Floor or Greatest Integer Function - Algebra-1

1. Fundamental Concepts

Definition: The floor function, denoted as $$\text{{floor}}(x)$$ or $$\lfloor x \rfloor$$, is a function that returns the greatest integer less than or equal to $$x$$. For example, $$\lfloor 3.7 \rfloor = 3$$ and $$\lfloor -2.3 \rfloor = -3$$.
Graphical Representation: The graph of the floor function consists of horizontal line segments with jumps at integer values.
Properties: The floor function is not continuous at integer points and has a step-like appearance.

Evaluation Rule: $$\lfloor x \rfloor = n \quad \text{if} \quad n \leq x < n+1$$

Application: Used in various fields including computer science, number theory, and discrete mathematics.

Identities: $$\lfloor x + n \rfloor = \lfloor x \rfloor + n$$ for any integer $$n$$

Problem: Evaluate $$\lfloor 4.5 \rfloor$$

Validation: Since 4 ≤ 4.5 < 5, the solution is correct.

Problem: Evaluate $$\lfloor -3.2 \rfloor$$

Validation: Since -4 ≤ -3.2 < -3, the solution is correct.

Visual Strategy: Use a number line to visualize the position of the number and identify the greatest integer less than or equal to it.
Error-Proofing: Always check if your answer satisfies the condition $$n \leq x < n+1$$.
Concept Reinforcement: Practice with a variety of numbers, both positive and negative, to solidify understanding.