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

⌊x⌋\lfloor x\rfloor (written [x][x] in NDA papers) is the greatest integer not exceeding xx — round down on the number line. The graph is a staircase: flat on each [n,n+1)[n,n+1), then a jump up by 1 at the next integer.

Definition

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

  • [x]≤x<[x]+1[x]\le x<[x]+1, and [x]=x[x]=x exactly when xx 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)(n,n+1).
  • Range is Z\mathbb{Z}; domain is R\mathbb{R}.
xy[x] = n on [n, n+1) · jumps up by 1 at each integer

Worked example

Evaluate [2.7][2.7], [−1.3][-1.3], and [5][5].
Practice this concept4 quick reps

From the bank · past-year question

Example 1FunctionsMODERATE
Consider the following statements in respect of the function y=[x]y=[x], x∈(−1,1)x\in(-1,1) where [.][.] is the greatest integer function: 1. Its derivative is 0 at x=0.5x=0.5 2. It is continuous at x=0x=0 Which of the above statements is/are correct?

[Q56 · Apr · 2022]

Floor rounds DOWN, so negatives go further from zero

[−1.3]=−2[-1.3]=-2, not −1-1: 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]\{x\}=x-[x]. It is the leftover after rounding down, so it always lands in [0,1)[0,1) — even for negative xx.

Definition

{x}=x−[x]∈[0,1)\{x\}=x-[x]\in[0,1) for every real xx. Equivalently [x]−x=−{x}∈(−1,0][x]-x=-\{x\}\in(-1,0]. It is periodic with period 1. For a non-integer xx, [x]−x[x]-x is strictly between −1-1 and 00, so [[x]−x]=−1\big[[x]-x\big]=-1.

Worked example

Find {−2.3}\{-2.3\}.
Practice this conceptself-check

From the bank · past-year question

Example 2FunctionsMODERATE
Let z=[y]z = [y] and y=[x]−xy = [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][x], treat [x][x] as an integer unknown nn, solve the ordinary equation for nn, then translate each integer back to the interval [n,n+1)[n,n+1). For floor sums, count how many terms share each floor value.

Definition

Key idea: [x]=n  ⟺  x∈[n,n+1)[x]=n\iff x\in[n,n+1). So a polynomial equation in [x][x] becomes a polynomial in the integer nn; 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[x]^2-5[x]+6=0 for real xx.
Practice this conceptself-check

From the bank · past-year question

Example 3FunctionsMODERATE
If f(x)=[x]2−30[x]+221=0f(x)=[x]^2-30[x]+221=0, where [x][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][x]; each one unpacks to a whole interval [n,n+1)[n,n+1) of real xx. Reporting only the integers x=nx=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.

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