|
|
|
|
|
|
|
One can imagine that we would have evolved a strategy of using the letter A to represent ten; or we might have used IIIIIIIIII to represent that idea. The Romans used X. The Arabic system, which we use, makes use of position in conjunction with numerals to represent values. The first (right-most) column is used for ones, and the next column is used for tens. Thus, the number fifteen is represented as 15 (read one, five); that is, 1 ten and 5 ones. |
|
|
|
|
|
|
|
|
Certain rules emerge, from which some generalizations can be made: |
|
|
|
|
|
|
|
|
1. Base 10 uses the digits 09. |
|
|
|
|
|
|
|
|
2. The columns are powers of ten: 1s, 10s, 100s, and so on. |
|
|
|
|
|
|
|
|
3. If the third column is 100, the largest number you can make with two columns is 99. More generally, with n columns you can represent 0 to (10n-1). Thus, with 3 columns you can represent 0 to (103-1) or 0999. |
|
|
|
|
|
|
|
|
It is not a coincidence that we use base 10; we have 10 fingers. One can imagine a different base, however. Using the rules found in base 10, you can describe base 8: |
|
|
|
|
|
|
|
|
1. The digits used in base 8 are 07. |
|
|
|
|
|
|
|
|
2. The columns are powers of 8: 1s, 8s, 64s, and so on. |
|
|
|
|
|
|
|
|
3. With n columns you can represent 0 to (8n-1). |
|
|
|
|
|
|
|
|
To distinguish numbers written in each base, write the base as a subscript next to the number. The number fifteen in base 10 would be written as 1510 and read as one, five, base ten. |
|
|
|
|
|
|
|
|
Thus, to represent the number 1510 in base 8 you would write 178. This is read one, seven, base eight. Note that it can also be read fifteen as that is the number it continues to represent. |
|
|
|
|
|
|
|
|
Why 17? The 1 means 1 eight, and the 7 means 7 ones. One eight plus seven ones equals fifteen. Consider fifteen asterisks: |
|
|
|
|
|
|
|
|
The natural tendency is to make two groups, a group of ten asterisks and another of five. This would be represented in decimal as 15 (1 ten and 5 ones). You can also group the asterisks as |
|
|
|
|
|