12 multiple-choice questions, progressively harder.
Evaluate ⌊−0.5⌋\lfloor -0.5 \rfloor⌊−0.5⌋.
Solution
Correct answer: B
The floor rounds down to the nearest integer to the left.
−0.5 lies between −1 and 0, so ⌊−0.5⌋=−1-0.5 \text{ lies between } -1 \text{ and } 0, \text{ so } \lfloor -0.5 \rfloor = -1−0.5 lies between −1 and 0, so ⌊−0.5⌋=−1
Rounding down from −0.5-0.5−0.5 moves away from zero, landing on −1-1−1.
Evaluate ⌈−0.5⌉\lceil -0.5 \rceil⌈−0.5⌉.
Correct answer: A
The ceiling is the nearest integer to the right.
−0.5 lies between −1 and 0, so ⌈−0.5⌉=0-0.5 \text{ lies between } -1 \text{ and } 0, \text{ so } \lceil -0.5 \rceil = 0−0.5 lies between −1 and 0, so ⌈−0.5⌉=0
Rounding up from −0.5-0.5−0.5 moves toward zero, landing on 000.
Evaluate ⌊−7.2⌋\lfloor -7.2 \rfloor⌊−7.2⌋.
Correct answer: D
The floor rounds toward negative infinity.
−7.2 lies between −8 and −7, so ⌊−7.2⌋=−8-7.2 \text{ lies between } -8 \text{ and } -7, \text{ so } \lfloor -7.2 \rfloor = -8−7.2 lies between −8 and −7, so ⌊−7.2⌋=−8
The greatest integer at or below −7.2-7.2−7.2 is −8-8−8, not −7-7−7.
Evaluate ⌈−7.2⌉\lceil -7.2 \rceil⌈−7.2⌉.
Correct answer: C
The ceiling is the least integer at or above the input.
−7.2 lies between −8 and −7, so ⌈−7.2⌉=−7-7.2 \text{ lies between } -8 \text{ and } -7, \text{ so } \lceil -7.2 \rceil = -7−7.2 lies between −8 and −7, so ⌈−7.2⌉=−7
The nearest integer to the right of −7.2-7.2−7.2 is −7-7−7.
Solve ⌈x⌉=3\lceil x \rceil = 3⌈x⌉=3 for all real xxx.
The ceiling equals 333 on the step that ends at 333, using n−1<x≤nn - 1 < x \le nn−1<x≤n.
2<x≤32 < x \le 32<x≤3
The left end 222 is excluded (its ceiling is 222) and the right end 333 is included.
Solve ⌈x⌉=−2\lceil x \rceil = -2⌈x⌉=−2 for all real xxx.
The ceiling equals −2-2−2 on the step ending at −2-2−2, using n−1<x≤nn - 1 < x \le nn−1<x≤n.
−3<x≤−2-3 < x \le -2−3<x≤−2
The left end −3-3−3 is excluded and the right end −2-2−2 is included.
Compute the fractional part {4.3}\{4.3\}{4.3}, where {x}=x−⌊x⌋\{x\} = x - \lfloor x \rfloor{x}=x−⌊x⌋.
Subtract the floor from the number.
{4.3}=4.3−⌊4.3⌋=4.3−4=0.3\{4.3\} = 4.3 - \lfloor 4.3 \rfloor = 4.3 - 4 = 0.3{4.3}=4.3−⌊4.3⌋=4.3−4=0.3
For a positive input the fractional part is just the digits after the decimal point.
Compute the fractional part {−4.3}\{-4.3\}{−4.3}, where {x}=x−⌊x⌋\{x\} = x - \lfloor x \rfloor{x}=x−⌊x⌋.
The floor of −4.3-4.3−4.3 rounds down to −5-5−5, so subtract that.
{−4.3}=−4.3−⌊−4.3⌋=−4.3−(−5)=0.7\{-4.3\} = -4.3 - \lfloor -4.3 \rfloor = -4.3 - (-5) = 0.7{−4.3}=−4.3−⌊−4.3⌋=−4.3−(−5)=0.7
The fractional part is never negative, so the answer must land in [0,1)[0, 1)[0,1).
If ⌊x⌋=6\lfloor x \rfloor = 6⌊x⌋=6 and xxx is not an integer, what is ⌈x⌉\lceil x \rceil⌈x⌉?
For a non-integer input the ceiling is one more than the floor.
⌈x⌉=⌊x⌋+1=6+1=7\lceil x \rceil = \lfloor x \rfloor + 1 = 6 + 1 = 7⌈x⌉=⌊x⌋+1=6+1=7
Since xxx sits strictly between 666 and 777, rounding up gives 777.
On the graph of y=⌊x⌋y = \lfloor x \rfloory=⌊x⌋, the step covering 2≤x<32 \le x < 32≤x<3 has its filled (closed) dot at which point?
Each floor step is closed on its left end, where the function actually equals that height.
⌊2⌋=2, so the closed dot is at (2,2)\lfloor 2 \rfloor = 2, \text{ so the closed dot is at } (2, 2)⌊2⌋=2, so the closed dot is at (2,2)
The right end (3,2)(3, 2)(3,2) is open, because at x=3x = 3x=3 the floor has jumped to 333.
A group of 303030 people travels in cars that each seat 888. How many cars are needed?
Every leftover person still needs a car, so round up with a ceiling.
⌈30/8⌉=⌈3.75⌉=4\lceil 30 / 8 \rceil = \lceil 3.75 \rceil = 4⌈30/8⌉=⌈3.75⌉=4
Three cars seat 242424 people; the remaining 666 need a fourth car.
Which inequality is true for every real number xxx?
The floor sits at or below xxx and the ceiling sits at or above it.
⌊x⌋≤x≤⌈x⌉\lfloor x \rfloor \le x \le \lceil x \rceil⌊x⌋≤x≤⌈x⌉
Both become equalities exactly when xxx is an integer.
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