Question about Relatively Prime numbers

[Johnny5]

Currently, I am trying to make my way through Euclid's book seven, which is on number theory.

 

I am having trouble with his very first theorem, can anyone here help?

 

I think the reason I am having trouble with it, is that the English translation isn't as good as it could be.

 

Here is a link to proposition 1 of book seven:

 

Euclid's elements, Book 7, Proposition 1

 

The translation that you see at the website, is not the same as is in Heath. I think Dr. Joyce tried to translate it more clearly on his own.

 

Anyway, I was reading through a graduate text on linear algebra, and around the time they begin to discuss rings, there is a discussion on "relative primeness" and fields.

 

So I want to be able to grasp all of this ultimately, and so I think there is a link of the modern information, to Euclid here.

 

So if anyone can help me decipher this first theorem, that will help me get going on the rest.

 

Thanks

 

 

PS: I was doing this yesterday, and a few things were confusing to me. First, I do not currently understand what is meant by "one number measures another number" does euclid mean "divide evenly"

 

I think one good example is all it will take.

 

Let the given numbers be 49,50.

 

He starts off with:

 

The less of two unequal numbers AB and CD being continually subtracted from the greater, let the number which is left never measure the one before it until a unit is left.

 

Actually, I am in the process of answering my own question here.

 

This is the Euclidean algorithm that Dave went over, a few weeks ago.

 

I still remember the algorithm.

 

For some reason Joyce is calling it "antenaresis," don't know where he came up with that.

 

 

Joyce clearly says that in modern terminology we say 'divides' not 'measures'.

 

 

And the notation is

 

a|b

 

for a divides b

 

Joyce goes on to say:

 

This proposition assumes that 1 is the result of an antenaresis process. Antenaresis, also called the Euclidean algorithm, is a kind of reciprocal subtraction. Beginning with two numbers, the smaller, whichever it is, is repeatedly subtracted from the larger.

 

So start with 7,50.

 

50-7=43

43-7=36

36-7=29

29-7=22

22-7=15

15-7=8

8-7=1