Numerical Equivalents
The Progression of Binary to Decimal Systems
A: In the Decimal (Deci) System of numbering, there are Ten digit values. The values from 0 to 9 can be counted with a single digit. (this is the 1's column) To count beyond the number 9, the place value to the left (the 10's column) must increase by 1 for each 10 values counted.
If we were counting or numbering with a system of only Nine digit values, (0 to 8) we could only count up to the value of 8 before having to increase the value of the next column to the left by 1.
The same rules would apply to numbering systems with even fewer digits. Since Zero must always be the first digit, (representing Space) the highest number that can be counted in any system is always one less than the total number of digits used in that counting system. An Eight digit system (octa) would only allow counting as far as 7 then. A Seven digit system could only count as far as 6, and so on...The simplest counting system that can be used is the Binary (Bina) System. With only the Zero and One to work with. This, of course, is the system that DC voltage based computers must use to manipulate information.
B: The place values increase by multiples of the base number of the system. With the Decimal System the first place value to the left has been shown to be the number 10. (10x1=10) The next place value in this system will then be 10x10=100. The next place value after that will be 10x100=1000, then 10x1000=10000 and so on...
All of the other numbering systems operate the same way. The Nine base system must increase the next place value at 9x1=9, 9x9=81, 9x81=729 and so on... To find the value at which the place value must change next: Multiply the base by One first, then continue to multiply the result by the base.
The Binary System must then increase the place value as 2x1=2, 2x2=4, 2x4=8 and so on...It becomes apparent in the table below how it quickly generates numbers with many place values.
Decimal through Binary:
(Table represents values from Zero to 37)
Deci: Noni: Octa: Sept: Hexa: Pent: Quad: Trin: Bina: 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 10 3 3 3 3 3 3 3 10 11 4 4 4 4 4 4 10 11 100 5 5 5 5 5 10 11 12 101 6 6 6 6 10 11 12 20 110 7 7 7 10 11 12 13 21 111 8 8 10 11 12 13 20 22 1000 9 10 11 12 13 14 21 100 1001 10 11 12 13 14 20 22 101 1010 11 12 13 14 15 21 23 102 1011 12 13 14 15 20 22 30 110 1100 13 14 15 16 21 23 31 111 1101 14 15 16 20 22 24 32 112 1110 15 16 17 21 23 30 33 120 1111 16 17 20 22 24 31 100 121 10000 17 18 21 23 25 32 101 122 10001 18 20 22 24 30 33 102 200 10010 19 21 23 25 31 34 103 201 10011 20 22 24 26 32 40 110 202 10100 21 23 25 30 33 41 111 210 10101 22 24 26 31 34 42 112 211 10110 23 25 27 32 35 43 113 212 10111 24 26 30 33 40 44 120 220 11000 25 27 31 34 41 100 121 221 11001 26 28 32 35 42 101 122 222 11010 27 30 33 36 43 102 123 1000 11011 28 31 34 40 44 103 130 1001 11100 29 32 35 41 45 104 131 1002 11101 30 33 36 42 50 110 132 1010 11110 31 34 37 43 51 111 133 1011 11111 32 35 40 44 52 112 200 1012 100000 33 36 41 45 53 113 201 1020 100001 34 37 42 46 54 114 202 1021 100010 35 38 43 50 55 120 203 1022 100011 36 40 44 51 60 121 210 1100 100100 37 41 45 52 61 122 211 1101 100101