- Number Representation System
- Basic Idea
- Converting Whole Numbers in Base 10 to Another Base
- Converting a Number in Base b to a Decimal Number
- Converting a Fraction in Base 10 to Base b
- Converting a Fraction Base b to Base 10

So that each digit is in 0..9, and its position determines which power of 10 we need to multiply it by. Two important symbols in our decimal system are 0, which has been used extensively only in the last few hundred years in Europe, and ".", the decimal point which marks the whole numbers from the fractions (something which was absent in the Babylonian system, and which causes some confusion.. when, for instance, we cannot distinguish 0.25 and 25!)

In general, we represent the whole numbers in our base 10 system in the following way:

[1.2]

Where a is a number, such as 201, and the a's with subscripts are numbers between 0 and 9.

Generalising even further, we can represent numbers in the base, b, as follows:

[1.3]

If we divide a number a, by b continuously, we find that the remainders of division are the digits of our new number base b.

When converting from one base to another (excluding base 10) we need to do arithmetic in a base other than ten. For this reason it is better to convert from one base to another through the intermediary of base 10; that is, we convert one number in base b

For instance, we wish to convert 10 in base 10, to a number in base 3.

Convert 10, base 10 to base 3 | ||

Number of 3's | Remainder | |

10= | 3•3+ | 1 |

3= | 3•1+ | 0 |

1= | 3•0+ | 1 |

We read off the new number, base 3 from the remainders: [1,0,1], or 101

Remark, we could have noticed that because the 1 in the last but one line is less than 3, we could have read the number off beginning with this obvious next remainder, and the other remainders: 101, base 3.

As a second example, we can convert 102 in Base 10 to Base 16 (or hexadecimal).

Convert 102, base 10 to base 16 | ||

Number of 3's | Remainder | |

102= | 16•6+ | 4 |

6= | 16•0+ | 6 |

Reading off the remainders, we note that 102

1

And for 64 in base 16:

6

And 101, base 3 gives:

1

[2.1]

In general, a fractional number, a, in base b can be written:

[2.1]

In decreasing powers of ten, with the first position being 10

Or, alternatively written as:

[3.2]

In converting a decimal in base 10 to a fraction in base b, we do the opposite of converting the whole numbers: we multiply the number in base 10 by b repeatedly, and our remainder is the integer part of the resulting number.

For instance, 0.5 in base 10 is 4•0.5=2, or 0.5 in base 4 is 0.2 in base 4

Convert 0.5 Base 10 to Base 4 | ||

Integer, Number of 4's | Fraction | |

0.5•4 | 2 | 0 |

Convert 0.5 Base 10 to Base 3 | ||

Integer Number of 3's | Fraction | |

0.5•3= | 1 | .5 |

0.5•3= | 1 | .5 |

0.5

Convert 0.16 Base 10 to Base 8 | ||

Integer, Number of 8's | Fraction | |

0.16•8= | 1 | .28 |

0.28•8= | 2 | .24 |

0.24•8= | 1 | .92 |

0.92•8= | 7 | .36 |

0.36•8= | 2 | .88 |

0.88•8= | 7 | .64 |

0.16 base 10 is 0.121727... base 8

[3.2, repeated]

For instance, 0.2 in base 4 is 2/4 in base 10, or 0.5.

Another example:

0.121727 in base 8 is (to base 10) is, by spreadsheet:

Number Part |
Result of: |
Dividing by |
divisor on the left
is 8 to the
power) |

1 | 0.125 | 8 | 1 |

2 | 0.03125 | 64 (8^{2}) |
2 |

1 | 0.0019531 | 512 (8^{3 }) |
3 |

7 | 0.001709 | 4096 (8^{4 }) |
4 |

2 | 6.104E-05 | 32768(8^{5 }) |
5 |

7 | 2.67E-05 | 262144(8^{6 }) |
6 |

0.121727_{8}= |
0.160000 | ( to 6 decimals) |

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