The binairy arithmatic or calculus is invented by Leibnitz around 1694 and is presumed to be the first one working with this idea.

One wonders what would have been happend if Leibnitz had worked on it with more enthusiasm and not had fled to the orient to persue other interests.

As we all know our counting system is based on the principle of decimal arithmetic. That
has not always been the case. We start to count with 0 to 9, a maximum of 10 symbols,
inclusive the zero.

With these symbols, or digits or ciphers, you can form any number by
permutation ( sequence of numbers) of these symbols. It looks a bit overdone to say this
here but not every one realizes that this was not the only way to calculate. Fact is that
all great mathematicians (Leibnitz, Pascal) are very aware of this in the 17th century
this story plays. But most of them find the decimal system the most convenient.

Leibnitz has more than fleeting interest in this binary system. Only two digits are used:
0 and 1. Hence the binary (two some) system. And yet it is possible to express any number
in binary. This system will later become the basis on which the entire computer industry
will be based, not all kind computers as you will read later on but most of them will.

It looks a bit overdone to say this here but not every one realizes that this was not the
only way to calculate. Fact is that all great mathematicians (Leibnitz, Pascal) are very aware of this in the 17th
century when this story plays. But most of them find the decimal system the most
convenient. Though there was quite a resistance from the clerics.

Leibnitz has more than a fleeting interest in this binary system.

The binairy system

In the binairy system only two digits are used: 0 and 1. Hence the binary (two some) system.
And yet it is possible to express any number in binary. This system will later become the
basis on which the entire computer industry will be based, not all kind computers as you
will read later on but most of them will.

How does it work?

Our decimal one is still written as 1 and our zero is still 0.

0 | 0 |

1 | 1 |

For computers this is very easy to go by. Because, for the time being, they only know two status: off or on. This looks very convenient.

But what about two three four etc. Since we deal with a binary system there is nothing else left than 0 and 1. There is no symbol for 2 or 3 or 4 here.

But the binary system is also a positional system and you can just count on: like two will become 10, there will be 11 etc. Counting can now be done by a computer. The only thing man had to invent is how to calculate beyond 0 and 1. Thus became the register and it was possible to expand the amount of digits beyond 2:

0 | 0000 |

1 | 0001 |

2 | 0010 |

3 | 0011 |

4 | 0100 |

5 | 0101 |

6 | 0110 |

7 | 0111 |

8 | 1000 |

9 | 1001 |

For very large numbers one just has to expand the registers to contain larger binairy strings. And voila!

011101101011011010110101010110111101010100100100100001011101101010101010101

This binary idea can be used to represent numbers for computing and to make it easy it does so with a base of 2. For example where X can be any number.

To represent 256 one can have
or 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2

But 8 bit can only go to 255. So to represent 256 we need 1 bit more:

255 | -11111111 |

256 |
100000000 |

When each digit means a 0 or 1 there are now 8 digits representing
255. Placing another 8 digits next to it you can form the
number 65536 =
(remember: a register of 16 bits can hold 2

65535 | 1000000000000000 |

65536 | 1000000000000001 |

65537 | 1000000000000010 |

etc. |

The large string of zero's and one's above could represent a number of over

Most computers of available now calculate with registers of 64 or 32 digits. But in a few years this will be 128 or higher.

Last Updated on 19 October, 2005 | For suggestions please mail the editors |

Footnotes & References

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