Number Systems with Bases other than 10

 

Counting in Different Bases Decimal (10) Binary (2) Octal (8) Hex (16) 0 0 0 0 1 1 1 1 2 10 2 2 3 11 3 3 4 100 4 4 5 101 5 5 6 110 6 6 7 111 7 7 8 1000 10 8 9 1001 11 9 10 1010 12 A 11 1011 13 B 12 1100 14 C 13 1101 15 D 14 1110 16 E 15 1111 17 F 16 10000 20 10 17 10001 21 11 18 10010 22 12 19 10011 23 13 20 10100 24 14

Binary | Octal | Hexadecimal   Using different numbers bases "fell out" of the mathematics curriculum for a period of time as people felt that the base 10 was the only base used in everyday life. However, with the prevalence of technology and computers, being able to move easily in computations from one base system to another is almost as similar as in moving easily from speaking in one language to another. We will focus on the three number bases used by computers: the binary, the octal, and hexadecimal, for: "Three numeration systems, binary, octal and hexadecimal, are used when dealing with the internal workings of computers. To be a knowledgeable, efficient programmer requires at least a rudimentary understanding of how they work and how they are interrelated." Numeration Systems with tables for binary, octal, and hexadecimal systems http://www.macdonald.egate.net/CompSci/Pascal/hnumeration.html

 

Binary or Base 2

The binary number base uses only 2 symbols: 0 and 1. The numbers are then:

Binary Number	Decimal Number	A base 2 number would be written in expanded form as: Or converting the above number to decimal would be: 8 + 0 + 0 + 0 = 8 To add two binary numbers, the following chart is true: Binary Addition + 0 1 0 0 1 1 1 10 So, the two possibilities are: A ) OR B) which means: each time "carrying" over the "1."
1	1
10	2
11	3
100	4
101	5
110	6
111	7
1000	8
1001	9
1010	10
1011	11
1100	12
1101	13
1110	14
1111	15
10000	16
10001	17
10010	18
10011	19
10100	20

 

Octal or Base 8

The octal number base uses only 8 symbols: 0, 1, 2, 3, 4, 5, 6, and 7. The numbers are then:

Octal Number	Decimal Number	A base 8 number would be written in expanded form as: which would convert to decimal form as: 16 + 1 = 17 The octal addition chart is: + 0 1 2 3 4 5 6 7 ------------------------------------------------------------------ 0 | | | | | | | | | | 0 1 2 3 4 5 6 7 1 1 2 3 4 5 6 7 10 2 2 3 4 5 6 7 10 11 3 3 4 5 6 7 10 11 12 4 4 5 6 7 10 11 12 13 5 5 6 7 10 11 12 13 14 6 6 7 10 11 12 13 14 15 7 7 10 11 12 13 14 15 16 Using the above table: Check if your calculator offers the option of different bases. If it does not, then you can always use the calculator at: Converting binary to octal to hexadecimal numbers http://www.terra.cc.oh.us/dlm270/session1.htm
1	1
2	2
3	3
4	4
5	5
6	6
7	7
10	8
11	9
12	10
13	11
14	12
15	13
16	14
17	15
20	16
21	17
22	18
23	19
24	20

Hexadecimal or Base 16

The hexadecimal number base uses 16 symbols. Since the decimal system only has 10 symbols,
6 more must be added. Those six are taken from the first letters of the alphabet,
so the symbols are: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, and F. The numbers are then:

Hexadecimal Number	Decimal Number	A base 16 number would be written in expanded form as: 21 = (2 x 16) + ( 1 + 1) which would convert to decimal from as: 32 + 1 = 33   The hexadecimal system is most often seen in designing web pages, where all color combinations are represented by hexadecimal numbers. There are 216 cross platform colors where each RGB color (red, green blue) is expressed in a two digit code. For this page, the background has color code ="FFFFFF" which is white. Thus, this code is broken in three position codes, since each color ranges from 0 - 255. So white is made up of FF for red, FF for green, and FF for blue. FF = 255 which is the maximum amount of color possible. The links on this page have color code 003366. What would that color be? i. e. how much red? how much green? how much blue?
1	1
2	2
3	3
4	4
5	5
6	6
7	7
8	8
9	9
A	10
B	11
C	12
D	13
E	14
F	15
10	16
11	17
12	18
13	19
14	20

  Week 3