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EXPONENTSPM [B07] The partner of ππ
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The heavenly Inspired Rendezvous
While cos ππ and ππ sinππare 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 β 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 Γ 1024 kg and that of the red blood cell is written as 1 Γ 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.
Exponentor index or power
Base or base number
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ππ
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 Γ 3 = 9
In words, 32 is called β3 to the power of 2, or β3 squaredβ or β2 to the second powerβ. For 3 Γ 3 Γ 3 Γ 3 Γ 3 Γ 3, it is simpler to write and easier to read as 36 .For any numbers, the general form is:
πππ₯π₯
This base can be anything
This power can be anything
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Graphs of π¦π¦ = πππ₯π₯
The diagram shows the function ππ π₯π₯ = πππ₯π₯ for several values of a.
ππ is the value of ππ such that the gradient of ππ π₯π₯ = πππ₯π₯ at π₯π₯ =0 equals 1. This is the blue curve, πππ₯π₯.
Functions 2π₯π₯ (dotted curve) and 4π₯π₯(dashed curve) are also shown; they are not tangent to the line of slope 1 (red). Wikipedia
Its value of ππ can be obtained in many ways. The closiest value is ππ = 2.718281828459 . . .
Picture source: Wikipedia
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Why us ππ chosen?
In more analytical terms, this means that this is the value of ππwhich makes the derivative of πππ₯π₯ equal to πππ₯π₯, rather than a constant multiple of πππ₯π₯.
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 ππ
This number e is of eminent importance in mathematics, alongside 0, 1, ππ and ππ. 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 ππ TO FORM EULER VECTORTo be continued in PM [B08]