Let's look at numbers with the same digit in different places and see if we can determine some relationship.
Consider the number 20.
Now, consider the number 200, which has the 2 in the location just to the left of where it is in 20. You're expect to observe that the number 200 is <em>ten times</em> the number 20.
Consider the number with the 2 in the position to the right of where it is in 20. That number is 2. You are expected to observe that the number 2 is <em>one-tenth</em> the number 20.
The place-value of a digit increases by a factor of 10 when moved one place left, and is reduced by a factor of 1/10 when moved one place right.
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This is what makes a place-value number system work. In Roman Numerals, for example, the value of a character is changed by ...
- putting it ahead of or after a higher-value character: IV, VI
- changing the character: I, V, X, L, C, D, M
Place-value number systems don't have to have 10 as their base. We use 60 for the base in (minutes):(seconds), both for time and angle measures. We use 2, 8, or 16 as the base in the binary, octal, and hexadecimal numbers used by computer systems. These other place-value systems have the same characteristic: the value of a digit is increased by a factor of the base when moved to the left, and decreased by a factor of the base when moved to the right. (The hexadecimal value A7C0 has 16 times the value of A7C, for example, and 1/16th the value of A7C00.)
