Polar Form

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Using modulus and argument to write complex numbers in the form rcisθ and reiθ, which make multiplication and division easier.

Want a deeper conceptual understanding? Try our interactive lesson!

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🚫

Distance in the complex plane

AHL 1.12

The distance in the complex plane between two points representing z and w is given by ∣z−w∣, which is the same as ∣w−z∣.

This is a reflection of the familiar distance formula

d=√(x2−x1)2+(y2−y1)2,

except that x1 and x2 are the real components of z and w, and y1 and y2 the imaginary components of z and w.

Angle between complex numbers in the plane

AHL AA 1.14

The angle between to complex numbers on an Argand diagram is the absolute value of the difference of their arguments:

Nice work completing Polar Form, here's a quick recap of what we covered:

Skills covered

Mixed Practice

Exercises checked off

I'm Plex, here to help you understand this concept!

Powers of Complex Numbers

Complex Numbers

Polar Form

0 of 0 exercises completed

Using modulus and argument to write complex numbers in the form rcisθ and reiθ, which make multiplication and division easier.

Want a deeper conceptual understanding? Try our interactive lesson!

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🚫

Distance in the complex plane

AHL 1.12

The distance in the complex plane between two points representing z and w is given by ∣z−w∣, which is the same as ∣w−z∣.

This is a reflection of the familiar distance formula

d=√(x2−x1)2+(y2−y1)2,

except that x1 and x2 are the real components of z and w, and y1 and y2 the imaginary components of z and w.

Angle between complex numbers in the plane

AHL AA 1.14

The angle between to complex numbers on an Argand diagram is the absolute value of the difference of their arguments:

Nice work completing Polar Form, here's a quick recap of what we covered:

Skills covered

Mixed Practice

Exercises checked off

I'm Plex, here to help you understand this concept!

Powers of Complex Numbers

Complex Numbers

Polar Form

0 of 0 exercises completed

Using modulus and argument to write complex numbers in the form rcisθ and reiθ, which make multiplication and division easier.

Want a deeper conceptual understanding? Try our interactive lesson!

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🚫

Distance in the complex plane

AHL 1.12

The distance in the complex plane between two points representing z and w is given by ∣z−w∣, which is the same as ∣w−z∣.

This is a reflection of the familiar distance formula

d=√(x2−x1)2+(y2−y1)2,

except that x1 and x2 are the real components of z and w, and y1 and y2 the imaginary components of z and w.

Angle between complex numbers in the plane

AHL AA 1.14

The angle between to complex numbers on an Argand diagram is the absolute value of the difference of their arguments:

Nice work completing Polar Form, here's a quick recap of what we covered:

Skills covered

Mixed Practice

Exercises checked off

I'm Plex, here to help you understand this concept!

Generating starter questions...

Powers of Complex Numbers

Complex Numbers

Polar Form

0 of 0 exercises completed

Using modulus and argument to write complex numbers in the form rcisθ and reiθ, which make multiplication and division easier.

Want a deeper conceptual understanding? Try our interactive lesson!

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🚫

Distance in the complex plane

AHL 1.12

The distance in the complex plane between two points representing z and w is given by ∣z−w∣, which is the same as ∣w−z∣.

This is a reflection of the familiar distance formula

d=√(x2−x1)2+(y2−y1)2,

except that x1 and x2 are the real components of z and w, and y1 and y2 the imaginary components of z and w.

Angle between complex numbers in the plane

AHL AA 1.14

The angle between to complex numbers on an Argand diagram is the absolute value of the difference of their arguments:

Nice work completing Polar Form, here's a quick recap of what we covered:

Skills covered

Mixed Practice

Exercises checked off

I'm Plex, here to help you understand this concept!

Generating starter questions...