What is the floor division formula?

For any real numbers a and b (b ≠ 0), floor division is defined as ⌊a/b⌋ = ⌊a ÷ b⌋, where ⌊x⌋ is the greatest integer less than or equal to x. The remainder r = a - ⌊a/b⌋ x b, which always satisfies 0 <= r < |b|.

Floor Division

Q: What is floor division?

Floor division (also called integer division rounding down) computes ⌊a/b⌋ -- the largest integer less than or equal to the exact division a/b. Unlike truncation (which rounds toward zero), floor division always rounds down to the nearest integer. For example, ⌊7/3⌋ = 2 and ⌊-7/3⌋ = -3 (while truncation gives -2).

Q: How is floor division different from regular division?

Regular division gives the exact floating-point result (e.g., 7/3 = 2.333...), while floor division returns only the integer part rounded down (⌊7/3⌋ = 2). Floor division is commonly used in programming languages like Python (// operator), where it ensures consistent behavior with negative numbers.

Q: Where is floor division used in programming?

Floor division is used extensively in programming for array indexing, pagination, time calculations, binary search midpoint computation, modular arithmetic, and generating hash values. Languages like Python, Ruby, and Dart use // for floor division, while C, Java, and JavaScript use truncating division.

Q: What is the remainder in floor division?

In floor division, the remainder satisfies a = b x q + r where q = ⌊a/b⌋ and 0 <= r < |b|. For example, with a = 10 and b = 3: ⌊10/3⌋ = 3, remainder = 10 - 3x3 = 1. With a = -10 and b = 3: ⌊-10/3⌋ = -4, remainder = -10 - (-4)x3 = 2.

Q: How does floor division handle negative numbers?

Floor division of negative numbers rounds down, not toward zero. For example, ⌊-10/3⌋ = -4 (not -3) because -4 x 3 = -12 -10. This ensures the remainder is always non-negative, which is important for modulo arithmetic.

Compute floor division ⌊a/b⌋ (integer division rounding down) of two numbers instantly. Free online floor division calculator returns quotient, remainder, and exact result.

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About This Calculator

This Floor Division Calculator computes ⌊a/b⌋ -- the largest integer less than or equal to the exact division a/b. Unlike truncation (which rounds toward zero), floor division consistently rounds down to the nearest integer, making it essential for modular arithmetic and certain programming paradigms.

Floor division is a core operation in many programming languages. Python uses the // operator (e.g., 7 // 3 = 2), and the behavior differs from C/Java/JavaScript's truncating division when dealing with negative numbers. For example, in Python -7 // 3 = -3 (floor division), while in JavaScript -7 / 3 | 0 = -2 (truncation).

This calculator is useful for programmers learning language differences, students studying modular arithmetic, mathematicians working with integer division properties, and anyone who needs to understand the distinction between floor and truncating division.

Frequently Asked Questions

What is floor division?

Floor division computes ⌊a/b⌋ -- the largest integer less than or equal to the exact division a/b. Unlike truncation (rounding toward zero), floor division always rounds down. For example, ⌊7/3⌋ = 2 and ⌊-7/3⌋ = -3.

How is floor division different from regular division?

Regular division gives the exact floating-point result (e.g., 7/3 = 2.333...), while floor division returns only the integer part rounded down (⌊7/3⌋ = 2). Floor division is commonly used in Python with the // operator.

How does floor division handle negative numbers?

Floor division of negative numbers rounds down, not toward zero. For example, ⌊-10/3⌋ = -4 (not -3) because -4 x 3 = -12 <= -10. This ensures the remainder is always non-negative.

Where is floor division used in programming?

Floor division is used in programming for array indexing, pagination, time calculations, binary search, modular arithmetic, and hash generation. Languages like Python, Ruby, and Dart use // for floor division.

What is the remainder in floor division?

In floor division, a = b x q + r where q = ⌊a/b⌋ and 0 <= r < |b|. For example, with a = -10 and b = 3: ⌊-10/3⌋ = -4, remainder = -10 - (-4)x3 = 2.