Ordinary vs. Binary numbers

What is the difference between ordinary numbers and bi­nary numbers and what are the advantages of each?

The ordinary numbers we use are "ten-based". That is, they are written as powers of ten. What we write as 7291 is really 7X10³ plus 2x 10² plus 9X10¹ plus 1x10⁰. Remem­ber that 10³=10x 10X10=1000; that 10²=10x 10=100; that 10¹=10 and 10⁰—1, so that 7291 is 7x1000 plus 2x100 plus 9x 10 plus 1. We say this when we read the number aloud. It is "seven thousand two hundred ninety (nine tens) one".

We have grown so accustomed to the use of powers of ten that we just write the digits by which they are multiplied, 7291 in this case, and ignore the rest.

But there is no magic about powers of ten. The power of any other number higher than one would do. Suppose, for instance, we wanted to write number 7291 in terms of powers of eight. Remember that 8⁰=1; 8¹=8; 8²=8x8=64; 8³= =8X8X8=512; and 8⁴=8x8x8x8=4096. The number 7291 can then be written as 1 X8⁴ plus 6x8³ plus 1 X8² plus 7X8¹ plus 3x8⁰. (Work it out and see for yourself). If we write only the digits we have 16173. We can say, then, that (8-based) =7291 (10-based).

The advantage of the 8-based system is that you only need to memorize seven digits besides 0. If you try to use the digit 8, you might have 8x8³ which is equal to 1 X8⁴, so you can always use a 1 instead of an 8. Thus 8 (10-based) =10 (8-based); 89 (10-based) = 131 (8-based) and so on. On the other hand, there are more total digits to the number in the 8-based system than in the 10-based system. The smaller the base, the fewer different digits but the more total digits.

If you used a 20-based system, the number 7291 becomes 18 X 20² plus 4X20¹ plus 11x20⁰. If you wrote 18 as # and 11 as % you could say that # 4% (20-based)=7291 (10- based). You would have to have 19 different digits in a 20- based system but you would have fewer total digits per num­ber.

Ten is a convenient base. It gives us not-too-many dif­ferent digits to remember and not-too-many separate digits in a given number.

What about a number based on powers of two — a 2- based number? It is this which is a "binary number", from a Latin word meaning "two at a time".

The number 7291 equals 1X2¹² plus lx2^u plus lx2¹⁰ plus 0X2⁹ plus 0x2⁸ plus 0x2⁷ plus lX2⁶ plus lx2⁵ plus 1x2⁴ plus 1X2³ plus 0x2² plus 1X2¹ plus 1x2⁰. (Work it out and see, remembering that 2⁹, for instance, is nine two's multiplied together: 2x2x2x2x2x2x2x2x2=512). If we write only the digits we have 1110001111011 (2-based) =7291 (10-based).

Binary numbers contain only l's and 0's, so that addi­tion and multiplication are fantastically simple. However, there are so many digits altogether in even small numbers like 7291 that it is fantastically easy for the human mind to become confused. A computer, however, can use a two-way switch. In one direction, current-on, it can symbolize a 1; in the other di­rection, current-off, a 0. By manipulating the circuits so that the switches turn on and off in accordance with binary rules of addition and multiplication, the computer can perform arithmetical computations very quickly. It can do it much more quickly than if it had to work with gears marked from 0 to 9, as in ordinary desk calculators based on the decimal or 10-based system.