# Integration of Modular Arithmetic Functions

• Igora
In summary, the author is looking for information on how to integrate a modular arithmetic function of the reverse, but does not seem to have success so far.
Igora
Hello, I have been searching and can't seem to find anything on the topic of integrating modular arithmetic functions. So far I have created an equation for a function in the form of f(x)=mod(x,a):
$\int mod (x,a) dx=\frac{a(x-mod (x,a))+mod (x,a)^2}{2}+c$
But, now I am investigating how to integrate a modular arithmetic function of the reverse: mod(a,x). Does anybody have any ideas or information on the topic?

Igora said:
Hello, I have been searching and can't seem to find anything on the topic of integrating modular arithmetic functions. So far I have created an equation for a function in the form of f(x)=mod(x,a):
$\int mod (x,a) dx=\frac{a(x-mod (x,a))+mod (x,a)^2}{2}+c$
But, now I am investigating how to integrate a modular arithmetic function of the reverse: mod(a,x). Does anybody have any ideas or information on the topic?

How do you define mod for non-integer x?

One reason there's not much literature on this (besides mod only being defined for integers) is that mod is not a binary operation in math, it's an equivalence relation. It's only in programming languages that we see 5 mod 3 = 2.

In math we would say that 5 = 2 (mod 3), but we would also say that 5 = -4 (mod 3). So you can only make mod into a function if you say that n mod a is the least positive member of the set of integers congruent to n mod a. In math it's rare to use that formulation.

Also, integrals only operate on continuous functions.

mod usually operates on integers or natural numbers. f(n):natural numbers-> real numbers is by definition not a continuous function.

There are a few ways to extend the modulo operation to the real numbers. The greatest integer function or floor function is one natural way.

a newMod b =floor(a) ( mod floor(b) )
But as the floor function is very much like a staircase, it's not particularly useful in this context.

If you ignore negative a's and x's, then $f(x) = \{^{a-x \; \mbox{if} \; x \le a}_{a \; \mbox{if} \; x > a}$. The easy way to integrate that, is to just integrate each piece of the piecewise.

but modular arithmetic can be pseudo continuous if the mod is prime, because every unit has an inverse.

## What is the definition of Modular Arithmetic?

Modular Arithmetic is a type of arithmetic that involves performing calculations on numbers with the consideration of a fixed modulus. This means that after a certain number is reached, the calculations "wrap around" and start from the beginning again.

## What are some real-world applications of Modular Arithmetic?

Modular Arithmetic has many applications in computer science, including cryptography, error-correcting codes, and data compression. It is also used in fields such as engineering and physics to find solutions to periodic problems.

## What is the Chinese Remainder Theorem and how does it relate to Modular Arithmetic?

The Chinese Remainder Theorem is a mathematical theorem that states that if we have a set of congruence equations with pairwise relatively prime moduli, then there exists a unique solution that satisfies all of the equations. This theorem is often used in modular arithmetic to solve systems of equations more efficiently.

## What is the difference between Modular Arithmetic and regular arithmetic?

The main difference between Modular Arithmetic and regular arithmetic is the inclusion of a modulus in the calculations. In regular arithmetic, all numbers are treated as absolute values, while in Modular Arithmetic, numbers are reduced to their remainder when divided by the modulus. This allows for more efficient calculations and solving of certain problems.

## What are some challenges when working with Modular Arithmetic?

One challenge when working with Modular Arithmetic is that it can be difficult to find solutions to certain problems without the use of a computer. Additionally, it can be challenging to determine the correct modulus to use for a given problem. Another challenge is that operations such as division and finding multiplicative inverses can be more complex in modular arithmetic compared to regular arithmetic.

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