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Our decimal number system is known as a *positional* number
system, because the value of the number depends on the position
of the digits. For example, the number **123** has a very different
value than the number **321**, although the same digits are
used in both numbers.

(Although we are accustomed to our decimal number system, which is positional, other ancient number systems, such as the Egyptian number system were not positional, but rather used many additional symbols to represent larger values.)

In a positional number system, the value of each digit is determined by which place it appears in the full number. The lowest place value is the rightmost position, and each successive position to the left has a higher place value.

In our decimal number system, the rightmost position represents
the "ones" column, the next position represents the
"tens" column, the next position represents "hundreds",
etc. Therefore, the number **123** represents **1** hundred
and **2** tens and **3** ones, whereas the number **321**
represents **3** hundreds and **2** tens and **1** one.

The values of each position correspond to powers of the base of the number system. So for our decimal number system, which uses base 10, the place values correspond to powers of 10:

... | 1000 | 100 | 10 | 1 |

... | 10^3 | 10^2 | 10^1 | 10^0 |

Other number systems use different bases. The **binary**
number system uses base 2, so the
place values of the digits of a binary number correspond to powers
of 2. For example, the value of the
binary number 10011 is determined
by computing the place value of each of the digits of the number:

1 | 0 | 0 | 1 | 1 | the binary number |

2^4 | 2^3 | 2^2 | 2^1 | 2^0 | place values |

So the binary number 10011 represents the value

(1 * 2^4) | + | (0 * 2^3) | + | (0 * 2^2) | + | (1 * 2^1) | + | (1 * 2^0) | |

= | 16 | + | 0 | + | 0 | + | 2 | + | 1 |

= | 19 |

The same principle applies to any number base. For example, the number 2132 base 5 corresponds to

2 | 1 | 3 | 2 | number in base 5 |

5^3 | 5^2 | 5^1 | 5^0 | place values |

So the value of the number is

(2 * 5^3) | + | (1 * 5^2) | + | (3 * 5^1) | + | (2 * 5^0) | |

= | (2 * 125) | + | (1 * 25) | + | (3 * 5) | + | (2 * 1) |

= | 250 | + | 25 | + | 15 | + | 2 |

= | 292 |

In order to convert a decimal number into its representation
in a different number base, we have to be able to express the
number in terms of powers of the other base. For example, if we
wish to convert the decimal number **100 **to base 4,
we must figure out how to express **100** as the sum of powers
of 4.

100 | = | (1 * 64) | + | (2 * 16) | + | (1 * 4) | + | (0 * 1) | |

= | (1 * 4^3) | + | (2 * 4^2) | + | (1 * 4^1) | + | (0 * 4^0) | ||

Then we use the coefficients of the powers of 4 to form the number as represented in base 4: | |||||||||

100 | = | 1 2 1 0 | base 4 |

One way to do this is to repeatedly divide the decimal number by the base in which it is to be converted, until the quotient becomes zero. As the number is divided, the remainders - in reverse order - form the digits of the number in the other base.

**Example:** Convert the decimal number **82** to base
6:

82/6 | = | 13 | remainder 4 |

13/6 | = | 2 | remainder 1 |

2/6 | = | 0 | remainder 2 |

The answer is formed by taking the remainders in reverse order: 2 1 4 base 6