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## November 17, 2005

### Two

The number Two has always played an important part in man's thinking. We have two feet, two legs, two hands, two eyes, two ears. There are two sexes. Noah took the animals in two's into his ark. Quite early man must have also discovered that each child has two parents, four grandparents, eight grand-grandparents and so forth. Writing this down in the respective numbers 1, 2, 4, 8, 16, 32, 64 ... leads to the simplest of the so-called geometric series.

In ancient Egypt this concept was already successfully incorporated in doing calculations. The Egyptians were able to do all their multiplications by successive doubling, halving and adding up as is shown in the following example.

Let's multiply 11 by 19. You would successively double the first factor and halve the second, taking only the integral half of any odd number until you reach 1. Then you would add up only those doubles which correspond to odd halves:

11 19*

22 9*

44 4

88 2

176 1*

That adds up to 11+22+176=209.

Another approach is writing 19 in a binary way:

19 = 1*16 + 0*8 + 0*4 + 1*2 + 1*1 => 10011

Now you can double 11 again and put the binary number of 19 in the second column from bottom to top. You then add up those doubles which correspond to a "1" in the right column:

11 1

22 1

44 0

88 0

176 1

It's quite fascinating, especially if you show the Egyptian Multiplication to other people, physicists and mathematicians excluded, of course. They will instantly try to think up an example to prove you wrong and usually will not give up before the fifth or sixth try.

Have fun!

Posted by Mausi at November 17, 2005 10:23 PM

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## Comments

Very interesting. Thanks!!! I love stuff like this.

Posted by: vw bug at November 17, 2005 01:00 PM

Your examples are very interesting! - This high standard of calculations (here used by Egyt's) should be explained to pupils in history instead of teaching accurate dates of battles as to obtain respect about the high level of ancient cultures.

Posted by: fuexel at November 20, 2005 07:52 PM