# BASE NUMBER SYSTEMS

#### #29 BASE NUMBER SYSTEMS

Well Base Number Systems are the whole entire system that our computers run on, Binary is a Base System. In this case Base 2.

Before the days of languages, such as C++, C, Visual Basic and so on, there was actually Intelligent people slaving away at computers typing up endless pages of machine code. If you want to know more about the history of Machine code and etc. Well the point I’m trying to make is that the average computer programmer, back then had done Degrees in mathematics. 80% of the time the programmers had degrees anyway, This degree in mathematics helped them understand the way a computer works, eg. the electronic pulses caused by the computers output of 1’s and 0’s. This being Binary.

The Language of Numbers.

At bottom, Computers understand only one language — the binary code of ones and zero’s that represent on-off electronic pulses. Because this code is so difficult for humans, programmers have built more concise ways of expressing the binary numbers that constitute, for example the contents of a computer’s memory or the address in memory of each piece of data. Two numbering systems that can serve as convenient short hand for the binary (base 2) are octal (base 8) and Hexadecimal (base 16). Hexadecimal is sometimes known as “Hex” by programmers. Because 8 is raised 2 to the third power ( 8 = 2 * 2 * 2) , one octal digit is the equivalent of three binary digits, similarly, one hexadecimal digit represents 4 binary digits ( 16 is raised to the forth power). The tables below list the decimal numbers 0 through 16 and there binary, octal, hexadecimal equivalents. In each system, the value of a digit is determined by the value of its place column. The letters A through F in hexadecimal represents the 11th through 16th digits in that system.

The process of Conversion

Decimal to Binary

Subtract the highest possible power of 2 from the decimal number – here, 4 from 5 – and continue subtracting the highest possible power from the remainder, marking a 1 in each binary place column where subtraction occurs and a 0 where it doesn’t . Here. one 4, no 2 and one 1 gives binary 101.

Binary to Decimal

Add the values of all the binary places occupied by 1s. Here, to convert the 12- digit binary number 100101101001, add the place values of 2048,256,64,32,8 and 1. The result is the decimal number 2409.

Binary to Octal

Starting with the rightmost digit, group the binary digits in threes, treating each three as a separate binary number with the place values of 4,2 and 1. The sum of each of trio’s place values equals one octal digit. Here, the sums of the values of each of the four groups are 4,5, 5 and 1, making octal 4551.

Again from the right, group the binary digits in fours, treating each four as 1 binary number with the place values 8, 4, 2 and 1. The sum of each group’s place values equals one hexadecimal digit. Here, the sums of the three groups are 9, 6 and 9, making hexadecimal 969.