Date: Sep 1, 2012 1:58 AM
Author: Jonathan Crabtree
Subject: Non-Euclidean Arithmetic

Arithmetic has been flawed ever since Euclid edited the Elements around 300 BCE.

The reason is the incorrect definition of multiplication included in Book VII Def. XV

"Def. 15. A number is said to multiply a number when the latter is added as many times as there are units in the former."


I am grateful to Professor David Joyce for changing the definition on the above page from the previous definition,

"Def. 15. A number is said to multiply a number when that which is multiplied is added to ITSELF (my caps) as many times as there are units in the other."


David updated Euclid's definition of multiplication on his wonderful Euclid website after an exchange of emails in which I pointed out that Euclid's definition of multiplication was wrong.

I wrote about this here

I'm a bit of a heretic suggesting arithmetic has been built on shaky foundations, yet Euclid can be forgiven as he wrote prior to the acceptance of both zero as a number and and negative numbers. He would have learnt much from Diophantus and more from Brahmagupta of Indian fame.

I have finally found another mathematician who has stated that mk = m added to itself k-1 times.

His name is S. F. Lacroix, whio wrote about multiplcation, "... by adding the number to itself as many times, wanting one, as is to be repeated.

For instance, by the following addition,

- --

the number 16 is repeated four times, and added to itself three times." ie k-1 times

So the non Euclidean definition I have created is,

"mk = the combination of m either added to or subtracted from zero k times"

>From this correct definition of multiplication, all rational arithmetic becomes easy, including integer and fraction multiplication. The number lines can now be used for fractions as well as multiplying negatives.

The whole Devlin MIRA debate has been a red herring.

Until now, multiplication itself has never been correctly defined for the rational numbers.

Jonathan Crabtree
1st September 2012

P.S. Think of a photocopying machine. You have one letter and you need four altogether. So what button do you press on the multiplication machine? Three! ie k-1 Press the four button and you end up with five.