Modular arithmetic, also known as the Gaussian number system for its discoverer, is similar to more commonly used types of aritmetic and is quite easy to learn. ; In the most widely used system, the Euclidian system, a number will continue to grow after it has exceeded its top numeral. ; That is, 9 will become 10. ; In the Gaussian number system, a number switches back to the lowest number, instead of continuing to grow. ; That means that 9 would be incremented back to 0 if 1 was added. ; There are more applications for this type of mathematics than are readily apparent.

There are two commonly used methods of writing modular numbers. ; The second is merely a simplification of the first, as is seen below:

This notation means that the integer 5, base 10 is equal to 5. This is rather straightforward.

This is a larger number that can demonstrate the difference between Gaussian and Euclidian number systems. ; This demonstrates that once the base limit, in this case 10, is reached, a Gaussian number will reset to its lowest value and begin counting again.<

A different, clearer way of writing the answers to modular equations like those listed above is to include the base of the number system in the answer, like so.

This keeps clearer track of how high the number can grow before it must be reset.

The simplest type of Gaussian mathematics to learn is addition. ; The key is to remember that the number is reset after a certain value. ; Therefore, simply add the two numbers normally, then subtract the base until doing so is no longer possible.

In Gaussian mathematics, negative numbers do not really exist. ; Instead, one simply continues to count backwards from 0. ; Since 0 is not only 0, but also the Gaussian equivalent of 10, simply go to the next highest number. That is, in a base 10 system, the number below 0, that is to say -1, would be 9.

Gaussian subtraction is also an extremely simply concept. ; The only difficulty is to remember that when one is counting backwards and reaches 0, one simply starts at the highest number again, as explained in ;quot;Negative Numbers. ;quot; ; For example,

Like addition and subtraction, multiplication can be mastered using the Euclidian techniques learned in elementary school. ; Simply place two numbers together inside a single set of brackets and multiply them. ; Then subtract the base until it is no longer possible, and the final product is the solution to the equation. ; In the case of numbers with two different bases, it is possible to choose either base as the one for the final equation if the whole numbers are given; that is, [32]

This is the only truly difficult set of concepts to grasp in modular arithmetic. ; There are several increaslingly complicated operations, two examples of which are shown here.

A is relatively prime to b if the two share no common factors. ; Therefore, is not relatively prime while [5]

This algorithm is used to find the Greatest Common Divisor(GCD) of two numbers in a set. ; The steps for this algorithm are the following.

A. Put the highest number of the two on the left-hand side of the set.

B. If the right-hand number is 0, then the left-hand number is the GCD, so the algorithm may be exited.

C. Subtract the right-hand number from the left until doing so is no longer possible.

D. Switch the two numbers.

E. Repeat, starting at B.