Modular Arithmetic - Page 3

ppo · 2005-10-25, 11:55

ok Bob, I should have not mixed software issues in the math forum, but it seemed to me that the behavior of the software was due to the mathematical definition used by the programmers, and I think than that was wrong.
Beside that, the definition in the wikipedia is not correct, and someone should take care of that.

mfgoode · 2005-10-29, 17:54

Answer to Bob Silverman.

Thank you Bob for your very precise and ultra modern definition of congruence. It is pregnant in meaning and more than meets the eye

I found an inconsistency in Oystein Ore’s book “Number Theory and its History” published in 1948 pages 211-213.

I quote: "The properties of congruences: Gauss introduces his congruences

Through the following definition:

Two integers a and b shall be said to be congruent for the modulus m when their difference a-b is divisible by the integer m. [Niven and Zimmerman say the same]

This he expresses in the symbolic statement

a=b (mod m ) call this (9 – 1) "

He goes on to say:

“One can state the congruence slightly differently by saying that b is congruent to a when it differs from a by a multiple of m

Therefore b = a +km call this (9 – 2 ) “

Well and good. This is more like your definition.

He explains that congruence is called an equivalence relation.

“The best known example of such a relation is the ordinary equality a-b

“It may be of interest to observe (here comes the crux) that the equality may

Itself be considered to be a congruence namely for the modulus 0

Since according to (9 - 2) the congruence

a = b (mod 0) signifies that a = b.

“This artificial terminology is not in use”

No wonder! as division by 0 is forbidden

Now to circumvent the discrepancy your definition of ‘integral multiple’ though he has

Given it in ( 9 – 2) should have been emphasised and not to have stuck to Gauss’ and his

Definition.

No! For the case of (mod 0) the division algorithm is not permitted by his definition for

a= b (mod 0)

Mally.

akruppa · 2005-11-29, 11:12

Quote:

Originally Posted by ppo

wikipedia has the following definition:

similar, but looking incorrect to me.
So, why the Windows calculator gives me an "undefined" error message for 5 mod 0 ?

I changed the Wikipedia article at http://en.wikipedia.org/wiki/Modular_arithmetic
Is it ok like this?

Alex

ppo · 2005-11-29, 16:37

Quote:

Originally Posted by akruppa

I changed the Wikipedia article at http://en.wikipedia.org/wiki/Modular_arithmetic
Is it ok like this?

Alex

Very good, thanks

Pierpaolo

akruppa · 2005-11-30, 07:48

They changed most of it back - there's a brief note about zero moduli further down which notes that Z/Z0 is isomorphic to Z. They did leave my change to "multiple of n" instead of "divisible by n", so that definition is correct for n=0 now.

Alex

mfgoode · 2005-11-30, 15:41

Quote:

Originally Posted by akruppa

They changed most of it back - there's a brief note about zero moduli further down which notes that Z/Z0 is isomorphic to Z. They did leave my change to "multiple of n" instead of "divisible by n", so that definition is correct for n=0 now.

Alex

:surprised. I think the credit should go to Bob Silverman.

Mally

2005-10-25, 11:55	#23
ppo Aug 2004 italy 113 Posts	ok Bob, I should have not mixed software issues in the math forum, but it seemed to me that the behavior of the software was due to the mathematical definition used by the programmers, and I think than that was wrong. Beside that, the definition in the wikipedia is not correct, and someone should take care of that. Last fiddled with by ppo on 2005-10-25 at 12:00

2005-10-29, 17:54	#24
mfgoode Bronze Medalist Jan 2004 Mumbai,India 4004₈ Posts	Modular Arithmetic Answer to Bob Silverman. Thank you Bob for your very precise and ultra modern definition of congruence. It is pregnant in meaning and more than meets the eye I found an inconsistency in Oystein Ore’s book “Number Theory and its History” published in 1948 pages 211-213. I quote: "The properties of congruences: Gauss introduces his congruences Through the following definition: Two integers a and b shall be said to be congruent for the modulus m when their difference a-b is divisible by the integer m. [Niven and Zimmerman say the same] This he expresses in the symbolic statement a=b (mod m ) call this (9 – 1) " He goes on to say: “One can state the congruence slightly differently by saying that b is congruent to a when it differs from a by a multiple of m Therefore b = a +km call this (9 – 2 ) “ Well and good. This is more like your definition. He explains that congruence is called an equivalence relation. “The best known example of such a relation is the ordinary equality a-b “It may be of interest to observe (here comes the crux) that the equality may Itself be considered to be a congruence namely for the modulus 0 Since according to (9 - 2) the congruence a = b (mod 0) signifies that a = b. “This artificial terminology is not in use” No wonder! as division by 0 is forbidden Now to circumvent the discrepancy your definition of ‘integral multiple’ though he has Given it in ( 9 – 2) should have been emphasised and not to have stuck to Gauss’ and his Definition. No! For the case of (mod 0) the division algorithm is not permitted by his definition for a= b (mod 0) Mally.

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2005-11-30, 07:48	#27
akruppa "Nancy" Aug 2002 Alexandria 9A3₁₆ Posts	They changed most of it back - there's a brief note about zero moduli further down which notes that Z/Z0 is isomorphic to Z. They did leave my change to "multiple of n" instead of "divisible by n", so that definition is correct for n=0 now. Alex