Interactive Modular Arithmetic Wheel

The modular arithmetic wheel is a wheel of numbers starting from 0 and counting upwards: 0, 1, 2, 3, and so on. The wheel is divided into a number of sections that is controlled by the slider. The wheel is “modular” since the number of sections corresponds to modular arithmetic and visually represents “modulo ” where is the number of sections. The individual numbers in the wheel can be selected to demonstrate the result of .

For example, the number wheel below is in modulo 3, since the wheel is divided into three parts. Additionally the section labeled with the number 8 is selected and so the math expression that is being visualized is: .

This visually demonstrates the modulus operator, or mod for short, since the mod operator returns the remainder of dividing some number by another number . In this case, the remainder of dividing 8 by 3 is 2. This is visualized by the wheel since one full rotation will always be equal to and so the answer will always lie in the section of the inner band of the wheel that the selected number belongs too.


The number of sections of the wheel corresponds to the modulo number. So in the general case of the number of sections corresponds to the variable .

The inner band of the wheel shows all the possible results for the current modulo number. For example, for any integer the possible results are 0, 1, and 2.

The answer to will always be found in the inner band of the section that belongs to.

Each section of the wheel contains a group of numbers that are congruent to eachother. This is denoted using the triple bar equivelant sign . For example, .