The inscrutable imaginary number, so useful and yet so intriguing. Explain why this is so and how important it is to quantum mechanics, resulting in the ultimate quantum.

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THE COMPLEX COORDINATE SYSTEM
PM [B03]
An imaginary number is unreal
and does not belong to this world.
But It has been proven so useful in
science and engineering that
mathematicians and scientists
embrace the concept like a real
valueless jewel.

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𝑖3
𝑖0
𝑖1
𝑖2
𝑖-2
𝑖-3
𝑖-4
𝑖5
𝑖4
𝑖6
The Eternal Cycle
It seems that the cyclicality of
the 𝑖-multiplication can go on
forever in both directions from
the negative realm to the
positive realm.

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The Essential Quadrant
But the basic elements are only
four and this group creates a
rigid configuration with its
components orthogonal to each
other.
i1
i0
i3
i2
i4

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The Argand plane
𝑖0
= 1 unit for the +x axis
𝑖1
= 𝑖 unit for the +y-axis
𝑖2
= 𝑖 × 𝑖 = −1 unit for the –x axis
𝑖3
= i × 𝑖 × 𝑖 = −𝑖 unit for the –y axis
𝑖4
= 1 back to the +x-axis
1 = 𝑖0
𝑖 = 𝑖1
−1 = 𝑖2
−𝑖 = 𝑖3
1 = 𝑖4
This cyclic feature of the imaginary number
enabled Jean-Robert Argand (1768-1822) to set
up a coordinate system after his name – the
Argand plane:
The Argand plane

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The Real & Imaginary
Axes
In mathematics an Argand coordinate
system is made up of a standard
number line lying horizontally as the 𝑥
axis. The numbers increases positively
in magnitude to the right, and
negatively increase in magnitude to the
left. This axis is called the ‘real axis’.
At 0 on this 𝑥-axis, we can erect an
imaginary line as the 𝑦-axis.
Now ‘positive’ imaginary numbers grow
in magnitude upwards, and "negative"
imaginary numbers dwindle in
magnitude downwards. This vertical
axis is called the ‘imaginary axis’ and is
denoted by 𝒊 or simply 𝐼𝑚.
Real axis
Imaginaryaxis
𝑖
-𝑖
r-r

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The Complex Number
On the Argand plane, any vector
can be written as:
𝑂𝐴 = 𝑎 + 𝑖𝑏
By tradition, 𝑎 is called the real
part; b with the 𝑖 is called the
imaginary part.
𝑎 + 𝑖𝑏
Is called a complex number.
Real axis
Imaginary axis
𝑖
-𝑖
r-r
𝜃
O
A
𝑎
𝑖𝑏

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A spooky world
So a complex coordinate systems
can be broken up into three parts:
1. The totally real.
2. The totally imaginary.
3. The complex.
The most importance difference
between a complex number and an
ordinary Cartesian vector is that
except where the thin horizontal
line is, all the space are occupied
by unreal or semi-unreal beings.
The purely
imaginary
The purely real
The complex The complex
The complexThe complex

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PLANE POLAR COORDINATES
To be continued on [PM B04]