pm [b07] exponent partner

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EXPONENTSPM [B07] The partner of 𝑖𝑖


The heavenly Inspired Rendezvous

While cos 𝜃𝜃 and 𝑖𝑖 sin𝜃𝜃are proceeding fast to their destination . . . .

There is another element hurrying on its way to meet them . . .


Exponents

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


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.


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.

Exponentor index or power

Base or base number


𝟔𝟔

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


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:

𝑎𝑎𝑥𝑥

This base can be anything

This power can be anything


Graphs of 𝑦𝑦 = 𝑎𝑎𝑥𝑥

The diagram shows the function 𝑓𝑓 𝑥𝑥 = 𝑎𝑎𝑥𝑥 for several values of a.

𝑒𝑒 is the value of 𝑎𝑎 such that the gradient of 𝑓𝑓 𝑥𝑥 = 𝑎𝑎𝑥𝑥 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


Why us 𝑒𝑒 chosen?

In more analytical terms, this means that this is the value of 𝑎𝑎which makes the derivative of 𝑒𝑒𝑥𝑥 equal to 𝑒𝑒𝑥𝑥, rather than a constant multiple of 𝑒𝑒𝑥𝑥.

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


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


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

pm [b07] exponent partner

Science