NDA Maths · Functions

The Greatest Integer (Floor) Function

⌊x⌋ rounds down to the nearest integer; its staircase graph, fractional part, and floor-equations are the testable pieces.

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Why this matters

Seven PYQs, and not one is EASY — every question here is MODERATE or HARD, so the floor function punches above its weight. The traps cluster around negatives (⌊−1.3⌋ = −2, not −1), the jump-down behaviour at integers, and the fractional part {x} = x − ⌊x⌋. Solving floor-equations and floor-sums rounds it out.

Concept 1 of 3

Definition, graph, and behaviour at integers

Intuition

\lfloor x\rfloor

(written

[x]

in NDA papers) is the **greatest integer not exceeding

x

— round down** on the number line. The graph is a staircase: flat on each

[n,n+1)

, then a jump up by 1 at the next integer.

Definition

$[x]=n$ where $n$ is the unique integer with $n\le x<n+1$ . Properties:

$[x]\le x<[x]+1$ , and $[x]=x$ exactly when $x$ is an integer.
Discontinuous (jump of 1) at every integer; constant in between, so its derivative is 0 on each open interval $(n,n+1)$ .
Range is $\mathbb{Z}$ ; domain is $\mathbb{R}$ .

Worked example

Evaluate

[2.7]

[-1.3]

, and

[5]

$[2.7]$ : greatest integer $\le 2.7$ is $2$ .
$[-1.3]$ : greatest integer $\le -1.3$ is $-2$ (round down, not toward zero).
$[5]$ : $5$ is already an integer, so $[5]=5$ .

Answer:

[2.7]=2,\ [-1.3]=-2,\ [5]=5

Practice this concept4 quick reps

Practice — Level 1 (4 reps)

Quick reps to lock in the method. Try each, then check.

1.
$[3.99]$ ?
2.
$[-0.5]$ ?
3.
Is $[x]$ continuous at $x=2$ ?
4.
Derivative of $[x]$ at $x=2.4$ ?

From the bank · past-year question

Example 1FunctionsMODERATE

Consider the following statements in respect of the function

y=[x]

x\in(-1,1)

where

[.]

is the greatest integer function: 1. Its derivative is 0 at

x=0.5

2. It is continuous at

x=0

Which of the above statements is/are correct?

[Q56 · Apr · 2022]

Floor rounds DOWN, so negatives go further from zero

[-1.3]=-2

, not

-1

: you must go to the integer below. Truncating toward zero is the most common floor mistake on negative inputs.

Drill 1 more on definition, graph, and behaviour at integers

Concept 2 of 3

The fractional part {x} = x − [x]

Intuition

Whatever the floor throws away is the fractional part

\{x\}=x-[x]

. It is the leftover after rounding down, so it always lands in

[0,1)

— even for negative

x

Definition

$\{x\}=x-[x]\in[0,1)$ for every real $x$ . Equivalently $[x]-x=-\{x\}\in(-1,0]$ . It is periodic with period 1. For a non-integer $x$ , $[x]-x$ is strictly between $-1$ and $0$ , so $\big[[x]-x\big]=-1$ .

Worked example

Find

\{-2.3\}

$[-2.3]=-3$ (floor rounds down).
$\{-2.3\}=-2.3-(-3)=0.7$ .
As expected, the result is in $[0,1)$ .

Answer:

\{-2.3\}=0.7

Practice this conceptself-check

Try it yourself

x

is positive and not an integer, what is

\big[[x]-x\big]

From the bank · past-year question

Example 2FunctionsMODERATE

Let

z = [y]

and

y = [x] - x

, where [.] is the greatest integer function. If x is not an integer but positive, then what is the value of z?

[Q71 · Sep · 2022]

Drill 1 more on the fractional part {x} = x − [x]

Concept 3 of 3

Floor equations and sums

Intuition

To solve an equation in

[x]

, treat

[x]

as an integer unknown

n

, solve the ordinary equation for

n

, then translate each integer back to the interval

[n,n+1)

. For floor sums, count how many terms share each floor value.

Definition

Key idea: $[x]=n\iff x\in[n,n+1)$ . So a polynomial equation in $[x]$ becomes a polynomial in the integer $n$ ; solve it, keep integer roots, and convert. For sums $\sum[\,\cdot\,]$ , group the index range by where the floor value changes.

Worked example

Solve

[x]^2-5[x]+6=0

for real

x

Let $n=[x]$ : $n^2-5n+6=0\Rightarrow(n-2)(n-3)=0$ , so $n=2$ or $3$ .
$[x]=2\Rightarrow x\in[2,3)$ ; $[x]=3\Rightarrow x\in[3,4)$ .
Union the intervals.

Answer:

x\in[2,4)

Practice this conceptself-check

Try it yourself

How many real

x

satisfy

[x]=4

? Describe them.

From the bank · past-year question

Example 3FunctionsMODERATE

f(x)=[x]^2-30[x]+221=0

, where

[x]

is the greatest integer function, then what is the sum of all integer solutions?

[Q77 · Sep · 2024]

[x] = n is an interval, not a single point

Solving a floor-equation gives **integer values of

[x]

**; each one unpacks to a whole interval

[n,n+1)

of real

x

. Reporting only the integers

x=n

misses the rest of the solution set.

Drill 2 more on floor equations and sums

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Watch out for (2)

Floor rounds DOWN, so negatives go further from zero
→ Definition, graph, and behaviour at integers
[x] = n is an interval, not a single point
→ Floor equations and sums

Mastery check — 4 interleaved questions

Try each one before clicking. Questions are interleaved across the concepts above, not grouped — interleaving sharpens transfer.

Example 1FunctionsHARD

Let

f(x)=[x]^2-x^2

What is

f(0.999)+f(1.001)

equal to?

[Q83 · Sep · 2024]

Example 2FunctionsMODERATE

Let

z=[y]

and

y=[x]-x

, where

[\cdot]

is the greatest integer function. If

x

is not an integer but positive, then what is the value of

z

[Q71 · Sep · 2024]

Example 3FunctionsMODERATE

Let

f(n) = \left[\dfrac{1}{4} + \dfrac{n}{1000}\right]

, where

[x]

denotes the integral part of

x

. Then the value of

\displaystyle\sum_{n=1}^{1000} f(n)

[Q87 · Sep · 2017]

Example 4FunctionsHARD

Let [x] denote the greatest integer function. What is the number of solutions of

x^2 - 4x + [x] = 0

[0, 2]

[Q29 · Apr · 2018]

Drill every past-year question on this subtopic

7 questions from the bank — paginated, with cart and Word-export support.

Drill the 7 questions