Norm Prokup

**Cornell University**

PhD. in Mathematics

Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.

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One of the most commonly used step functions is **the greatest integer function**. The greatest integer function has it's own notation and tells us to round whatever decimal number it is given down to the nearest integer, or the greatest integer that is less than the number. The graph of the greatest integer function resembles an ascending staircase.

I want to talk about a new parent function called the greatest integer function and it's sometimes also called the "floor function." And this is the symbol for the greatest integer of x and the definition is the greatest integer less than or equal to x. Let's do some let's evaluate it for a couple of numbers. The greatest integer less than or equal to 0.5. If you go on the number line and find 0.5 right here. The greatest integer less than or equal to 0.5 is 0, so it's equal 0. Greatest integer less than or equal to 0.99 0.99 is not quite 1, so the greatest integer less than or equal to 0.99 is also 0. And remember that it's the greatest integer less than or equal to, so the greatest integer less than or equal to zero is itself zero. So all of these numbers have the same greatest integer zero.

What about negative 0.5? Negative 0.5 is right here. The greatest integer less than or equal to negative 0.5 is -1. How about negative 0.01. That's ever so slightly to the left of zero. So the greatest integer less than or equal to negative 0.01 is negative 1. And of course, the greatest integer less than or equal to -1 is -1 so all three of theses numbers have the same greatest integer, -1.

What about something like root 2? Square root of 2 is about 1.41, 1.41 is in here so the greatest integer less than or equal to root 2 is 1.

How about pi? Pi is a little more than 3, 3.14. The greatest integer less than or equal to pi is 3 and negative pi. Watch out for the greatest integer when you're dealing with negative numbers. Remember you're always travelling to the left on the number line. Negative pi would be -1, -2, -3 would be over here, and so the greatest integer less than or equal to negative pi would actually be -4.

So next we're going to graph this function, it's a very interesting graph.