Cartesian form

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Arithmetic of complex numbers in cartesian form, including conjugates, and the Argand diagram (complex plane).

Want a deeper conceptual understanding? Try our interactive lesson!

Imaginary number i

AHL 1.12

The imaginary number i is the square root of −1:

i=√−1⇔i2=−1🚫

In general, imaginary numbers are of the form bi,b∈R∖{0}. Notice that

(bi)2=b2i2=−b2<0

Conclusion: the square of any imaginary number is negative.

Complex Numbers a+bi

AHL 1.12

A complex number

z=a+bi📖

is the sum of a real number a and an imaginary number bi. We call this the Cartesian form for a complex number.

Finding complex roots of quadratics

AHL 1.12

When a quadratic

ax2+bx+c=0

has

Δ=b2−4ac<0,

it has no real roots since the square root in

x=2a−b±√b2−4ac

is not a real number. Instead, the square root will give an imaginary number, making the roots complex.

Real and Imaginary parts

AHL 1.12

For a complex number z=a+bi, we call a the real part of z and b the imaginary part of z:

Re(z)Im(z)=a=b

For example,

z=2−3i

is a complex number with a real part Re(z)=2 and imaginary part Im(z)=−3.

The sets ℂ, ℝ and ℝ\ℂ

AHL 1.12

Real numbers are a subset of complex numbers a+bi where b=0. Imaginary numbers are also a subset of complex numbers with a=0.

Product of complex numbers

AHL 1.12

The product of two complex numbers in Cartesian form is

(a+bi)×(c+di) =ac+adi+bci+bdi2=ac+i(ad+bd)+bd⋅(−1)=ac−bd+(ad+bd)i

The complex plane

AHL 1.12

Complex numbers can be visualized in the complex plane, also known as the Argand Diagram.

To plot a complex number, the real part determines the x-coordinate and the imaginary part determines the y-coordinate. Therefore the complex number a+bi has coordinates (a,b) on the plane.

It is conventional to use arrows from the origin to the point (a,b) to represent complex numbers.

Nice work completing Cartesian 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!

Complex conjugate

Complex Numbers

Cartesian form

0 of 0 exercises completed

Arithmetic of complex numbers in cartesian form, including conjugates, and the Argand diagram (complex plane).

Want a deeper conceptual understanding? Try our interactive lesson!

Imaginary number i

AHL 1.12

The imaginary number i is the square root of −1:

i=√−1⇔i2=−1🚫

In general, imaginary numbers are of the form bi,b∈R∖{0}. Notice that

(bi)2=b2i2=−b2<0

Conclusion: the square of any imaginary number is negative.

Complex Numbers a+bi

AHL 1.12

A complex number

z=a+bi📖

is the sum of a real number a and an imaginary number bi. We call this the Cartesian form for a complex number.

Finding complex roots of quadratics

AHL 1.12

When a quadratic

ax2+bx+c=0

has

Δ=b2−4ac<0,

it has no real roots since the square root in

x=2a−b±√b2−4ac

is not a real number. Instead, the square root will give an imaginary number, making the roots complex.

Real and Imaginary parts

AHL 1.12

For a complex number z=a+bi, we call a the real part of z and b the imaginary part of z:

Re(z)Im(z)=a=b

For example,

z=2−3i

is a complex number with a real part Re(z)=2 and imaginary part Im(z)=−3.

The sets ℂ, ℝ and ℝ\ℂ

AHL 1.12

Real numbers are a subset of complex numbers a+bi where b=0. Imaginary numbers are also a subset of complex numbers with a=0.

Product of complex numbers

AHL 1.12

The product of two complex numbers in Cartesian form is

(a+bi)×(c+di) =ac+adi+bci+bdi2=ac+i(ad+bd)+bd⋅(−1)=ac−bd+(ad+bd)i

The complex plane

AHL 1.12

Complex numbers can be visualized in the complex plane, also known as the Argand Diagram.

To plot a complex number, the real part determines the x-coordinate and the imaginary part determines the y-coordinate. Therefore the complex number a+bi has coordinates (a,b) on the plane.

It is conventional to use arrows from the origin to the point (a,b) to represent complex numbers.

Nice work completing Cartesian 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!

Complex conjugate

Complex Numbers

Cartesian form

0 of 0 exercises completed

Arithmetic of complex numbers in cartesian form, including conjugates, and the Argand diagram (complex plane).

Want a deeper conceptual understanding? Try our interactive lesson!

Imaginary number i

AHL 1.12

The imaginary number i is the square root of −1:

i=√−1⇔i2=−1🚫

In general, imaginary numbers are of the form bi,b∈R∖{0}. Notice that

(bi)2=b2i2=−b2<0

Conclusion: the square of any imaginary number is negative.

Complex Numbers a+bi

AHL 1.12

A complex number

z=a+bi📖

is the sum of a real number a and an imaginary number bi. We call this the Cartesian form for a complex number.

Finding complex roots of quadratics

AHL 1.12

When a quadratic

ax2+bx+c=0

has

Δ=b2−4ac<0,

it has no real roots since the square root in

x=2a−b±√b2−4ac

is not a real number. Instead, the square root will give an imaginary number, making the roots complex.

Real and Imaginary parts

AHL 1.12

For a complex number z=a+bi, we call a the real part of z and b the imaginary part of z:

Re(z)Im(z)=a=b

For example,

z=2−3i

is a complex number with a real part Re(z)=2 and imaginary part Im(z)=−3.

The sets ℂ, ℝ and ℝ\ℂ

AHL 1.12

Real numbers are a subset of complex numbers a+bi where b=0. Imaginary numbers are also a subset of complex numbers with a=0.

Product of complex numbers

AHL 1.12

The product of two complex numbers in Cartesian form is

(a+bi)×(c+di) =ac+adi+bci+bdi2=ac+i(ad+bd)+bd⋅(−1)=ac−bd+(ad+bd)i

The complex plane

AHL 1.12

Complex numbers can be visualized in the complex plane, also known as the Argand Diagram.

To plot a complex number, the real part determines the x-coordinate and the imaginary part determines the y-coordinate. Therefore the complex number a+bi has coordinates (a,b) on the plane.

It is conventional to use arrows from the origin to the point (a,b) to represent complex numbers.

Nice work completing Cartesian 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...

Complex conjugate

Complex Numbers

Cartesian form

0 of 0 exercises completed

Arithmetic of complex numbers in cartesian form, including conjugates, and the Argand diagram (complex plane).

Want a deeper conceptual understanding? Try our interactive lesson!

Imaginary number i

AHL 1.12

The imaginary number i is the square root of −1:

i=√−1⇔i2=−1🚫

In general, imaginary numbers are of the form bi,b∈R∖{0}. Notice that

(bi)2=b2i2=−b2<0

Conclusion: the square of any imaginary number is negative.

Complex Numbers a+bi

AHL 1.12

A complex number

z=a+bi📖

is the sum of a real number a and an imaginary number bi. We call this the Cartesian form for a complex number.

Finding complex roots of quadratics

AHL 1.12

When a quadratic

ax2+bx+c=0

has

Δ=b2−4ac<0,

it has no real roots since the square root in

x=2a−b±√b2−4ac

is not a real number. Instead, the square root will give an imaginary number, making the roots complex.

Real and Imaginary parts

AHL 1.12

For a complex number z=a+bi, we call a the real part of z and b the imaginary part of z:

Re(z)Im(z)=a=b

For example,

z=2−3i

is a complex number with a real part Re(z)=2 and imaginary part Im(z)=−3.

The sets ℂ, ℝ and ℝ\ℂ

AHL 1.12

Real numbers are a subset of complex numbers a+bi where b=0. Imaginary numbers are also a subset of complex numbers with a=0.

Product of complex numbers

AHL 1.12

The product of two complex numbers in Cartesian form is

(a+bi)×(c+di) =ac+adi+bci+bdi2=ac+i(ad+bd)+bd⋅(−1)=ac−bd+(ad+bd)i

The complex plane

AHL 1.12

Complex numbers can be visualized in the complex plane, also known as the Argand Diagram.

To plot a complex number, the real part determines the x-coordinate and the imaginary part determines the y-coordinate. Therefore the complex number a+bi has coordinates (a,b) on the plane.

It is conventional to use arrows from the origin to the point (a,b) to represent complex numbers.

Nice work completing Cartesian 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...