Introduction to Math for computer programming

This page is currently only a very very small fraction of what I intend it to be. If you have idea for topics or the heavenly inspiration to contribute something great to this topic: Please click that "edit" button and do it!

Please try to keep this simple and easy to understand without prior knowledge of anything beyond introductory algebra unless the topic warrants it. If you know calculus: Good for you. That doesn't mean the average viewer of this page does.

Number systems

The following is a list of common number systems. The names of which are derived from latin.

Base	Name	Number set	Commonly used for
1	Unary	1	*Tallying ()**
2	Binary	01	Computers ()**
3	Ternary	012
4	Quaternary	0123
5	Quinary	01234
6	Senary	012345
7	Septenary	0123456
8	Octal	01234567	Computers ()*
9	Nonary	012345678
10	Decimal	0123456789	Ordinary Counting!
11	Undenary	0123456789A
12	Duodecimal	0123456789AB
16	Hexadecimal	0123456789ABCDEF	Computers ()
20	Vigesimal	Too long!
60	Sexagesimal	Too long!	Clocks

Unary is of very limited usefulness since only positive integers can be expressed in it. The same is more-or-less true for any numbering system whose absolute value is equal to one.
Binary is used to portray the specific values of on (1) and off (0) used in digital electronics like computers
Octal was more commonly used in the early days of computing and especially on computers whose byte size was divisible by three. It is still used in some computing cultures and for some specific purposes but its use has been largely supplanted by hexadecimal.
Hexadecimal is often used to portray binary numbers in a smaller, neater-seeming form

Determining number value

Because people tend to be most familiar with dealing with numbers in Decimal, that is the basis we will use to get a feel for a number's value.

You may determine the decimal value of a number in any given base using the following formula:

Where:

K = Decimal value of a given number
d = Decimal value of a value in a specified base
B = Decimal value of the base of the number being evaluated
i = Decimal value of the index at which the number currently being evaluated is located (0 = right-most; 1, 2, 3, etc. are 1, 2, 3, etc. places from the right-most)

In this scheme, the right-most digit in any number in any base is:

The digit place with the smallest potential value
Has an index (i) value of 0

It is common to determine the value of a number, starting with the right-most value. That's what I'm going to be doing in the following examples.

Converting decimal to decimal

Converting the decimal number 931 to decimal:

1 * 10^0 =   1
3 * 10^1 =  30
9 * 10^2 = 900

900 + 30 + 1 = 931

Converting the decimal number 23 to decimal:

3 * 10^0 =  3
2 * 10^1 = 20

20 + 3 = 23

Converting binary to decimal

Converting the binary number 0b01001100 to decimal:

0b00000000 = 0 * 2^0 =  0
0b00000000 = 0 * 2^1 =  0
0b00000100 = 1 * 2^2 =  4
0b00001000 = 1 * 2^3 =  8
0b00000000 = 0 * 2^4 =  0
0b00000000 = 0 * 2^5 =  0
0b01000000 = 1 * 2^6 = 64
0b00000000 = 0 * 2^7 =  0

64 + 8 + 4 = 76

Converting the binary number 0b11111111 to decimal:

0b00000001 = 1 * 2^0 =   1
0b00000010 = 1 * 2^1 =   2
0b00000100 = 1 * 2^2 =   4
0b00001000 = 1 * 2^3 =   8
0b00010000 = 1 * 2^4 =  16
0b00100000 = 1 * 2^5 =  32
0b01000000 = 1 * 2^6 =  64
0b10000000 = 1 * 2^7 = 128

128 + 64 + 32 + 16 + 8 + 4 + 2 + 1 = 255

Converting hexadecimal to decimal

Converting the hexadecimal number 0x16 to decimal:

6 * 16^0 = 6
1 * 16^1 = 16

16 + 6 = 22

Converting the hexadecimal number 0xBEEF to decimal:

F = 15 * 16^0 =     15
E = 14 * 16^1 =    224
E = 14 * 16^2 =  3,584
B = 11 * 16^3 = 45,056

45,056 + 3,584 + 224 + 15 = 48,879

Decimal

Place name	Decimal Value	Scientific notation
Millions	1,000,000	10⁶
Hundred-thousands	100,000	10⁵
Ten-thousands	10,000	10⁴
Thousands	1,000	10³
Hundreds	100	10²
Tens	10	10¹
Ones	1	10⁰

10⁶	+	10⁵	+	10⁴	+	10³	+	10²	+	10¹	+	10⁰	=	Expressed as	=	Decimal value
1		2		3		4		5		6		7		1234567		1234567
0		0		0		0		0		0		0		0		0
4		3		7		5		9		1		0		4,375,910		4,375,910
0		0		0		1		0		2		4		1,024		1,024
9		9		9		9		9		9		9		9,999,999		9,999,999

Binary

Binary relys on only two digits which are usually written 0 and 1, but are easilly encoded as "off" and "on" in switch circuits, which makes Binary the language of choice in low-level Computing and Digital Electronics.

Place name	Decimal Value	Scientific notation
One-twenty-eights	128	2⁷
Sixty-fours	64	2⁶
Thirty-twos	32	2⁵
Sixteens	16	2⁴
Eights	8	2³
Fours	4	2²
Twos	2	2¹
Ones	1	2⁰

2⁷	+	2⁶	+	2⁵	+	2⁴	+	2³	+	2²	+	2¹	+	2⁰	=	Expressed as	=	Decimal value
0		0		0		0		0		0		0		0		0b00000000		0
0		1		0		0		0		0		0		1		0b01000001		65
1		0		0		0		0		0		0		0		0b10000000		128
1		0		0		1		1		1		1		1		0b10011111		159
1		1		1		1		1		1		1		1		0b11111111		255

Hexadecimal

Hexadecimal is commonly used to represent binary data, as it occupies only one quarter of the space when written down.

Place name	Decimal Value	Scientific notation
No thanks!	16,777,216	16⁶
No thanks!	1,048,576	16⁵
No thanks!	65,536	16⁴
40-96s	4,096	16³
256s	256	16²
Sixteens	16	16¹
Ones	1	16⁰

16⁷	+	16⁶	+	16⁵	+	16⁴	+	16³	+	16²	+	16¹	+	16⁰	=	Expressed as	=	Decimal value
0		0		0		0		0		0		0		0		0x00000000		0
0		0		0		0		0		0		4		1		0x41		65
D		E		A		D		B		E		E		F		0xDEADBEEF		3,735,928,559
0		0		0		0		0		F		7		3		0xF73		3,955
F		F		F		F		F		F		F		F		0xFFFFFFFF		4,294,967,295

Corollary

In all number systems, prefixed zeros can be ignored (do not affect the value represented), however common practice is to prefix all octal numbers with a leading 0 (zero), all hexadecimal values with 0x and all binary numbers with zeros so that it contains a number of digits which is divisible by eight.

Decimal Value	Binary Value	Hexadecimal Value
000	00000000	00
001	00000001	01
002	00000010	02
003	00000011	03
004	00000100	04
005	00000101	05
006	00000110	06
007	00000111	07
008	00001000	08
009	00001001	09
010	00001010	0A
011	00001011	0B
012	00001100	0C
013	00001101	0D
014	00001110	0E
015	00001111	0F
016	00010000	10
017	00010001	11
018	00010010	12
019	00010011	13
020	00010100	14
021	00010101	15
022	00010110	16
023	00010111	17
024	00011000	18
---	--------	--
254	11111110	FE
255	11111111	FF

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