Contents:

1. Counting in Binary, Octal, and Hexadecimal
2. Base Conversion
3. Gray Codes
4. ASCII Codes

The table below shows numbers in counting order for the decimal, binary, octal, and hexadecimal systems. The left column, shown in black, contains the decimal numbers from 0 to 28. The next three columns show the binary equivalent of each decimal number. The left binary column, shown in red, shows the five bit binary numbers containing only 0s and 1s. Since binary numbers are sums of powers of two, they tend to get rather long even for reasonably small numbers. It is often easier to work with the number if the bits are organized into groups of 3 or groups of 4. With groups of three, shown in blue, each group can be interpreted directly to an octal digit (base 8) without any arithmetic computation for the conversion. With groups of four, shown in green, each group can be interpreted directly to a hexadecimal digit (base 16) without any arithmetic computation for the conversion.

Conversion from binary to decimal using decimal arithmetic is accomplished by simply summing the powers of 2 corresponding to 1s in the binary number. For example the binary number 10111 =b4 b3 b2 b1 b0 is converted to decimal by adding b4x16 + b3x8 + b2x4 + b1x2 +b0x1 = 1x16 + 0x8 + 1x4 + 1x2 +1x1 = 16+4+2+1=(23)decimal.

Conversion from decimal to binary using decimal arithmetic is accomplished by repeated division of the decimal number by two. After each division, the remainder is the next bit of the binary number (starting from the least significant). For example, the decimal number 26 is converted to binary as follows:

So (26)decimal = (11010)binary.

Conversion between binary and bases which are powers of 2( such as octal or hexadecimal) requires no arithmetic computation. For example, the decimal number 19 = 1x10 + 9x1 is written as a binary number with 10011=1x16 + 0x8 + 0x4 + 1x2 +1x1. If this number is interpreted in groups of three bits ( start the grouping from the right or the least significant bit) then each group of three can be interpreted as an octal digit multiplying a power of 8.

(10011)binary = 1x16 + 0x8 + 0x4 + 1x2 +1x1 = (1x16 +0x8) + (0x4 + 1x2 + 1x1) = 2x8 + 3x1 = (23)octal

Similarly, when groups of 4 bits are used, each group can be interpreted as a hexadecimal digit multiplying a power of 16.

(10011)binary = 1x16 + 0x8 + 0x4 + 1x2 +1x1 = (1x16) + (0x8 + 0x4 + 1x2 + 1x1) = 1x16 + 3x1 = (13)hexadecimal

A gray code is a sequence of binary numbers in which only one bit changes with each step through the sequence. A sequence of binary numbers with one bit must be only 0 or 1. A sequence of two bit numbers can have four values. If a leading zero is added to the one bit numbers, we have 00 and 01 for the first two elements of the sequence. If the leading bit is cahnged to 1, then the other bit must not change, so the original sequence goes in reverse order when the leading bit is 1. Thus, the four element sequence is 00, 01, 11, 10. This can be repeated to make an 8 element sequence of three bit numbers by adding a a leading 0 to the 2 bit numbers to make the first four and adding a leading 1 to the 2 bit numbers in reverse order to make the second four.

ASCII stands for the American National Standard Code for Information Interchange. It was developed for computer communication systems and was adopted in 1963. It replaced a 5 bit code with shift states called theBaudot code which was developed for multiplexed telegraph systems and was used in Telex machines.

Supplemental Material for ELEN 021, Logic Design

(c) Copyright 1998 Sally L, Wood