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The greatest integer function looks simple until a negative number turns up: [−2.3] is −3, not −2. Equations that mix [x] and {x} look harder still, but they almost all yield to one move: split x into its integer part and its fractional part. This post covers the definitions, the graphs, four types of equation, and the slips that cost marks.
Where is this in the syllabus?
The CBSE 2026-27 mathematics curriculum lists, in Class XI, "modulus, signum, exponential, logarithmic and greatest integer functions, with their graphs". NCERT's Class 11 Chapter 2, Relations and Functions (2026-27 reprint) defines the greatest integer function and draws its graph. The JEE (Advanced) 2026 brochure lists "greatest integer" among the special functions. The fractional part function is not in NCERT, but it follows directly from [x]. Check your exam's current documents.
What do [x] and {x} mean?
[x] is the greatest integer less than or equal to x. NCERT lists the pattern: [x] = −1 for −1 ≤ x < 0, [x] = 0 for 0 ≤ x < 1, [x] = 1 for 1 ≤ x < 2, and so on.
{x} = x − [x] is the fractional part. It always satisfies 0 ≤ {x} < 1.
For negative numbers: [−2.3] = −3, because −3 is the greatest integer not above −2.3, and {−2.3} = −2.3 − (−3) = 0.7.
Graphs. [x] is a staircase: flat steps of length 1, each with a filled dot at its left end and an open dot at its right end. {x} is a sawtooth: a line of slope 1 rising from 0 towards 1 on each interval [n, n + 1), then dropping back to 0. It repeats every 1 unit, so its period is 1.
Which properties do you use most?
[x] ≤ x < [x] + 1, which is the same as 0 ≤ {x} < 1.
[x + n] = [x] + n for any integer n.
[x] + [−x] is 0 if x is an integer and −1 otherwise.
[x + y] is not always [x] + [y]: [0.6 + 0.6] = 1, but [0.6] + [0.6] = 0.
[x] + [x + 1/2] = [2x]. Check with x = 0.7: 0 + 1 = 1 = [1.4].
Worked examples: four equation types
Example 1: a quadratic in [x]. Solve [x]² − 5[x] + 6 = 0.
Treat [x] as the unknown: ([x] − 2)([x] − 3) = 0, so [x] = 2 or [x] = 3. That means 2 ≤ x < 3 or 3 ≤ x < 4, so 2 ≤ x < 4. The answer is an interval, not two numbers.
Example 2: [x] of an expression. Solve [2x − 1] = 3.
This means 3 ≤ 2x − 1 < 4, so 4 ≤ 2x < 5, giving 2 ≤ x < 2.5.
Example 3: [x] and {x} together. Solve 4{x} = x + [x].
Write x = n + f. Then 4f = (n + f) + n, so 3f = 2n and f = 2n/3. Now use 0 ≤ f < 1: 0 ≤ 2n/3 < 1 gives 0 ≤ n < 1.5, so n = 0 or n = 1. These give f = 0 and f = 2/3, so x = 0 or x = 5/3.
Check x = 5/3: {5/3} = 2/3 and 4 × 2/3 = 8/3, while x + [x] = 5/3 + 1 = 8/3.
Example 4: count solutions with a graph. How many solutions does {x} = x/5 have?
Write x = n + f. Then f = (n + f)/5, so 4f = n and f = n/4. The condition 0 ≤ n/4 < 1 allows n = 0, 1, 2 and 3, giving x = 0, 5/4, 5/2 and 15/4: 4 solutions. On a graph, the line y = x/5 cuts the first four teeth of the sawtooth; for negative x the line is below zero, where {x} never goes.
Where do these functions show up later?
[x] jumps at every integer, so it is discontinuous there, which makes it a favourite in limits and continuity questions. In integration, you split the range at each integer; our post on definite integrals with modulus and floor functions shows how. And because {x} has period 1, it appears in periodic function questions.
Which mistakes cost the most marks?
Rounding towards zero for negatives: [−2.3] is −3.
Giving points instead of intervals in equations like example 1.
Forgetting 0 ≤ f < 1, especially that f = 1 is not allowed.
Splitting [x + y] into [x] + [y].
Drawing the steps with the wrong dots: filled on the left of each step, open on the right.
A practice routine
These questions come in a few types, and the skill is recognising which one you face. Taylor and Rohrer (2010) found that children who practised mixed types of maths problems did worse during practice but doubled their scores on a test a day later; an analysis of errors suggested mixing helped them pair each problem with the right procedure.
Draw both graphs from memory each day for a week, with the dots correct.
Evaluate eight values each day, half of them negative.
Solve a mixed set of the four equation types, labelling each type before solving.
When a split into n + f does not lead anywhere, the Fermi Concept Library covers every concept in physics, chemistry, biology, mathematics and computing, so you can review the definitions behind the step.
Today, solve [x]² = [x] + 2 and check one value from your answer interval.
