How do the floor and ceiling functions work on negative numbers?

It’s clear to me how these functions work on positive real numbers: you round up or down accordingly. But if you have to round a negative real number: to take 0.8 to 1, then do you take the floor of 0.8, or the ceiling?

That is, which of the following are true?

0.8=1

or
0.8=1

Answer

The first is the correct: you round “down” (i.e. the greatest integer LESS THAN 0.8).

In contrast, the ceiling function rounds “up” to the least integer GREATER THAN 0.8=0.

0.8=1since1<0.8<00.8=0since1<0.8<0

In general, we must have that xxxxR

And so it follows that 1=0.80.80.8=0


K.Stm’s suggestion is a nice, intuitive way to recall the relation between the floor and the ceiling of a real number x, especially when x<0. Using the “number line” idea and plotting 0.8 with the two closest integers that “sandwich” 0.8 gives us:

enter image description here

We see that the floor of x=0.8 is the first integer immediately to the left of 0.8, and the ceiling of x=0.8 is the first integer immediately to the right of 0.8, and this strategy can be used, whatever the value of a real number x.

Attribution
Source : Link , Question Author : Mirrana , Answer Author : amWhy

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