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Binary Calculus

 

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.

0 1 2 3 4 5 6 7 8 9

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.

0 and 1

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 2powerx.gif (1127 bytes)where X can be any number.

To represent 256 one can have 2 powered by 8 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 = 2 powered by 16 (remember: a register of 16 bits can hold 216 -1

 

65535 1000000000000000
65536 1000000000000001
65537 1000000000000010
etc.  

 

The large string of zero's and one's above could represent a number of over 2 raised to the power of 72

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

 

 

Go BackGo to main pagesearch this site Last Updated on 19 October, 2005 For suggestions please mail the editors 

 

Footnotes & References