third quadrant, which are not in the range of arctan. We therefore need to add or subtract π to get the correct argument:

When a<0:

arg(z)=arctan(ab)±π🚫

When z is in the second quadrant, we add π; when z is in the third quadrant, we subtract π.

Note on convention: By convention, the argument is usually given in radians in the range [−π,π]. It will be made clear by the IB which range is preferred in a given question.

Complex Argument when a=0 (purely imaginary)

AHL HL 1.12

If a=0, then tan(argz)=ab is undefined. tanθ is also undefined for θ=2π,23π… So when a=0 we have

arg(bi)=⎩⎪⎪⎪⎨⎪⎪⎪⎧2πb>0 −2πb<0🚫

This can be seen on the complex diagram by remembering that bi lies on the yi axis:

Note on convention: By convention, the argument is usually given in radians in the range [−π,π]. It will be made clear by the IB which range is preferred in a given question.

Polar form of complex numbers

AHL AA 1.13

The modulus ∣z∣ and argument argz uniquely define the complex number z. That means we can represent any complex number using its modulus and argument instead of a+bi:

It is conventional to call r=∣z∣ and θ=argz. Using trigonometry, we deduce that

z=r(cosθ+isinθ)📖

And we use the shorthand cisθ=cosθ+isinθ:

z=rcisθ📖

Euler's form

AHL AA 1.13

There is one more way to express complex numbers:

z=rcisθ=reiθ📖

For example, we can write z=−1=1⋅cis(π)=eiπ. This leads to the classic result

eiπ+1=0🚫

We call this Euler's form (or sometimes exponential form) because of the presence of Euler's number, e.

Multiplying in polar / Euler form

AHL AA 1.13

The main advantage of Euler's form is that it makes multiplying complex numbers much easier:

r1eiα⋅r2eiβ=r1r2ei(α+β)🚫

In words, when we multiply two complex numbers the arguments add and the moduli multiply.

Similarly for division:

r2eiβr1eiα=r2r1ei(α−β)🚫

In words, when we divide one complex number from another, we subtract the arguments and divide the moduli.

In polar form this becomes:

r1cis(α)⋅r2cis(β)=r1r2cis(α+β)🚫

Complex conjugate in polar form

AHL AA 1.13

If z=rcisθ=reiθ, then the conjugate z∗ is

z∗=rcis(−θ)=re−iθ🚫