Understanding binary numbers

Computers today handle data in digital format (this is why they are called digital computers). Digital computers handle data characters as group of binary digits. The term digital refers to a signal such as binary. Binary is identified by two specific states, values, or outcomes. Consider the following examples:

  • ON and OFF
  • Positive and negative
  • 1 and 0
  • hot and cold

In comparison, an analog signal varies through the entire range between two states. Confused? Consider an old TV set that has a knob for volume control. Suppose this knob lets you set the volume either to high or low. This is how binary digits are represented, one for high volume setting (or state) and the other for low volume setting (or state). On the other hand, a TV set that has volume control knob that lets you choose the volume between high and low is considered analog. With analog, you have a range of values, not just two.

When we say data in digital computers is represented in digital or binary format, we mean each character (such as a letter, number symbol, etc.) is a specific binary number. A binary number is a string of binary digits such as 00001, 00010, 0100, etc. Notice in binary notation only 0s and 1s are used, recall only two possible values or states. In a decimal system, on other hand, we use numerals 0 through 9 in our everyday use. Computers use binary values (or binary system) for processing our instructions.

Binary math

In a binary system a value is represented by summing the places marked by a 1. In other words, each place has a set value, the first place on the right has a value of 1, the second place has value of 2, the third place has a value of 4, the fourth place has value of 16, and so on. In this system, the value doubles with each place. See table 1 to visualize.

Table 1 binary system
Eighth Seventh Sixth Fifth Fourth Third Second First Place
256 128 64 32 16 4 2 1 Decimal value
10000000 01000000 0100000 010000 01000 0100 0010 0001 Binary equivalent

This does not mean the binary system only represents 256, 128, 64, 32, 16, 4, 2, and 1. Other values are also possible, as shown selected values in table 2.

Table 2 binary and decimal values
Binary Decimal
0001 1
0010 2
0011 3
0100 4
0101 5
0110 6
0111 7
1000 8
1001 9
1010 10
110010 50
1100011 99
1100100 100
1100101 101
1111101000 1000

As it may be evident from table 2, the value of a binary number is determined by adding up the values of the places that contain a 1. Note a 0 in a binary system is not counted! Let’s do some binary math to understand how we get binary digits (a term referring to the 1s and 0s that make up the binary numbers, as you see in the left column of table 2).

To get a binary digit equivalent to the decimal number 3, we will use the first two rows from table 2. In other words, if we add 1 + 2, we will get 3 in decimal; similarly, when we add the binary equivalent of these decimals digits we should get the binary digits. From the table we see, binary digits for decimal value of 1 is 0001 and 0010 for the number 2.

This is how our addition problem looks like in binary:

+ 0 0 0 1
0 0 1 0
  0 0 1 1

The decimal system equivalent is

+ 1
+ 1

Again, in a binary system we only add the value of the places that contain a 1. 0s on the left can be ignored. Thus 01, 001, 0001, 00001, 000001 is 1 regardless of how many 0s you add to the left. Similarly, in a decimal system adding 0s to the left of a number does not change its value such as 02000, 002000, 0002000 is the same as 2,000.

Posted on 10/2/2007
by Raj Singh