Integer and Fractional Part: Floor, Ceiling & Rounding

The Integer Part (Floor)

The integer part of a real number

x

, also called its floor, is the greatest integer that is less than or equal to

x

. It is written

\lfloor x \rfloor

.

The Integer Part (Floor)

\lfloor x \rfloor = n \quad \text{where } n \text{ is the integer with } n \leq x < n + 1

On the number line, the floor is found by moving left from

x

to the nearest integer, or staying put if

x

is already an integer.

Example

\lfloor 3.7 \rfloor = 3

,

\; \lfloor 5 \rfloor = 5

,

\; \lfloor \pi \rfloor = 3

,

\; \lfloor 0.99 \rfloor = 0

.

Pitfall: Dropping the decimals of a negative number

Writing

\lfloor -3.7 \rfloor = -3

. But

-3

is greater than

-3.7

, so it is not below it. The greatest integer below

-3.7

is

-4

.

The Ceiling

The ceiling is the mirror image of the floor: the least integer that is greater than or equal to

x

, written

\lceil x \rceil

.

The Ceiling

\lceil x \rceil = n \quad \text{where } n \text{ is the integer with } n - 1 < x \leq n

On the number line, the ceiling is found by moving right to the nearest integer, or staying put at an integer.

Example

\lceil 3.2 \rceil = 4

,

\; \lceil 5 \rceil = 5

,

\; \lceil -3.2 \rceil = -3

,

\; \lceil -0.9 \rceil = 0

.

For an integer, floor and ceiling are equal. For any other number, the ceiling is exactly one more than the floor:

\lceil x \rceil = \lfloor x \rfloor + 1

.

Pitfall: Adding 1 to an integer

Using

\lceil x \rceil = \lfloor x \rfloor + 1

for every

x

gives

\lceil 5 \rceil = 6

. At an integer nothing moves:

\lceil 5 \rceil = 5

.

The Fractional Part

The fractional part of

x

is what is left after the integer part is taken away. It is written

\{x\}

.

The Fractional Part

\{x\} = x - \lfloor x \rfloor, \qquad 0 \leq \{x\} < 1

Every real number splits into its two parts:

x = \lfloor x \rfloor + \{x\}

. The integer part says which unit interval

x

is in; the fractional part says how far

x

is from the left end of that interval.

Example

\{3.7\} = 3.7 - 3 = 0.7

,

\; \{5\} = 0

,

\; \{\pi\} = \pi - 3 \approx 0.14159

.

Pitfall: A negative fractional part

Writing

\{-3.7\} = -0.7

. With the floor

\lfloor -3.7 \rfloor = -4

, the fractional part is

-3.7 - (-4) = 0.3

, never negative.

Integer Part Notation

Notation

Integer Part Notation

Brackets that keep only one edge, braces for what is left over, and a rounding rule built from the floor — the page's four ways of cutting a real number to an integer.

The condition

n \leq x < n + 1

is a half-open interval, read as in interval notation; remainders come from divisibility notation.

\lfloor x \rfloor

floor of x — the integer part of x

The greatest integer at most

x

: the integer

n

with

n \leq x < n + 1

. Only the bottom edges of the brackets are drawn — the direction is down, toward

-\infty

, never toward zero.

CasesFor negative inputs, down means away from zero:

\lfloor -3.7 \rfloor = -4

, not

-3

. At an integer nothing moves:

\lfloor 5 \rfloor = 5

.

Also written

[x]

— older books, under the name greatest integer function.

\text{floor}(x)

,

\text{INT}(x)

— calculators, spreadsheets and code (section 10).

Do not confuseTruncation. Cutting off the decimals agrees with the floor only for

x \geq 0

: truncating

-3.7

gives

-3

, the floor gives

-4

.

Same glyph elsewhere

[x]

— the old floor — is also plain grouping brackets and, with two entries, the closed interval

[a, b]

. The one-edged brackets replaced it for that reason.

\lceil x \rceil

ceiling of x

The least integer at least

x

: the integer

n

with

n - 1 < x \leq n

. Top edges only — the direction is up, toward

+\infty

.

CasesOff the integers,

\lceil x \rceil = \lfloor x \rfloor + 1

; on an integer the two coincide,

\lceil 5 \rceil = 5

. The mirror rule

\lceil x \rceil = -\lfloor -x \rfloor

turns any ceiling into a floor.

Also written

\text{ceil}(x)

,

\text{CEILING}(x)

— calculators and code.

Do not confuseRounding away from zero. For negatives the ceiling moves toward zero:

\lceil -3.2 \rceil = -3

, not

-4

.

\{x\}

fractional part of x

What the floor leaves behind:

\{x\} = x - \lfloor x \rfloor

, always with

0 \leq \{x\} < 1

. The two brackets rebuild the number:

x = \lfloor x \rfloor + \{x\}

.

CasesNegative inputs:

\{-3.7\} = 0.3

, not

-0.7

— the digits after the point are not the fractional part. Every integer has

\{x\} = 0

.

Also written

\text{frac}(x)

;

x \bmod 1

— the remainder on division by

1

.

Do not confuseThe truncation version some calculators use, which returns

-0.7

for

-3.7

and breaks the range

0 \leq \{x\} < 1

.

Same glyph elsewhereSet braces:

\{x\}

is also the one-element set holding

x

. Where both could appear, write

\text{frac}(x)

.

\text{round}(x) = \left\lfloor x + \tfrac{1}{2} \right\rfloor

x rounded to the nearest integer

Rounding built from the floor: shift by one half, then floor. This version sends ties up:

2.5 \to 3

,

-2.5 \to -2

.

CasesThree tie rules are in use — half up, half away from zero, half to even (section 5). They disagree only at exact halves.

Also written

\lfloor x \rceil

— mixed brackets, occasionally used for the nearest integer;

\text{ROUND}(x, 0)

in spreadsheets.

Do not confuseRounding in code. Python sends

2.5

to

2

(half to even), while the formula above gives

3

.

All arithmetic symbols →↑ Arithmetic

Negative Numbers

For positive numbers, the floor is the same as dropping the digits after the decimal point. For negative numbers it is not, and this is where almost every mistake with floor and fractional part happens.

Dropping the decimals, called truncation, always moves toward zero. The floor always moves down. For positive numbers down is toward zero, so the two agree; for negative numbers down is away from zero, so they differ by one.

RULE

For a negative number that is not an integer, truncation gives the ceiling, not the floor: dropping the decimals of

-3.7

gives

-3 = \lceil -3.7 \rceil

.

The table compares the operations on pairs of opposite numbers. The fractional parts of

3.7

and

-3.7

are

0.7

and

0.3

, which add up to

1

; for every non-integer

x

,

\{x\} + \{-x\} = 1

.

The same difference splits modulo of negative numbers into two conventions: truncated division rounds the quotient toward zero, floored division rounds it down.

x	⌊x⌋ (floor)	⌈x⌉ (ceiling)	drop the decimals	{x} (fractional part)
3.7	3	4	3	0.7
−3.7	−4	−3	−3	0.3
0.5	0	1	0	0.5
−0.5	−1	0	0	0.5
5	5	5	5	0
−5	−5	−5	−5	0

Rounding to the Nearest Integer

Rounding picks the nearest integer. Floor and ceiling give the two candidates; rounding chooses whichever is closer.

Rounding to the Nearest Integer

\text{round}(x) = \left\lfloor x + \tfrac{1}{2} \right\rfloor

Adding

\frac{1}{2}

before taking the floor moves every number whose fractional part is at least

\frac{1}{2}

past the next integer. So

2.4

becomes

\lfloor 2.9 \rfloor = 2

and

2.6

becomes

\lfloor 3.1 \rfloor = 3

.

The only real choice is what to do with a tie, a number exactly halfway between two integers. Three rules are in common use:

Half up: ties go up, toward $+\infty$ . This is the formula above: $2.5 \to 3$ and $-2.5 \to -2$ .
Half away from zero: ties go away from zero, the usual school rule: $2.5 \to 3$ and $-2.5 \to -3$ .
Half to even: ties go to the even neighbour: $2.5 \to 2$ , $3.5 \to 4$ . Rounding a long list this way pushes up and down equally often, so totals do not drift; banks, and some programming languages such as Python, use it for that reason.

The rules disagree only on ties. For every other number, all three give the nearest integer.

Which numbers round to which integer

Each integer collects the numbers within ½ of it. Under round half up, the tie 2.5 belongs to the interval of 3: the filled dot at 2.5 is on the lower segment, and the hollow dot on the upper one marks that 2.5 does not round to 2.

x	floor	ceiling	toward zero	half up ⌊x + ½⌋	half away from zero	half to even
2.4	2	3	2	2	2	2
2.5	2	3	2	3	3	2
3.5	3	4	3	4	4	4
−2.5	−3	−2	−2	−2	−3	−2
−2.6	−3	−2	−2	−3	−3	−3

Properties of Floor, Ceiling and Fractional Part

The rules below hold for every real

x

and

y

and every integer

n

.

RULE

An integer can be moved in and out of the floor:

\lfloor x + n \rfloor = \lfloor x \rfloor + n

. A number that is not an integer cannot.

The second half of the rule is where errors come from. In general

\lfloor x + y \rfloor \neq \lfloor x \rfloor + \lfloor y \rfloor

: the two fractional parts may add up to more than

1

and carry an extra unit. For the same reason

\lfloor 2x \rfloor

is not

2\lfloor x \rfloor

:

\lfloor 2 \cdot 1.6 \rfloor = 3

, while

2\lfloor 1.6 \rfloor = 2

.

Property	Rule	Example
Bounds	⌊x⌋ ≤ x < ⌊x⌋ + 1, equivalently x − 1 < ⌊x⌋ ≤ x	⌊2.3⌋ = 2 ≤ 2.3 < 3
Range of the fractional part	0 ≤ {x} < 1	{−0.1} = 0.9
Integers are fixed	⌊n⌋ = ⌈n⌉ = n and {n} = 0 for integer n	⌊−4⌋ = −4
Adding an integer	⌊x + n⌋ = ⌊x⌋ + n, {x + n} = {x}	⌊7.2⌋ = ⌊0.2⌋ + 7 = 7
Floor and ceiling	⌈x⌉ = −⌊−x⌋	⌈2.3⌉ = −⌊−2.3⌋ = −(−3) = 3
Gap between them	⌈x⌉ − ⌊x⌋ = 0 for integers, 1 otherwise	⌈2.3⌉ − ⌊2.3⌋ = 1
Opposites	⌊x⌋ + ⌊−x⌋ = 0 for integers, −1 otherwise	⌊2.3⌋ + ⌊−2.3⌋ = 2 − 3 = −1
Sums	⌊x⌋ + ⌊y⌋ ≤ ⌊x + y⌋ ≤ ⌊x⌋ + ⌊y⌋ + 1	⌊0.6 + 0.6⌋ = 1, ⌊0.6⌋ + ⌊0.6⌋ = 0
Nested division	⌊⌊x⌋ / n⌋ = ⌊x / n⌋ for positive integer n	⌊⌊7.9⌋ / 2⌋ = ⌊3.5⌋ = 3 = ⌊3.95⌋

The Graphs

The graph of

y = \lfloor x \rfloor

is a staircase. On each interval

[n, n + 1)

the function is constant at

n

, so the graph is a horizontal step with a filled dot at the left end, which belongs to the step, and a hollow dot at the right end, which does not. At every integer the graph jumps up by

1

.

The graph of

y = \{x\}

is a sawtooth. On each interval

[n, n + 1)

it equals

x - n

: it rises with slope

1

from

0

toward

1

, then drops back to

0

at the next integer. It repeats with period

1

, and its range is

[0, 1)

.

Both functions are continuous everywhere except at the integers, where they have jump discontinuities: the limit from the left of

\lfloor x \rfloor

at

x = n

is

n - 1

, the limit from the right is

n

. Between the integers the floor is constant, so its derivative there is

0

.

The step functions section of the function families page and the step functions section of the piecewise functions page treat these graphs as functions.

The staircase

Each step covers one interval [n, n + 1) at height n. The filled dot shows that the left end belongs to the step; the hollow dot shows that the right end does not, because at the next integer the floor has already jumped up by 1.

The sawtooth

Between consecutive integers the fractional part rises with slope 1 from 0 toward 1, never reaching it. At each integer it drops back to 0. The graph to the left of 0 is identical to the graph on the right: {−0.3} = 0.7, the same height as {0.7}.

Floor and Division

Dividing an integer

a

by a positive integer

n

gives a quotient and a remainder. The floor computes both.

Floor and Division

q = \left\lfloor \frac{a}{n} \right\rfloor, \qquad r = a - n\left\lfloor \frac{a}{n} \right\rfloor = a \bmod n

Example

23 \div 5

:

\; q = \lfloor 4.6 \rfloor = 4

and

r = 23 - 5 \cdot 4 = 3

, so

23 = 5 \cdot 4 + 3

.

Example

-23 \div 5

:

\; q = \lfloor -4.6 \rfloor = -5

and

r = -23 - 5 \cdot (-5) = 2

, so

-23 = 5 \cdot (-5) + 2

.

The remainder is also the fractional part of the quotient, scaled back up:

r = n \left\{ \frac{a}{n} \right\}

. For

-23 \div 5

,

\{-4.6\} = 0.4

and

5 \cdot 0.4 = 2

.

Because the floor always rounds down, this remainder is always in

\{0, 1, \ldots, n - 1\}

, even for a negative

a

. That is the convention used for modulo in mathematics; the modulo calculator shows the steps for any pair of numbers.

Counting with Floor and Ceiling

Many counting questions ask how many whole things fit, or how many are needed. The first kind is answered by a floor, the second by a ceiling.

Example

How many multiples of

7

are there from

1

to

100

?

\; 100 \div 7 \approx 14.29

, and

\lfloor 14.29 \rfloor = 14

: they are

7, 14, \ldots, 98

.

Example

50

students travel in buses that seat

12

.

\; 50 \div 12 \approx 4.17

. Full buses:

\lfloor 4.17 \rfloor = 4

, carrying

48

students. Buses needed so that nobody is left behind:

\lceil 4.17 \rceil = 5

.

Example

How many digits does

4821

have?

\; \log_{10} 4821 \approx 3.68

, and

\lfloor 3.68 \rfloor + 1 = 4

. In general a positive integer

N

has

\lfloor \log_{10} N \rfloor + 1

digits.

The test for which one to use: if a part that does not fill a whole group is dropped, use the floor; if it still needs a group of its own, use the ceiling.

Floor counts full groups, ceiling counts groups needed

The tick marks are the multiples of 12. The amber dot, 50, lies between 48 and 60. Four full buses carry 48 students; the 2 left over still need a bus of their own, so the number of buses is the next multiple up: 5.

In Calculators and Programming Languages

Software offers floor, ceiling, truncation and rounding as separate operations, and the names do not always say which is which.

Floor and ceiling: $\texttt{floor}$ and $\texttt{ceil}$ in most languages, and $\texttt{INT}$ in spreadsheets, which rounds down: $\texttt{INT}(-3.7) = -4$ .
Truncation: converting to an integer type usually truncates. In Python, C and Java, the conversion of $-3.7$ gives $-3$ ; JavaScript has $\texttt{Math.trunc}$ .
Integer division: Python's $\texttt{//}$ floors, so $-7 \; \texttt{//} \; 2 = -4$ . In C and Java, integer division truncates and gives $-3$ . JavaScript has no integer division: $-7 / 2 = -3.5$ , and $\texttt{Math.trunc}$ turns it into $-3$ .
Rounding ties: Python's $\texttt{round}$ rounds half to even, so $\texttt{round}(2.5) = 2$ . JavaScript's $\texttt{Math.round}$ rounds half up, so $\texttt{Math.round}(-2.5) = -2$ .

Positive test inputs cannot tell these apart. Code that floors, truncates or rounds should always be tested with a negative number and with a value exactly halfway between two integers.

Common Pitfalls

Dropping the decimals of a negative number

The floor of a negative number is found by deleting the digits after the decimal point.

Wrong working:

\lfloor -3.7 \rfloor = -3

.

Right working: the floor is the greatest integer at most

-3.7

. Since

-3 > -3.7

, it does not qualify;

-4 \leq -3.7

does. So

\lfloor -3.7 \rfloor = -4

.

Why it happens: for positive numbers, deleting the decimals and taking the floor give the same result, so the floor is remembered as "delete the decimals".

How to check: the floor must not be greater than the number. On the number line, it is on the left of

x

or at

x

.

A negative fractional part

The fractional part of a negative number is read off as the digits after the decimal point, with a minus sign.

Wrong working:

\{-3.7\} = -0.7

.

Right working:

\{-3.7\} = -3.7 - \lfloor -3.7 \rfloor = -3.7 - (-4) = 0.3

.

Why it happens: the fractional part is thought of as "the decimal digits" rather than as

x - \lfloor x \rfloor

.

How to check: the fractional part is always in

[0, 1)

, and

\lfloor x \rfloor + \{x\}

must give back

x

:

-4 + 0.3 = -3.7

.

Adding 1 to an integer

The ceiling is computed as the floor plus one, without checking whether

x

is an integer.

Wrong working:

\lceil 5 \rceil = \lfloor 5 \rfloor + 1 = 6

, and so

\lceil 60 / 12 \rceil = 6

buses for

60

students in buses of

12

.

Right working: an integer is its own ceiling,

\lceil 5 \rceil = 5

: five buses are enough.

Why it happens:

\lceil x \rceil = \lfloor x \rfloor + 1

is true for every non-integer, so the exception is forgotten.

How to check: test the case where the division comes out exact.

Splitting the floor over a sum

The floor is distributed over a sum or pulled out of a multiple.

Wrong working:

\lfloor 0.6 + 0.6 \rfloor = \lfloor 0.6 \rfloor + \lfloor 0.6 \rfloor = 0

, and

\lfloor 2 \cdot 1.6 \rfloor = 2 \lfloor 1.6 \rfloor = 2

.

Right working:

\lfloor 1.2 \rfloor = 1

and

\lfloor 3.2 \rfloor = 3

. Only an integer term can be taken out:

\lfloor x + n \rfloor = \lfloor x \rfloor + n

.

Why it happens: the floor is treated like a linear operation, such as multiplying by a constant.

How to check: add the fractional parts. If they sum to

1

or more, the floor of the sum is one larger than the sum of the floors.

Summary: Integer and Fractional Part at a Glance

The definitions, the behaviour on negative numbers, the rules and the uses on this page reduce to the table below.

Idea	Statement	Example
Floor (integer part)	⌊x⌋ is the greatest integer ≤ x	⌊3.7⌋ = 3, ⌊−3.7⌋ = −4
Ceiling	⌈x⌉ is the least integer ≥ x	⌈3.2⌉ = 4, ⌈−3.2⌉ = −3
Fractional part	{x} = x − ⌊x⌋, always in [0, 1)	{3.7} = 0.7, {−3.7} = 0.3
Negative numbers	floor moves down, away from zero; dropping the decimals moves toward zero	⌊−3.7⌋ = −4, trunc(−3.7) = −3
Rounding	nearest integer; ties need a rule	2.5 → 3 (half up) or 2 (half to even)
Integers pass through	⌊x + n⌋ = ⌊x⌋ + n	⌊7.2⌋ = 7
Division	q = ⌊a / n⌋, r = a − n⌊a / n⌋	−23 ÷ 5: q = −5, r = 2
Counting	⌊N / k⌋ multiples of k up to N; ⌈N / k⌉ groups of k to hold N	⌈50 / 12⌉ = 5 buses
Graphs	⌊x⌋ is a staircase; {x} is a sawtooth; both jump at every integer	jump of 1 at x = 2

Integer and Fractional Part FAQ

What is the fractional part of a negative number?

+

With the standard definition {x} = x − ⌊x⌋, the fractional part is always between 0 and 1. For −3.7 it is −3.7 − (−4) = 0.3, not −0.7. Some calculators use a sign-keeping version instead and give −0.7.Read more →

What is the difference between floor and truncation?

+

Floor rounds down, toward negative infinity; truncation deletes the decimal digits, which rounds toward zero. They agree for positive numbers and differ by one for negative non-integers: ⌊−3.7⌋ = −4, while truncating −3.7 gives −3.Read more →

Why does 2.5 sometimes round to 2?

+

Under the rule round half to even, a number exactly halfway between two integers goes to the even one: 2.5 rounds to 2 and 3.5 to 4. Python's round function and many financial systems use this rule because it keeps totals from drifting upward.Read more →

Does the floor of a sum equal the sum of the floors?

+

No, not in general. The fractional parts can add up to

1

or more and carry an extra unit:

\lfloor 0.6 + 0.6 \rfloor = 1

, while

\lfloor 0.6 \rfloor + \lfloor 0.6 \rfloor = 0

. The floor of the sum is always equal to the sum of the floors or one more. Only an integer can be moved out freely,

\lfloor x + n \rfloor = \lfloor x \rfloor + n

, so

\lfloor 2 \cdot 1.6 \rfloor = 3

while

2\lfloor 1.6 \rfloor = 2

.Read more →

Should you use floor or ceiling in a word problem?

+

It depends on what happens to the part left over. If an incomplete group is simply dropped, use the floor; if it still needs a group of its own, use the ceiling. With

50

students and buses seating

12

,

50 \div 12 \approx 4.17

:

\lfloor 4.17 \rfloor = 4

buses are full, but

\lceil 4.17 \rceil = 5

buses are needed. When the division is exact, as with

60

students, both give

5

.Read more →

Does the INT function in a spreadsheet round negative numbers down?

+

Yes. Spreadsheet

\texttt{INT}

is the floor, so

\texttt{INT}(-3.7) = -4

, not

-3

. Converting

-3.7

to an integer type in Python, C or Java truncates instead and gives

-3

. Integer division splits the same way: Python's

\texttt{//}

floors, so

-7 \; \texttt{//} \; 2 = -4

, while C and Java truncate to

-3

. Positive inputs cannot tell these apart, so test with a negative number.Read more →

Integer and Fractional Part

Splitting a Number in Two

Key Terms

The Integer Part (Floor)

The Ceiling

The Fractional Part

Integer Part Notation

Integer Part Notation

Negative Numbers

Rounding to the Nearest Integer

Properties of Floor, Ceiling and Fractional Part

The Graphs

Floor and Division

Counting with Floor and Ceiling

In Calculators and Programming Languages

Common Pitfalls

Summary: Integer and Fractional Part at a Glance

Integer and Fractional Part FAQ

What is the fractional part of a negative number?

What is the difference between floor and truncation?

Why does 2.5 sometimes round to 2?

Does the floor of a sum equal the sum of the floors?

Should you use floor or ceiling in a word problem?

Does the INT function in a spreadsheet round negative numbers down?