Complex numbers
12
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Imaginary and complex numbers, real & imaginary part, norm, operations with complex numbers
Transcript of Complex numbers
Introduction to complex numbers
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Complex numbers
Introduction to complex numbers 2
Imagine a new number π with the property π2 = β1
The set π β π π β β is called the set of imaginary numbers π
β β© π = 0
β β π = β
β is the set of complex numbers. It is the cartesian product of β and π.This means that each element of β consists 2 numbers: a real number coupled to an imaginary number.
It can be written as a coordinate pair : z = π₯, π¦ with π₯, π¦ β β
It is customary to write as a sum: π§ = π₯ + ππ¦ with π₯, π¦ β β
Real & Imaginary part
Introduction to complex numbers 3
π§ = π₯ + ππ¦ is a complex number
With β¦a real part π π π§ = π₯an imaginary part Im π§ = π¦ π§1
π§2
Example
Real axis
Imaginary axis
Complex plane
π π π§1 = 2 πΌπ π§1 = 2
π π π§2 = 3 πΌπ π§2 = β4
Multiplication with a real number
Introduction to complex numbers 4
π§
π§ = π₯ + ππ¦ β β
π β β
π β π§ = π β π₯ + ππ¦ = ππ₯ + πππ¦ 3π§
Norm
Introduction to complex numbers 5
π§ = π₯ + ππ¦
π§1
π§2
Example
π§ = π₯2 + π¦2
The norm of a complex nr is a measure of its magnitude.It equals the distance from the origin.
π§1 = 22 + 22 = 2 2
π§2 = 32 + 42 = 5
5
Addition
Introduction to complex numbers 6
π§1 = π₯1 + ππ¦1 π§2 = π₯2 + ππ¦2
π§1 + π§2 = π₯1 + ππ¦1 + π₯2 + ππ¦2
π§1 + π§2 = π₯1 + π₯2 + π π¦1 + π¦2 π§1
π§2
π§1 + π§2
Compare head-to-tail method
in physics
Subtraction
Introduction to complex numbers 7
π§1 = π₯1 + ππ¦1 π§2 = π₯2 + ππ¦2
π§1
π§2
π§1 β π§2 = π₯1 β π₯2 + π π¦1 β π¦2
π§1 β π§2 = π₯1 + ππ¦1 β π₯2 + ππ¦2 π§1 β π§2
π§2 β π§1
Compare difference
vector in physics
Multiplication
Introduction to complex numbers 8
π§1 = π₯1 + ππ¦1 π§2 = π₯2 + ππ¦2
π§1 β π§2 = π₯1 + ππ¦1 β π₯2 + ππ¦2
π§1 β π§2 = π₯1π₯2 + ππ₯1π¦2 + ππ¦1π₯2 + π2π¦1π¦2 π§1
π§2
π§1 β π§2
π§1 β π§2 = π₯1π₯2 + ππ₯1π¦2 + ππ¦1π₯2 β π¦1π¦2
π§1 β π§2 = π₯1π₯2 β π¦1π¦2 + π π₯1π¦2 + π₯2π¦1
2 + 2π β 1 β 2π = 2 β 1 β 2 β β2 + π 2 β β2 + 1 β 2 = 6 β 2π
Example
Complex conjugate
Introduction to complex numbers 9
π§ = π₯ + ππ¦
π§1
π§2β
Example
The conjugate of a complex nr has a reversed imaginary part.
π§β = π₯ β ππ¦
π§1β
π§2
β2 + 2π β = β2 β 2π
3 β 4π β = 3 + 4π
Note:
1: π§β β = π§2: if π§β = π§ then π§ β β
Complex conjugate & Norm
Introduction to complex numbers 10
π§ = π₯ + ππ¦π§β
π§β = π₯ β ππ¦
π§
π§ β π§β = π₯ + ππ¦ β π₯ β ππ¦
π§ β π§β = π₯2 β ππ₯π¦ + ππ¦π₯ + ππ¦ βππ¦
π§ β π§β = π₯2 + π βπ π¦2
π§ β π§β = π₯2 + π¦2
π§ β π§β = π§ 2
Division
Introduction to complex numbers 11
π§1 = π₯1 + ππ¦1 π§2 = π₯2 + ππ¦2
π§1
π§2
π§1
π§2
π§1π§2
=π§1π§2
βπ§2β
π§2β =
π§1 β π§2β
π§22
Example
3 + 2π
1 β π=
3 + 2π β 1 + π
1 β π 2=
3 β 2 + π 3 + 2
2=
1
2+ 2
1
2π
1 β π
3 + 2π=
1 β π β 3 β 2π
3 + 2π 2=
3 β 2 + π 3 + 2
13=
1
13+
5
13π
π§2
π§1
END
Introduction to complex numbers 12
DisclaimerThis document is meant to be apprehended through professional teacher mediation (βlive in classβ) together with a mathematics text book, preferably on IB level.
