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 2^{1}=2, 2^{2}=4, 2^{3}=8, 2^{4}=16, 2^{5}=32, 2^{6}=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 2^{3} plus 1 of 2^{2} plus 0 of 2^{1} 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 | M¯V | 111110100000 |
5000 | ¯V | 1 0011 1000 1000 |
10,000 | ¯X | 10 0111 0001 0000 |
20,000 |
¯X¯X |
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