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The function x (mod 1)

So I've never encountered such a function before and I'm unsure of a few things. What could be said about the differentiability of this particular function? Can an inverse of this function be derived? I honestly have no clue..

Reply 1

Speleo

7

I would assume that y = x (mod 1) is equivalent to y = 0, but then again I know nothing.

Reply 2

DFranklin

18

Speleo

I would assume that y = x (mod 1) is equivalent to y = 0, but then again I know nothing.

In this context, I'm assuming x (mod 1) is the same as the fractional part of x for x positive. That is, 18.231 = 0.231 (mod 1).

The function is differentiable everywhere other than the integers (but discontinous on the integers). It is also invertible over any interval [a,b] where b-a <1. Plus it's invertible over the semi-closed intervals [a,b) with b-a = 1.

Reply 3

Speleo

7

I've usually seen fractions in modular arithmetic dealt with like this:
18.321 = 18321/1000
Since 1000 * 0 mod 1 = 18321 mod 1, 18.321 = 0 mod 1
But as I say I know nothing :tongue:

Reply 4

Sol.

OP

1

Yes, the function x (mod 1) is also known as the fraction part of x. I tried plotting a graph of this function with Mathematica and it turned out shaped like a freaky sawtooth, which somehow, sparked interest from me. Ah wells, thanks for the prompt replies!

Reply 5

DFranklin

18

Speleo

I've usually seen fractions in modular arithmetic dealt with like this:
18.321 = 18321/1000
Since 1000 * 0 mod 1 = 18321 mod 1, 18.321 = 0 mod 1
But as I say I know nothing :tongue:

In this context, it's not "proper" modular arithmetic; integer arithmetic (mod 1) doesn't really make no sense. In particular, we're using "mod 1" as a function, not a relation. (i.e. the distinction between 7 = 13 (mod 3), and the actual "value" of 7 mod 3, which is 1).

Conceptually, we can define a function mod(x, n) by

mod(x, n) = x - n \lfloor \frac{x}{n} \rfloor

. This behaves like we'd expect for integers: mod(7, 3) = 7 - 3 * 2 = 1, etc. But we can also use the same function to define mod for real values of x and n (though we generally insist n > 0).

Thanks

Sol.

So I've never encountered such a function before and I'm unsure of a few things. What could be said about the differentiability of this particular function? Can an inverse of this function be derived? I honestly have no clue..

Well, it's an interesting function! :p:

Its graph is called a sawtooth wave (for obvious reasons). http://en.wikipedia.org/wiki/Sawtooth_wave might be an interesting read (or not, but it might be a place to start if you want to know more), there are some applications of the sawtooth wave and a sound example :biggrin:

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The function x (mod 1)

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