The number system we are taught and with which we deal daily in our lives is decimal. The base of this number system is 10. We are taught early that the positional values for this number system are ones, tens, hundreds, thousands, etc. We are taught this by rote, but we are not taught the rule on how to calculate them. But that is easy. Number systems always start with the ones position. The positional value to the left of one will be one times the base of the number system or ten (1 X 10 = 10). The next positional value to the left will be that positional value times the base of the system or 100 (10 X 10 = 100). This is how the ones, ten, hundreds, thousands are calculated.

Each number system has one character multiplier values in it that range from zero to one less than the base of the number system. In decimal, the multipliers are 0 - 9.

When we look at the number 123, we just accept it as the decimal value, but in number system theory, it is the sum of all multipliers times the positional value for the multiplier. Follow this example:

3 X 1 = 3

2 X 10 = 20

1 X 100 = 100

Add up the numbers to the right of the equal signs (3 + 20 + 100) and you get 123.

You have heard that computers use binary. This is because in electrical circuits there is an on or on off value (two possible values). The base for binary is two (2). Follow the rule for multipliers that they will be from 0 to one less than the base of the system so the multipliers for binary are zero and one.

Follow the rule for determining the positional values for the system. Start at one and multiply it by the base (two) so the position to the left of the one position is two (1 X 2). The next position to the left is the two position times the base (2) or 4 (2 X 2). The next position to the left is the four position times two (4 X 2) or 8. Here are the first few positional values in binary:

256 - 128 - 64 - 32 - 16 - 8 - 4 - 2 - 1

Notice that each position is doubled the value to its right. This makes sense since the base is two and we are multiplying each positional value by two to get the value of the next position to the left.

Let's look at this binary number.

10011

To convert this to decimal, we multiply the positional value by the multiplier.

1 X 1 = 1

1 X 2 = 2

0 X 4 = 0

0 X 8 = 0

1 X 16 = 16

Now add up the values to the right of the equal sign and we get 19 decimal for the binary number 10011.

What is the decimal value of 111 binary? If you got 7 decimal then you understand how binary works.

1 X 1 = 1

1 X 2 = 2

1 X 4 = 4

1 + 2 + 4 = 7

Convert one more binary number - 1000000. If you got 64 decimal, you are right.

The size of the memory in computers is a binary number because of the binary multiplier in circuit design. A one meg memory is really 1,048,576 bytes. A 256 meg memory is really 268,435,456 bytes. The 256 or 512 that we use today are throwbacks to the days of smaller memory. The 256 KB memory (262, 144) was roughly 1,000 times the 256 memory and the 256 meg memory (268,435,456) is roughly 1,000,000 times the 256.

Congratulations! You are now an expert in binary numbers.

Copyright 2006 John Howe, Inc.