The essential partner of i. Together they form the most beautiful and powerful partnership in the universe.

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EXPONENTS
PM [B07] The partner of 𝑖𝑖

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The heavenly Inspired Rendezvous
While cos 𝜃𝜃 and 𝑖𝑖 sin 𝜃𝜃
are proceeding fast to
their destination . . . .
There is another
element hurrying on its
way to meet them . . .

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Exponents
This is the element of exponent
– another essential part of
numerical manipulation that
makes up the great vector.

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Large & Small Numbers
In studying nature, scientists
often come across very small
and very large numbers. For
example:
1. The mass of a red blood cell
is about 0.000,000,000,000,1
kilogram.
2. The mass of the earth is
about 5,980,000,000,000,
000,000,000,000 kilogram.

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Exponent Notations
To curtail the inconvenience of writing
a long chain of zeroes, scientists use
the exponential notations. The zeroes
are expressed as powers of 10.
Thus the mass of the earth is written
as 5.98 × 1024 kg and that of the red
blood cell is written as 1 × 10-12
kilogram.
The powers of ten are called the
exponents and they are there to
indicate the number of zeroes before
or after the decimal point, depending
on the positive or negative sign.
Exponent
or index
or power
Base or base number

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𝟔𝟔
It is equivalent to
moving the number
of times to the
power position

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Exponents in general
In general the exponent of a number is an
indication of how many times the number
is to multiply by itself.
In this example, 3 to the power 2:
32 = 3 × 3 = 9
In words, 32 is called “3 to the power of 2,
or “3 squared” or “2 to the second power”.
For 3 × 3 × 3 × 3 × 3 × 3, it is simpler to
write and easier to read as 36 .
For any numbers, the general form is:
𝑎𝑎 𝑥𝑥
This base can
be anything
This power can
be anything

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Graphs of 𝑦𝑦 = 𝑎𝑎 𝑥𝑥
The diagram shows the function
𝑓𝑓 𝑥𝑥 = 𝑎𝑎 𝑥𝑥
for several values of a.
𝑒𝑒 is the value of 𝑎𝑎 such that the
gradient of 𝑓𝑓 𝑥𝑥 = 𝑎𝑎 𝑥𝑥
at 𝑥𝑥 =
0 equals 1. This is the blue curve,
𝑒𝑒𝑥𝑥.
Functions 2𝑥𝑥 (dotted curve) and 4𝑥𝑥
(dashed curve) are also shown;
they are not tangent to the line of
slope 1 (red). Wikipedia
Its value of 𝑒𝑒 can be obtained in
many ways. The closiest value is
𝑒𝑒 = 2.718281828459 . . .
Picture source: Wikipedia

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Why us 𝑒𝑒 chosen?
In more analytical terms, this
means that this is the value of 𝑎𝑎
which makes the derivative of
𝑒𝑒 𝑥𝑥
equal to 𝑒𝑒 𝑥𝑥
, rather than a
constant multiple of 𝑒𝑒 𝑥𝑥
.
Consequently, the exponential
function with base e is
particularly suited to doing
calculus. Choosing e, as opposed
to some other number, as the
base of the exponential function
makes calculations involving the
derivative much simpler.
Wikipedia

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The significance of 𝑒𝑒
This number e is of eminent
importance in mathematics,
alongside 0, 1, 𝜋𝜋 and 𝑖𝑖. All five of
these numbers play important
and recurring roles across
mathematics, and are the five
constants appearing in one
formulation of Euler's identity.
Wikipedia

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UNION WITH 𝑖𝑖 TO FORM EULER VECTOR
To be continued in PM [B08]