The connection between quotient, remainder, modulo and floor functions motivates the following generalization of congruences:

# Definition: Modulo Operation for Real Numbers

Using the definition of the floor function, we define for any two real numbers $$x,y$$ with $$y\neq 0$$ the modulo operation "$$\operatorname{mod}$$" as follows:

$x\mod y:=x-y\cdot \left\lfloor \frac xy \right\rfloor.$

This definition leaves the case $$y=0$$ undefined, in order to avoid division by zero. For $$y=0$$, we set1

$x\mod 0:=x.$

The number $$y$$ after "$$\operatorname{mod}$$" is called the modulus.

### Example

In the following interactive figure you can study the graph of $$x\mod y$$ by manipulating the value of $$y$$:

Examples: 1

Github: ### References

#### Bibliography

1. Graham L. Ronald, Knuth E. Donald, Patashnik Oren: "Concrete Mathematics", Addison-Wesley, 1994, 2nd Edition

#### Footnotes

1. This convention preserves the property that $$x\operatorname{mod}y$$ always differs from $$x$$ by a multiple of $$y$$.