Modulus & Greatest IntegerJEE Main

Concept

Three piecewise workhorses. |x − a| = distance from a: a V with vertex at a. ⌊x⌋ = greatest integer ≤ x: a staircase jumping at every integer. {x} = x − ⌊x⌋: the sawtooth of fractional parts, periodic with period 1. JEE loves them because they break naive algebra — you must think in CASES.

Key formula

Derivation

|x−a| splits the line at a; every modulus equation/inequality becomes interval-by-interval linear algebra.

Floor: ⌊2.7⌋ = 2 but ⌊−2.3⌋ = −3 (NOT −2) — floor rounds DOWN, always. Sum of distances |x| + |x−a| is minimised (value |a|) on the entire segment between the two points — no calculus, pure geometry.

Scenarios to explore

• Modulus & Greatest Integer — Kinks, steps and sawtooths — |x| and [x] graph surgery.

Real-world applications

• Solving |x−1| + |x−3| = 4 type equations by zones.
• Number theory: ⌊n/p⌋ counts multiples (Legendre's formula).
• Sawtooth waves in signal processing.

JEE exam tips

• ⌊x⌋ + ⌊−x⌋ = 0 for integers, −1 otherwise — a favourite identity.
• |x−a| < r ⟺ a − r < x < a + r — the open interval translation.
• {x} is periodic ⇒ ∫₀ⁿ {x}dx = n/2, instantly.

Common mistakes

• ⌊−2.3⌋ = −2 (wrong — it's −3).
• |x|² = x² is fine, but √(x²) = |x|, NOT x.
• Solving |f(x)| = g(x) without demanding g(x) ≥ 0.

Exam traps to avoid

• ⌊x⌋ is discontinuous at every integer but RIGHT-continuous.
• Equations with ⌊x⌋: substitute ⌊x⌋ = n (integer), solve, then verify n ≤ x < n+1.