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Modular Arithmetic terminology

In modular arithmetic, why do we say 'For a positive integer n, two integers a and b are said to be congruent modulo n, written: '

What is the significance of the word, 'congruent' there? How would it make a difference if I said 'a is equivalent to b mod n'?
Original post by Aaradhana


What is the significance of the word, 'congruent' there? How would it make a difference if I said 'a is equivalent to b mod n'?


Equivalence is used in relations and to say a is equivalent to b mod n, means a and b belong to the same equivalence class viz. [a]=, which is true: n would normally be stated beforehand rather than being part of the equation. This implies a theoretic framework defining equivalence relations, which is admittedly pretty basic, and covered in first year uni.

Congruent is the correct term. Meaning n | a-b

Mathematics is a language, it has its own syntax, grammer, vocabulary and rules; to be understood you do need to use the exact term.
(edited 10 years ago)
Reply 2
Original post by ghostwalker
means a and b belong to the same equivalence class viz. [a]=, which is true: n would normally be stated beforehand rather than being part of the equation. This implies a theoretic framework defining equivalence relations, which is admittedly pretty basic, and covered in first year uni.

Congruent is the correct term. Meaning n | a-b


I'm sorry but I didn't understand any of that.

Modulo is a new concept to me and I've been understanding it all by myself and I'm still in 12th so it would be nice if you could explain it in simpler terms, like to a kid.
Reply 3
Original post by Aaradhana
I'm sorry but I didn't understand any of that.

Modulo is a new concept to me and I've been understanding it all by myself and I'm still in 12th so it would be nice if you could explain it in simpler terms, like to a kid.


It's just the terminology that we use:

"a is congruent to b (mod n)" means that a and b leave the same remainder when divided by n, so for example 7 is congruent to 2 mod 5 because 7 leaves remainder 2 when divided by 5.

"equivalent" is a bit of a more general term, so you could use it, but it could be misinterpreted :smile:
Original post by Aaradhana
I'm sorry but I didn't understand any of that.


Sorry, ignore my comments, they were pitched at the wrong level for you.

As davros said.
Reply 5
Thanks but I think I wasn't very clear with my question...

I know what modulo is and I also know a little modular arithmetic but I have a confusion with the term 'congruent' (till now I have only heard of it in the connection of similar shapes and congruent shapes).

When I googled it, I read about the clock example (how it wraps around) and I also read that congruences have their own properties and are important in number theory. I can connect the clock thing to modulo but I have no idea how congruence comes in.
Reply 6
Original post by Aaradhana
Thanks but I think I wasn't very clear with my question...

I know what modulo is and I also know a little modular arithmetic but I have a confusion with the term 'congruent' (till now I have only heard of it in the connection of similar shapes and congruent shapes).

When I googled it, I read about the clock example (how it wraps around) and I also read that congruences have their own properties and are important in number theory. I can connect the clock thing to modulo but I have no idea how congruence comes in.


I'm not quite sure what you're asking...as I explained, "congruent" is just the term that is used in modular arithmetic when 2 quantities (numbers or expressions) leave the same remainder on division by a certain quantity - it doesn't relate to geometrical congruence as far as I am aware!


You could say "a is equi-remaindorial to b" if you preferred :smile:
Reply 7
But it has a more important implicit meaning, right... I mean why just that word and not the word 'equivalence', which is normally used for ≡?
Is it not of my level?
Reply 8
Original post by Aaradhana
But it has a more important implicit meaning, right... I mean why just that word and not the word 'equivalence', which is normally used for ≡?
Is it not of my level?


It's nothing to do with your 'level'!

The symbol \equiv has 2 meanings in mathematics:

when it relates to algebraic expressions e.g. (x+1)2x2+2x+1(x+1)^2 \equiv x^2 + 2x + 1 it means "is identically equal to", in other words, both sides have the same value whatever value of x we put in.

When we use it in modular arithmetic, it has the meaning I explained earlier ab(modn)a \equiv b (mod n) means that a and b have the same remainder when divided by n.

It's not uncommon to use the same symbol in mathematics to mean different things depending on the context :smile:
Reply 9
O, I get it!
They're two totally different things with different meanings...
Thanks! :smile:

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