“The Mathematics of Computers and Logic Circuits”

I.Binary Code

A long time a go, mankind wanted to find another system. Once electricity was invented, man thought up a new number system that would do more. The people of India and the Mayan Indians are thought to have found the zero. Both of these groups used a base system of twenty. Eskimos counted with a base of five. And back to electricity, it only can exist in two stages- on and off. This would use an base of two. The only numbers that are used are 1 and 0. This system was originally discovered but then lost 4,000 years ago in China. There is a country between New Guinea and Cape York of Australia that used a two base system.

Every time a digit is shifted one place, it is multiplied by two. It starts with one value and works it's way up. 2, 4, 6, 8, 16, 32, …….. In order to reach 15, you must use 8, 4, 2, and 1. 13. The numbers roll over every two digits. One is 1, two is 10, three is 11, four is 100, five is 101, and so on. On the computer, these binary digits are called bits. 17

If you move digits to the left the number increases by powers of two compared by decimal increments of ten. The number 435 is the sum of 4×100, 3×10, and 5×1. 19 Binary number is 1×4, 1×2, and 1×1. That equals 7.

To represent a quantity, 0 or 1 is multiplied by it's respective bit position. Once you have done this, you add the numbers to get the binary number.

There are two methods to do binary to decimal conversion. These methods are called expansion and the double-dable method.

Binary to decimal

Thefirst step in expansion is to write down the power of the binary number in a column format begining with the least significant digit. You then add the values. The double-dabble method is a short cut method. First double the highest binary digit (leftmost) and add it to the next digit to it's right. Then double the sun and add one or zero depending on whether the next digit t…