Modular Arithmetic: Introduction to Congruences & Interesting Applications

[matqkks]

What the best way to introduce congruences in a number theory course? I am looking for something which will have an impact. What are the really interesting applications of congruent mathematics?

 

[Evgeny.Makarov]

What the best way to introduce congruences in a number theory course? I am looking for something which will have an impact. What are the really interesting applications of congruent mathematics?

Ask your students this: If it rains at midnight, what is the probability that it will be sunny in 72 hours?

 

[Deveno]

Numbers (and by this I mean positive integers), as anyone ought to know, can be really big. Large numbers are quite time-consuming for us poor humans to deal with.

Often, however, we want to know just some PROPERTY of a number, like whether or not it is even or odd.

Well, it turns out that there is an arithmetic for that, and it works in much the same way as regular arithmetic.

In fact, we can STILL do arithmetic on a number of finite systems, by "looping" the number line (the integers embedded in the reals) around to 0, at some given integer n. This changes the "linear" aspect of arithmetical operations to a cyclical one. And everything still works (well, almost everything...division is problematic if n is not prime...for pretty much the same reason as we don't allow division by 0).

For example, if we are "counting by 6's", the fact that 6 = 2*3 suggests we should "count by 3's" and "count by 2's" and compare the results. And indeed, this works (this is a simplified explanation of the basis for the Chinese Remainder Theorem)!

Of course, what we are actually doing is taking an infinite cyclic integral domain, and forming the cyclic quotient ring, modulo a principal ideal. But that sounds so technical...and obscures the simple underlying idea: we tame infinity by chopping it up into finite bits.

And the payoff is: facts involving very BIG numbers can be deduced from facts involving much SMALLER numbers, and everybody wins!

(Evgeny's observation is pertinent: the "infinite line of time" can be tamed by use of a clock, which repeats itself every so often...the usefulness of this has been known from antiquity...the Babylonians took this idea and ran with it, basing their number system on "highly divisible numbers", forming the basis of our present calender, radial measurements, and time-keeping system...ever wonder where all those 12's, 30's, 60's and 360's come from?).

Mathematics is in many ways is the story of two simple concepts: the line and the circle. Each has many stories to tell, and secrets to teach us.

 

[matqkks]

Thanks for this detailed reply. It is exactly what I was looking for.

 

[HallsofIvy]

Ask your students this: If it rains at midnight, what is the probability that it will be sunny in 72 hours?

Cute, but "in 72 hours" can reasonably be interpreted as "72 hours from now". Better wording "If it rains at midnight, what is the probability it will be sunny 72 hour later?