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 IMAGINARY NUMBER
PM [B01]

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Solutions to −1
When we try to calculate the
square root of a negative
number, it becomes clear that it
cannot be done with either a
positive or a negative number.
For example:
−1 ≠ ±1
since (−1)2
= +1.
By mathematical definition,
there is no square root to a
negative number.
𝟐𝟐
−𝟏𝟏 =?
−𝟏𝟏

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Imaginary Number 𝒊𝒊
Mathematicians had to invent a
special number 𝑖𝑖, such that
−1 = 𝑖𝑖 (engineers use 𝑗𝑗), so
that the square roots can be
expressed as:
2
−9 =
2
𝑖𝑖29 = 𝑖𝑖
2
9 = 𝑖𝑖 𝑖
2
−25 = 𝑖𝑖
2
25 = 𝑖𝑖 𝑖

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The magical number 𝒊𝒊
Mathematician called this 𝑖𝑖 the
imaginary number.
Thus an imaginary number 𝑖𝑖 may
be defined as a number whose
square is a real number less than
zero.
When an imaginary number is
squared, the result is always
negative, that is:
𝑖𝑖2 = −1
And:
𝑖𝑖 𝑖 2 = −9
𝑖𝑖2
= 𝑖𝑖 × 𝑖𝑖
= −1

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The magical number 𝒊𝒊
An imaginary number 𝑖𝑖 is an
unreal entity. It does not exist in
this world but only in our
imagination. Anything associated
with it will become unreal and
non-existent.

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ORTHOGONAL ROTATION
To be continued in PM [B02]:
Being condemned to
eternal unreality.
ABCC