12 multiple-choice questions, progressively harder.
The solution set of ⌊x⌋=4\lfloor x \rfloor = 4⌊x⌋=4 is an interval. What is its length?
Solution
Correct answer: D
The floor equals 444 exactly on 4≤x<54 \le x < 54≤x<5, one full step.
length=5−4=1\text{length} = 5 - 4 = 1length=5−4=1
Every floor level set is a half-open interval of length 111.
Solve ⌈x⌉=0\lceil x \rceil = 0⌈x⌉=0 for all real xxx.
Correct answer: A
The ceiling equals 000 on the step ending at 000, using n−1<x≤nn - 1 < x \le nn−1<x≤n.
−1<x≤0-1 < x \le 0−1<x≤0
The left end −1-1−1 is excluded and the right end 000 is included.
Evaluate ⌊−0.001⌋\lfloor -0.001 \rfloor⌊−0.001⌋.
Correct answer: C
The input is just below zero, so rounding down lands on −1-1−1.
−0.001 lies between −1 and 0, so ⌊−0.001⌋=−1-0.001 \text{ lies between } -1 \text{ and } 0, \text{ so } \lfloor -0.001 \rfloor = -1−0.001 lies between −1 and 0, so ⌊−0.001⌋=−1
Even a tiny step below 000 drops the floor to −1-1−1.
Compute the fractional part {2.0}\{2.0\}{2.0}, where {x}=x−⌊x⌋\{x\} = x - \lfloor x \rfloor{x}=x−⌊x⌋.
The input is already an integer, so its floor is itself.
{2.0}=2−⌊2⌋=2−2=0\{2.0\} = 2 - \lfloor 2 \rfloor = 2 - 2 = 0{2.0}=2−⌊2⌋=2−2=0
The fractional part is 000 exactly on the integers.
For which real numbers xxx is ⌊x⌋=⌈x⌉\lfloor x \rfloor = \lceil x \rceil⌊x⌋=⌈x⌉?
The floor and ceiling agree only where there is no rounding to do.
⌊x⌋=⌈x⌉ ⟺ x is an integer\lfloor x \rfloor = \lceil x \rceil \iff x \text{ is an integer}⌊x⌋=⌈x⌉⟺x is an integer
For a non-integer, ⌈x⌉=⌊x⌋+1\lceil x \rceil = \lfloor x \rfloor + 1⌈x⌉=⌊x⌋+1, so they differ by 111.
From those 535353 eggs packed 121212 to a carton, how many cartons come out completely full?
Correct answer: B
Only complete cartons count, so round down with a floor.
⌊53/12⌋=⌊4.41…⌋=4\lfloor 53 / 12 \rfloor = \lfloor 4.41\ldots \rfloor = 4⌊53/12⌋=⌊4.41…⌋=4
Four cartons are full at 484848 eggs; the remaining 555 do not fill a fifth.
With 100100100 people seated 121212 to a row, the last row is only partly full. How many people are in it?
Subtract off the people in the full rows, ⌊100/12⌋=8\lfloor 100/12 \rfloor = 8⌊100/12⌋=8 of them.
100−12×8=100−96=4100 - 12 \times 8 = 100 - 96 = 4100−12×8=100−96=4
The leftover is exactly what the complete rows do not use.
How many of the 535353 eggs are left in the last, partly filled carton (12 to a carton)?
Subtract the eggs used by the full cartons, ⌊53/12⌋=4\lfloor 53/12 \rfloor = 4⌊53/12⌋=4 of them.
53−12×4=53−48=553 - 12 \times 4 = 53 - 48 = 553−12×4=53−48=5
The leftover is what the complete cartons do not use.
Evaluate ⌈−4⌉−⌊−4⌋\lceil -4 \rceil - \lfloor -4 \rfloor⌈−4⌉−⌊−4⌋.
The input −4-4−4 is an integer, so the floor and ceiling both return it.
⌈−4⌉−⌊−4⌋=−4−(−4)=0\lceil -4 \rceil - \lfloor -4 \rfloor = -4 - (-4) = 0⌈−4⌉−⌊−4⌋=−4−(−4)=0
The difference is 000 because −4-4−4 needs no rounding.
Evaluate ⌊−2.3⌋+⌈2.3⌉\lfloor -2.3 \rfloor + \lceil 2.3 \rceil⌊−2.3⌋+⌈2.3⌉.
Round −2.3-2.3−2.3 down and 2.32.32.3 up, then add.
⌊−2.3⌋+⌈2.3⌉=−3+3=0\lfloor -2.3 \rfloor + \lceil 2.3 \rceil = -3 + 3 = 0⌊−2.3⌋+⌈2.3⌉=−3+3=0
The floor of −2.3-2.3−2.3 is −3-3−3 and the ceiling of 2.32.32.3 is 333, so they cancel.
For which real numbers xxx is ⌊x⌋=x\lfloor x \rfloor = x⌊x⌋=x?
The floor equals its input only when no rounding down is needed.
⌊x⌋=x ⟺ x is an integer\lfloor x \rfloor = x \iff x \text{ is an integer}⌊x⌋=x⟺x is an integer
For any non-integer, ⌊x⌋\lfloor x \rfloor⌊x⌋ is strictly less than xxx.
Rounding 3.53.53.5 to the nearest integer with ⌊x+12⌋\lfloor x + \tfrac{1}{2} \rfloor⌊x+21⌋, evaluate ⌊3.5+12⌋\lfloor 3.5 + \tfrac{1}{2} \rfloor⌊3.5+21⌋.
Add one half, then take the floor.
⌊3.5+12⌋=⌊4⌋=4\lfloor 3.5 + \tfrac{1}{2} \rfloor = \lfloor 4 \rfloor = 4⌊3.5+21⌋=⌊4⌋=4
The half-step pushes 3.53.53.5 up to 444 before the floor rounds it.
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