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This essential trains for: AMC-8, AMC-10, AMC-12, AIME.

A positional system of numeration is a system of writing numbers in which each digit has an intrinsic value that is multiplied by a "place value," or positional value. The number of digits used by such a system is called a base.

For example, the digit 2 has the different values 2, 20, 200 and 2000 in the following numbers:

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Since the Arabic system of numeration — which is the one we use — has 10 digits (0, 1, 2, 3, 4, 5, 6, 7, 8 and 9), we say that we use "base 10."

Students with an interest in ancient civilizations sometimes ask "what base do Roman numerals use?". Unfortunately, the answer isn't very exciting: since the Roman system of writing number is not a positional system of numeration, the question of "base" does not apply. The Roman system of numeration is an additive system with relatively complex calculation rules, but each digit always represents the same value regardless of position.

A number written in base 10 is said to be written in decimal notation. This is the written form of numbers that we deal with most usually.

Note that the value of the base itself is not represented by a digit in that base. For example, there is no "digit 10" in base 10. The highest digit in base 10 is "digit 9." There are 10 digits in total in base 10, but they start with "digit 0," and so "digit 9" is the last digit in base 10. The value of the base always has the representation "10" in any base. That is, the digit sequence "10" represents the integer 2 if it is in base 2, 3 if it is in base 3, and so on.

The fewer digits a base has, the longer the appearance of the number. For example, the number 100, if written in base 2, looks like this:

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and in base 100 it looks like this:

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Note that when we say "10 in base 100," there is the assumption that the "100" part is written in base 10, in the same way that we assume any number is written in base 10 if the base is not specified.

The reason bases of numeration are studied is that there are bases other than 10 that have historic or practical value. For example, most digital electronic signals use the base 2 with the digits 0 and 1. Base two is used because electrical signaling is based on the presence or absence of "voltage" between a pair of wires. Sometimes "voltage off" corresponds to 0 and "voltage on" corresponds to 1, and sometimes it's the other way around.

Numbers get very long in base 2 (binary), so computer engineers frequently use base 8 (octal) or base 16 (hexadecimal). Each octal digit corresponds to a group of three binary digits, while each hexadecimal digit corresponds to a group of four binary digits. This "grouping" happens because 8 and 16 are powers of 2. 10 is not a power of 2, so it is not possible to convert between base 2 and base 10 by such a simple method as converting groups of digits at a time.

Around 3000 BC, Mesopotamians used base 60.

The base of a number is usually mentioned in parenthesis at the right bottom corner of the number. For example, this is a number in base 14:

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Converting from base 10 to another base:

Divide the number successively (again and again) by b until you get the quotient zero. Then, collect all the remainders from last to first — they are the digits of the new number from left to right.

Example 1: Convert the base 10 number 486 to base 5.

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Example 2: Convert the base 10 number 486 to base 14. Note that, since we do not have numeric symbols for all the 14 digits, we may have to use the symbols A, B, C and D for the digits corresponding to 10, 11, 12 and 13, respectively.

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Converting from another base to base 10:

This is most directly achieved by writing the number in expanded form and doing the calculations.

Example 3: Convert the base 13 number 6B7 to base 10:

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Example 4: Convert the base 5 number 3341 to base 10:

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While the method of evaluating the expanded form is easier to understand, there is a somewhat faster algorithm. Starting with the result 0, work through the input number left to right. For each digit of the input number, multiply the result by the base and then add the digit to the result. If the calculations are done in base 10, the result will be in base 10. Intriguingly, if the calculations are done in any other base, the result will be in that base. So, in principle, this method could be used to convert from base 10 to other bases, too. In practice, it is almost never used for this purpose because people find it more difficult to perform multiplications and additions in other bases than to perform divisions in base 10. Also, calculators are designed to assist calculations in base 10, and not in other bases.

All the operations with numbers that are possible in base 10, are also possible in other bases.

Example 5: In base 4, write the repeating decimal 0.2222... in fraction form.

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Observation: The smallest base of numeration is 2 (binary). It is impossible to have a positional system of numeration with fewer than 2 digits. This is because the only digit would be "0," which has no value in any place. However, it is possible to write numbers in "unary," for example, by drawing as many pebbles as the value of the number. Numbers written in unary are very easy to add: simply erase the "plus" sign between the piles of pebbles. Despite the conceptual simplicity of unary, it is not used because the representation of large numbers is too long.