Fraction Calculator

Definition

The fraction calculator performs the four arithmetic operations on ordinary fractions a/b and c/d, plus simplification of a single fraction. Numerators and denominators are integers; a denominator cannot be zero. The result is always reduced to lowest terms: after the operation, numerator and denominator are divided by their greatest common divisor (GCD, Euclid’s algorithm). A negative denominator’s sign is moved to the numerator so the result denominator is positive.

Addition and subtraction: a/b ± c/d = (a·d ± c·b)/(b·d), then reduce. Multiplication: (a/b)·(c/d) = (a·c)/(b·d). Division: (a/b)÷(c/d) = (a·d)/(b·c) when c ≠ 0. Simplifying a/b alone is (a/g)/(b/g) with g = gcd(|a|, |b|). These rules are standard school arithmetic and the arithmetic of the field ℚ of rationals.

In classroom practice it helps to separate the mathematical formula from floating-point limits: this calculator reports the browser’s floating-point computation (about 15–16 significant digits), not a symbolic identity on paper. When inputs are extreme, check the domain first, then the floating-point range.

Historical and conceptual context

Ordinary fractions model the field of rationals: equivalence classes of integer pairs (p, q) with q ≠ 0 under (p, q) ∼ (p′, q′) ⇔ p·q′ = p′·q. Euclid’s GCD algorithm is ancient (Elements, Book VII) and remains the basis of cancellation. MathWorld (Fraction, Euclidean Algorithm) and Knuth, TAOCP §4.5, cover both the arithmetic and GCD complexity.

Conceptually the calculator does not operate on decimal fractions or symbolic expressions. You enter four (or two) integer fields; the output is again an integer-coefficient fraction. The parallel decimal display p/q is only a floating-point arithmetic preview, not a change of model.

In classroom practice it helps to separate the mathematical formula from floating-point limits: this calculator reports the browser’s floating-point computation (about 15–16 significant digits), not a symbolic identity on paper. When inputs are extreme, check the domain first, then the floating-point range.

Formal properties

Domain: a, b, c, d ∈ ℤ, b ≠ 0, and for division c ≠ 0 (with d ≠ 0 as the second denominator). Codomain: a reduced fraction with positive denominator. Results lie in ℚ.

GCD satisfies gcd(a, b) = gcd(b, a mod b). LCM(a, b) = |a·b|/gcd(a, b) for a, b ≠ 0; the panel may show the LCM of denominators when adding. Identities: (a/b) + (c/b) = (a + c)/b; (a/b)·(b/a) = 1 when a, b ≠ 0.

Reduction is unique up to sign: if p/q is reduced with q > 0, every equivalent fraction has the form (k·p)/(k·q) for k ∈ ℤ \ {0}.

In classroom practice it helps to separate the mathematical formula from floating-point limits: this calculator reports the browser’s floating-point computation (about 15–16 significant digits), not a symbolic identity on paper. When inputs are extreme, check the domain first, then the floating-point range.

Full derivation

Step 1 (sign normalization). If b < 0, replace (a, b) by (−a, −b). Likewise for the second fraction.

Step 2 (operation). For a sum: num = a·d + c·b, den = b·d. For a difference: num = a·d − c·b. For a product: num = a·c, den = b·d. For a quotient: num = a·d, den = b·c (requires c ≠ 0).

Step 3 (reduction). g = gcd(|num|, |den|); result = (num/g)/(den/g) with positive den. Euclid’s algorithm terminates because remainders decrease in ℕ.

Step 4 (correctness). Equality in ℚ follows from the class definition: (a/b) + (c/d) = (ad + cb)/(bd) is independent of representative after reduction. Axiomatic treatments: Lang, Undergraduate Algebra; Hardy & Wright for GCD.

In classroom practice it helps to separate the mathematical formula from floating-point limits: this calculator reports the browser’s floating-point computation (about 15–16 significant digits), not a symbolic identity on paper. When inputs are extreme, check the domain first, then the floating-point range.

Special cases

Examples: 1/2 + 1/3 = 5/6; 3/4 − 1/4 = 1/2; (2/3)·(3/5) = 2/5; (1/2)÷(1/4) = 2/1; simplify 12/18 → 2/3. Mixed numbers (e.g. 1 1/2) are a display of |p| = q·whole + remainder, not a separate data type.

Division by 0/d or by c = 0 is rejected. Negative numerators are allowed; −2/4 reduces to −1/2. Two zero numerators with a nonzero denominator yield 0/1.

In classroom practice it helps to separate the mathematical formula from floating-point limits: this calculator reports the browser’s floating-point computation (about 15–16 significant digits), not a symbolic identity on paper. When inputs are extreme, check the domain first, then the floating-point range.

Numerical and computational notes

Fields accept integers only (validation ^-?\d+$). Intermediate products a·d may leave the range of exactly representable integers; with extreme values floating-point arithmetic loses integer exactness above 2⁵³. For typical school tasks the range is enough.

A comma in an integer field is rejected (this is not the decimal converter). The panel shows a decimal form, the GCD, and optionally LCM and a mixed form.

In classroom practice it helps to separate the mathematical formula from floating-point limits: this calculator reports the browser’s floating-point computation (about 15–16 significant digits), not a symbolic identity on paper. When inputs are extreme, check the domain first, then the floating-point range.

Related concepts and authorities

Related: decimal-to-fraction (the reverse direction via continued fractions), floor, and fractional part. This calculator does not compute fractional powers or roots.

Authorities: Euclid, Elements VII; Knuth, TAOCP §4.5; MathWorld — Fraction, Euclidean Algorithm; NIST DLMF §1.1. Educational tool — not a substitute for written exams.

In classroom practice it helps to separate the mathematical formula from floating-point limits: this calculator reports the browser’s floating-point computation (about 15–16 significant digits), not a symbolic identity on paper. When inputs are extreme, check the domain first, then the floating-point range.