I recall my study of Maths in high school:-
1. First we learn about Integers : 0, 1, 2, 3,.. positive and negative
2. Then about Decimals : 0.1, 0.23, 3.5 etc
3. Follow by Fractions in the form of a/b where a and b are integers.
4. By converting fractions to decimals, we discover infinite but
repetitive sequence
e.g. 2/9 = 0.2222....; 17/27 = 0.629629629...with infinite repetition
of 629
5. And the study of geometry and algebra introduce Irrational numbers.
e.g. square root of 2 = 1.414213562373.... to infinity small without
any repetitive sequence.
Basically, an irrational number is one that cannot be expressed by a
fraction of integers.
And any numbers that can be expressed by a fraction is called rational
number.
6. But most irrational numbers can be obtained from solving a polynomial
equations
e.g. x**2-2 = 0 gives rise to x = +/- sqrt root 2.
7. And we learn about Imaginary number from solving equation such as
X** 2 + 4 = 0 gives rise to x = +/- 2i
8. Finally, Pi and "e" were introduced as Transcendental numbers :-
Those irrational numbers that cannot be derived from the roots of any
Polynomial Equations!
Integers, Decimals, Fractions, Irrational, and even Transcendental
Numbers, they
are all FINITE and PRECISE (can be precisely defined). Such properties
hold irregard
what sort of Numeric Representation (such as Binary, Hexadecimal etc).
9. Euler had marvellously combined all the above into one equation
"e" to the power of i (imaginary) * PI = -1.
Thanks to my Maths teachers for showing the wonders of the Numbering
System
HweeBoon
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