Number bases/Introduction

Note: incomplete

Prerequisites

Natural numbers

Notating numbers

Let's assume that you want to denote all natural numbers with a finite number of symbols (let's say, 5- feel free to use other numbers instead). We may start by assigning symbols to letters, as so:

0, 1, 2, 3, 4... Oops! We're in danger of exceeding our limit. However, this is fine- we can just add another digit, counting the number of 5's in the number:

0, 1, 2, 3, 4, 10, 11, 12, 13, 14, 20, 21, 22, 23, 24, 30, 31, 32, 33, 34, 40, 41, 42, 43, 44... We've encountered the same problem again! But this is fine; when we arrive at five fives, we can notate that with a third digit, as so:

0, 1, 2, 3, 4, 10, 11, 12, 13, 14, 20, 21, 22, 23, 24, 30, 31, 32, 33, 34, 40, 41, 42, 43, 44, 100, 101, 102, 103, 104, 110, 111, 112, 113, 114, 120, 121, 122, 123, 124, 130, 131, 132, 133, 134, 140, 141, 142, 143, 144, 200, 201, 202, 203, 204, 210, 211, 212, 213, 214, 220, 221, 222, 223, 224, 230, 231, 232, 233, 234, 240, 241, 242, 243, 244, 300, 301, 302, 303, 304, 310, 311, 312, 313, 314, 320, 321, 322, 323, 324, 330, 331, 332, 333, 334, 340, 341, 342, 343, 344, 400, 401, 402, 403, 404, 410, 411, 412, 413, 414, 420, 421, 422, 423, 424, 430, 431, 432, 433,434, 440, 441, 442, 443, 444... We've encountered the same problem again, but it's clear that we can just keep adding another digit.

This is called base 5. However, there's nothing special about the number 5 here. We could just as easily use any other number; computers use base 2 (symbols: 0, 1) and programmers use base 16 (symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F). The most common system is base 10, or decimal. This fits well with how we naturally count in English, as shown below:

four hundred	and thirty	two
4	3	2

Place value charts

This correspondence allows us to use 'place value' charts to denote numbers:

UK:

...	billi-						milli-						-
...	-ards			-ons			-ards			-ons			thousands			ones
...	hundreds	tens	ones	hundreds	tens	ones	hundreds	tens	ones	hundreds	tens	ones	hundreds	tens	ones	hundreds	tens	ones
...

Or in abbreviated form:

...	sext-						quint-						quadr-						trillions						billions						millions						ones
...	-illiard			-illion			-illiard			-illion			-illiard			-illion			trilliards			trillions			billiards			billions			milliards			millions			thousands			ones
...	H	T	O	H	T	O	H	T	O	H	T	O	H	T	O	H	T	O	H	T	O	H	T	O	H	T	O	H	T	O	H	T	O	H	T	O	H	T	O	H	T	O
...

Essentially, the first part of the number tells you how many powers of a million, and the second part tells you if you should add one half. Examples include (note- prescribe the examples in a random order):

Billion: bi=2, so it's 1,000,000,000,000 (twice as many zeros as a million)

Billiard: bi=2, -ard counts as 1/2, so it's 1,000,000,000,000,000 (two-and-a-half times as many zeros as a million)

Trillion: tri=3, so it's 1,000,000,000,000,000,000 (thrice as many zeros as a million)

Trilliard: tri=3, -ard counts as 1/2, so it's 1,000,000,000,000,000,000,000 (three-and-a-half times as many zeros as a million)

Quadrillion: quad=4, so it's 1,000,000,000,000,000,000,000,000 (four times as many zeros as a million)

Quadrilliard: tri=4, -ard counts as 1/2, so it's 1,000,000,000,000,000,000,000,000,000 (four-and-a-half times as many zeros as a million)

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