Binary (and other) Number Systems

Presently, all digital computers rely on the binary number system. The simple reason is that present-day technology allows to store information reliably only by turning something on or off or on a CD to make a small indentation into the surface or not.
Mathematically we state this in terms of a zero (0) or a (1) and call a storage unit which can change its state between 0 and 1 a "bit". Eight consecutive (in memory) bits are called a byte and all computations computers perform are conducted in terms of this binary ( 0 and 1 ) number system.

How does it work ? Well, there are a lot of similarities between the usual decimal and the binary system :
In the decimal system, when we count some things and write as we count, we would write down all the non-zero digits ( 1, 2, ..., 9) and once we have exhausted the non-zero digits we proceed with a 1 infront of a 0 ( as in 10 ). Then we keep on playing with the non-zero digits as we keep on counting until we reach 99 and then 100, again adding a zero.
In the binary system we do the same keeping in mind that we have only a single non-zero digit, namely the 1. As a consequence the decimal numbers one through nine become :

1, 10, 11, 100, 101, 110 , 111, 1000, 1001
Addition, subtraction, multiplication, and division can be handled now as well ( 100 + 111 = 1011 , using carry-overs as in the decimal system )
An alternate way of looking at numbers in the decimal system is :
1507 = 1*103 + 5*102 + 0*101 + 7*100
Remember, 100 = 1. In the very same fashion we can write out binary numbers, this time using the 2 as base :
10111100011 = 1*210 + 0*29 + 1*28 + 1*27 + 1*26 + 1*25 + 0*24 + 0*23 + 0*22 + 1*21 + 1*20


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Zig Herzog © 2014
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Last revised: 08/22/13