Binary Numbers

The purpose of this little article is to better explain the binary number system, how to convert from the binary to decimal number systems, and how to convert from the decimal to binary number systems.

The Binary Number System

To better understand the binary number system, it helps to review the decimal number system first. The decimal number system uses ten different symbols (the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9) arranged using positional notation. Positional notation is used when a number larger than 9 needs to be represented; each position of a digit signifies how many groups of 10, 100, 1000, etc. are contained in that number. Specifically, each position of a digit represents a group that is ten times as large as the group represented by the the position to the right. Thus, the number 1,065 represents a number that contains one group of 1000, zero groups of 100, six groups of 10, and 5 "groups" of 1. Notice that the size of the groups represented by each position in the number correspond to a power of 10:

Along similar lines, the binary number system uses only two different symbols (the digits 0 and 1) that are also arranged using positional notation. When a number larger than 1 needs to be represented, positional notation is used to represent how many groups of 2, 4, 8, etc. are contained in that number. Each position in a number represents a group that is two times as large as the group represented by the the position to the right. Thus, in the binary number system, 1011 represents a number that contains one group of 8, zero groups of 4, one group of 2, and 1 "group" of 1(which adds up to 8 + 2 + 1 = 11). Notice that the size of the groups represented by each position in the number correspond to a power of 2 (see the table below for more powers of 2). Also notice that it is hard to tell what number system you are talking about when you are dealing with numbers like 1011 and 11. To avoid this type of confusion, a decimal subscript is often used to specify exactly what number system is being used. Thus, we write 1011₂ = 11₁₀.

Notice that eight entries are recorded in the table above. That is because computer scientists often group information into eight digit binary numbers, called a byte (a single binary digit is called a bit). A byte can represent any number from 0 ( = 00000000₂) to 255 ( = 11111111₂). This explains why the number 255 turns up a lot with computers, such as a setting choice of 255 colors for your monitor.

Binary to Decimal Conversion

To convert from a number represented in the binary number system to a number represented in the decimal number system, all you need to do is add up the groups (whose size is always a power of two) represented by each position in the binary number. It helps to put things into a column to see the pattern. Thus, to convert the number 10101101₂ into the decimal number system:

For practice, you should see if you can show that 1100110₂ = 102₁₀ and that 10001010111₂ = 1111₁₀.

Decimal to Binary Conversion

To convert from a number represented in the decimal number system to a number represented in the binary number system, all you need to do is repeatedly subtract the largest power of two that you can from the decimal number, then place ones or zeros in the correct position to represent these groups. To see what this means, let's look at how to do this with decimal numbers and then binary numbers:

How many stars are in the picture below? It's hard to say offhand...

 

It's easier to count the stars if they are put into groups of 1, 10, 100, ... It's easy to see that the picture contains: 1 group 100 3 groups of 10 2 "groups" of 1. Thus, there are 132 stars.

 

To count in binary, look for groups of 1, 2, 4, 8, 16, 32, 64, 128, ... It's easy to see that the picture contains: 1 group 128 0 groups of 64 0 groups of 32 0 groups of 16 0 groups of 8 1 group of 4 0 groups of 2 0 "groups" of 1. Thus, there are 100001002 stars. Therefore, 13210 = 100001002

To convert from decimal to binary by hand it is again easier to use a columnar format to keep things organized. The table below whows how to convert 201 into binary. Once you understand the example, try to see if you can show that 196₁₀ = 11000100₂ and that 1967₁₀ = 11110101111₂.

	© 2003 by Kevin Mirus (kmirus@madison.edu or kjmirus@execpc.com).
	This document was last modified .