Number Systems

History of Number

Binary has two symbols (including the zero: no quantity but vital for position), 0 and 1. It is a number system of magnitude by position, as is the Hindu-Arabic decimal system with any of 10 symbols (including the zero) to any position. The Roman system needs counting along, where some symbols reduce the size of the list. This is why they justify to the left whereas both decimal and binary justify to the right. Another system is hexadecimal (below to 255 decimal), also justified to the right, which uses 0 to 9 and then A to F.
To convert a decimal number to binary, change it to groups of powers of 2 as in 21=2, 22=4, 23=8, 24=16, 25=32, 26=64, and so on). Go for the highest number to the 2 power contained within first and get the lesser powers downwards subsequently until nothing or one remainder is left (0 or 1 unit). For example, 13 is one of 8 plus one of 4 plus nothing of 2 plus one of 1 remaining (or 1 of 23 plus 1 of 22 plus 0 of 21 plus 1). Each power of two is doubling and means an extra 0 in position and therefore this example is 1000 (8) plus 100 (4) plus 00 plus 1 (1) or therefore (slotting them in - but look at the emboldened ones and zero in the brackets above!) 1101.

Decimal Roman Binary
1 I 1
2 II 10
3 III 11
4 IV 100
5 V 101
6 VI 110
7 VII 111
8 VIII 1000
9 IX 1001
10 X 1010
11 XI 1011
12 XII 1100
13 XIII 1101
14 XIV 1110
15 XV 1111
16 XVI 1 0000
17 XVII 1 0001
18 XVIII 1 0010
19 XIX 1 0011
20 XX 1 0100
21 XXI 1 0101
29 XXIX 1 1101
30 XXX 1 1110
32 XXXII 10 0000
40 XL 10 1000
50 L 11 0010
60 LX 11 1100
64 LXIV 100 0000
90 XC 101 1010
99 XCIX 110 0011
100 C 1100100
128 CXXVIII 1000 0000
200 CC 1100 1000
256 CCLVI 1 0000 0000
300 CCC 1 0010 1100
400 CD 1 1001 0000
500 D 1 1111 0100
512 DXII 10 0000 0000
600 DC 10 0101 1000
900 CM 11 1000 0100
1000 M 11 1110 1000
1024 MXXIV 100 0000 0000
1500 MD 101 1101 1100
2000 MM 111 1101 0000
4000 MV 111110100000
5000 V 1 0011 1000 1000
10,000 X 10 0111 0001 0000
20,000 XX
100 1110 0010 0000
100,000 C 1100 0011 01010 0000
Note: the Roman symbols for 5000, 10,000 and 100,000 have a bar above the V, X and C respectively, here shown with the bar just to the left. Also note that the comma in the decimal system is for ease of sight only to delimit thousands at and above ten thousand and has no mathematical quality. Equally, the sometimes used convention of grouping binary numbers into blocks of four is for seeing purposes only.

 

Adrian Worsfold