Stay updated on our latest
articles with RSS
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.
Lets 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.
Now, with this background in number system rules and examples, lets move on to the octal numbering system. As the name implies, the base for the octal numbering system is eight (8). Since the rule for the multiplier values says that the multipliers will be zero through one less than the base, the multipliers will be 0 - 7.
The positional values start with one. The next position to the left is one time the base (8). 1 X 8 = 8. The next position is the 8 position times 8 or 64. The first few positional values for octal are:
2,097,152 - 262,144 - 32,768 - 4,096 - 512 - 64 - 8 - 1
Convert this octal number to decimal:
7532
Multiply each multiplier by its positional value.
2 X 1 = 2
3 X 8 = 24
5 X 64 = 320
7 X 512 = 3,584
Add up the values to the right of the equal signs and you get 3,930 decimal.
Another reason that octal is used is that each three positions of a binary number can be expressed in one octal position so it is much easier to write a large number in octal than in binary.
Lets look at the binary value 111. If you use the rules, this will be 1+2+4 = 7. Seven is the highest multiplier value in octal.
Lets convert a binary number to octal. Start at the ones position moving from right to left, and break the binary number into three position groups. Express each three position group as its octal value
Binary 1 011 110 010 010 111
Octal 1 3 6 2 2 7
So 1011110010010111 in binary equals 13622 in octal.
To double check that the two are equal follow the rules and convert both number systems to decimal. If you came up with 48,279 you are right.
Octal was used in computers until the hexadecimal numbering system was developed and "hex" has largely replaced it, but you may still run into situations where the octal numbering system is still used. Just remember the rules from this article and you will have no problem understanding the octal numbering system.
Copyright 2006 John Howe, Inc.
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.
Lets 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.
Now, with this background in number system rules and examples, lets move on to the octal numbering system. As the name implies, the base for the octal numbering system is eight (8). Since the rule for the multiplier values says that the multipliers will be zero through one less than the base, the multipliers will be 0 - 7.
The positional values start with one. The next position to the left is one time the base (8). 1 X 8 = 8. The next position is the 8 position times 8 or 64. The first few positional values for octal are:
2,097,152 - 262,144 - 32,768 - 4,096 - 512 - 64 - 8 - 1
Convert this octal number to decimal:
7532
Multiply each multiplier by its positional value.
2 X 1 = 2
3 X 8 = 24
5 X 64 = 320
7 X 512 = 3,584
Add up the values to the right of the equal signs and you get 3,930 decimal.
Another reason that octal is used is that each three positions of a binary number can be expressed in one octal position so it is much easier to write a large number in octal than in binary.
Lets look at the binary value 111. If you use the rules, this will be 1+2+4 = 7. Seven is the highest multiplier value in octal.
Lets convert a binary number to octal. Start at the ones position moving from right to left, and break the binary number into three position groups. Express each three position group as its octal value
Binary 1 011 110 010 010 111
Octal 1 3 6 2 2 7
So 1011110010010111 in binary equals 13622 in octal.
To double check that the two are equal follow the rules and convert both number systems to decimal. If you came up with 48,279 you are right.
Octal was used in computers until the hexadecimal numbering system was developed and "hex" has largely replaced it, but you may still run into situations where the octal numbering system is still used. Just remember the rules from this article and you will have no problem understanding the octal numbering system.
Copyright 2006 John Howe, Inc.