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A fraction represents a part of a whole or, more generally, any number of equal parts of a unit. The numerator (top number) indicates how many parts are being considered, while the denominator (bottom number) indicates how many equal parts the whole is divided into. For example, in 3/4, the whole is divided into 4 equal parts and we are referring to 3 of them. **Types of Fractions:** - Proper fraction: numerator is less than denominator (e.g., 3/4, 2/7) — value is between 0 and 1. - Improper fraction: numerator is greater than or equal to denominator (e.g., 7/4, 9/3) — value is 1 or greater. - Mixed number: combination of a whole number and a proper fraction (e.g., 1 3/4) — equivalent to an improper fraction. - Unit fraction: numerator equals 1 (e.g., 1/5, 1/12) — the basic building block of Egyptian fraction arithmetic. - Equivalent fractions: different fractions that represent the same value (e.g., 1/2 = 2/4 = 3/6 = 50/100). **Core Arithmetic Operations:** **Addition and Subtraction:** To add or subtract fractions, both must share a common denominator. The Least Common Denominator (LCD) is the smallest number both denominators divide into evenly. Formula: a/b ± c/d = (ad ± bc) / (bd), then simplify. Example: 1/3 + 1/4 = 4/12 + 3/12 = 7/12. **Multiplication:** Multiply the numerators together and the denominators together, then simplify. Formula: a/b × c/d = (a×c) / (b×d). Example: 2/3 × 3/5 = 6/15 = 2/5. **Division:** Multiply by the reciprocal of the divisor (flip the second fraction). Formula: a/b ÷ c/d = a/b × d/c = (a×d) / (b×c). Example: 3/4 ÷ 1/2 = 3/4 × 2/1 = 6/4 = 3/2 = 1 1/2. **Key Concepts and Algorithms:** **Greatest Common Divisor (GCD):** The largest integer that divides both the numerator and denominator. Used to simplify fractions to lowest terms. For 12/18, GCD(12,18) = 6, so 12/18 = 2/3. **Least Common Multiple (LCM):** Used to find the LCD when adding or subtracting fractions with different denominators. LCM(4, 6) = 12, so to add 1/4 + 1/6 we use twelfths: 3/12 + 2/12 = 5/12. **Converting Between Forms:** - Mixed number to improper fraction: multiply whole by denominator and add numerator. Example: 2 3/4 = (2×4 + 3)/4 = 11/4. - Improper fraction to mixed number: divide numerator by denominator, remainder becomes new numerator. Example: 11/4 = 2 remainder 3 → 2 3/4. - Fraction to decimal: divide numerator by denominator. Example: 3/8 = 0.375. - Repeating decimals: 1/3 = 0.333..., 1/7 = 0.142857142857... (the bar notation indicates repetition). **Simplifying Fractions:** A fraction is in lowest terms (fully simplified) when GCD(numerator, denominator) = 1. To simplify, divide both by their GCD repeatedly, or use the Euclidean algorithm to find the GCD in one step. Example: 48/72 → GCD = 24 → 2/3. **Applications:** Fractions appear in cooking (1/2 cup, 3/4 teaspoon), construction (2 1/4 inch lumber, 5/8 inch drywall), music (quarter notes, eighth notes), probability (3 chances in 8 = 3/8), and finance (interest rates as fractions of percentages).

Fractions have been used for over 4,000 years, making them one of the oldest mathematical concepts still in everyday use. Ancient Egyptians relied heavily on unit fractions — fractions with a numerator of 1 — expressing all other fractions as sums of distinct unit fractions. For example, they wrote 2/3 as 1/2 + 1/6. The Rhind Mathematical Papyrus (c. 1650 BCE) contains extensive tables of fraction decompositions used by scribes for calculating grain rations, land areas, and tax assessments. The Babylonians of ancient Mesopotamia approached fractions differently, using a base-60 (sexagesimal) positional number system. This made fractions with denominators that are powers of 60 very natural to express. Remnants of this system survive today in our divisions of time (60 seconds per minute, 60 minutes per hour) and angles (360 degrees in a circle). Ancient Greek mathematicians, including Euclid and Archimedes, developed rigorous proofs about ratios and proportions — essentially the theory of fractions — in works such as Euclid's Elements (c. 300 BCE). Greek mathematicians distinguished between commensurable and incommensurable quantities, laying groundwork for understanding irrational numbers. The modern fraction notation using a horizontal bar (called the vinculum) was developed by Arab mathematicians and scholars in the medieval Islamic world. Abu Bakr al-Karaji and Al-Hassar wrote fractions in the form we recognize today around the 10th–12th centuries. This notation was introduced to European mathematics by Leonardo of Pisa (Fibonacci) in his landmark work Liber Abaci (1202), which helped standardize arithmetic across Europe. The diagonal slash notation (a/b) appeared in print during the 18th century as a typographical convenience when horizontal bars were difficult to typeset. Both notations are still used today. Decimal fractions, pioneered by Simon Stevin in his 1585 pamphlet De Thiende ("The Tenth"), offered an alternative way to express parts of a whole. Stevin argued that decimals would simplify commerce and science, and his ideas eventually led to the metric system. However, fractions remain indispensable for exact arithmetic — the fraction 1/3 is exact, while its decimal equivalent 0.333... never terminates or repeats finitely. Today, fractions are foundational in mathematics education, engineering, cooking, music theory, probability, and computer science, where rational arithmetic is essential for precise calculations.