## The Maple floor Function

In mathematics, the function that takes a real number as input and returns its integer part is called the greatest integer function. The notation used is and the formal definition is that is the largest integer n satisfying . Another common name for this function is the floor function, , and that is the name used by Maple. See the examples below.

```  > floor(21/4);
```

```  > floor(13.543);
```

```  > plot(floor(x),x=0..4);
```

```  > plot(floor(x),x=0..10);
```

Next, consider the plot generated by the following Maple command.

```  > plot(x-floor(x),x=0..4);
```

To help you understand the plot, recall that if , then so . Now, if then so . In fact, we could describe with the following formula

There is one thing to watch out for, though. Compare the plots generated by the following commands.

```  > plot(x-floor(x),x=0..8);
```

```  > plot(x-floor(x),x=0..8,numpoints=1000);
```

The first plot generated is not an accurate plot of . You can tell this because the peaks are not at the same height, as they should be. The second command uses the numpoints parameter to tell Maple to plot more points. There is nothing special about the value 1000, however, you should just increase the value of this parameter until you get a plot that looks right.

Now, suppose that L is a positive real number and consider the following expression.

You should be able to convince yourself that the following formula holds.

You can try the following Maple commands if you need more help visualizing how behaves.

```  > plot(x-2*floor(x/2),x=0..4);
```

```  > plot(x-6*floor(x/6),x=0..24,numpoints=250);
```

Notice how the value of L affects not only the interval at which the graph repeats, but also the maximum value attained.

Finally, suppose we have a function and we define a new function by

After a little thought, you should be able to convince yourself that

for any value of x. This means that the function only depends on the values of for . In fact the graph of for consists of copies of the graph of for translated to the right by multiples of L. As an example, consider the function . Try the following Maple commands.

```  > f := x -> x^2+1;
```

```  > g := x -> f(x-floor(x));
```

```  > plot(g(x),x=0..5,numpoints=200);
```

```  > h := x -> f(x-2*floor(x/2));
```

```  > plot(h(x),x=0..5);
```

```  > h3 :=  x -> f(x-3*floor(x/3));
```

```  > plot(h3(x),x=0..6);
```