A fraction was written in several ways before it reached the form we use today. In Egypt it was a sum of parts with numerator 1, in Mesopotamia it was written in base 60, and the horizontal bar is attributed to al-Ḥaṣṣār, around 1200. The bar came after unit fractions and base 60, and before the ÷ sign (1659) and decimals in Europe (Stevin, 1585).
| Milestone | What is claimed | Kind of support |
|---|---|---|
| Rhind papyrus | Fractions written as sums of parts with numerator 1, with the exception of 2/3 | Documentary evidence |
| Base 60 | Fractions whose denominators are powers of 60 | Documentary evidence; the reason for choosing 60 is a hypothesis |
| Horizontal bar | Appears in the work of al-Ḥaṣṣār, around 1200 | Attribution by historians |
| Liber Abaci | Spreads Hindu-Arabic numerals in Europe (1202) | Documentary evidence |
| De Thiende | Introduces decimals into European mathematics (1585) | Documentary evidence |
| The ÷ sign | First use for division in a book by Rahn (1659) | Documentary evidence; who created the sign is disputed |
The last column shows how much weight each claim carries. “Documentary evidence” means a surviving text shows it, while “attribution” or “hypothesis” means it is a reading by historians. You can try the first milestone with the Egyptian fractions tool.
Egypt: sums of unit fractions
According to MacTutor, the Egyptians used unit fractions, with the exception of 2/3, so any other quantity was written as a sum of distinct unit fractions. The Rhind papyrus shows this: the scribe Ahmes wrote it around the middle of the 16th century BC and, in his own words, it copies texts about three hundred years older. It contains a table for doubling unit fractions 1/n for odd n from 5 to 101.
Historians disagree about which rules the scribes followed to choose each decomposition. One way to get a decomposition today is the greedy method, which always takes the largest unit fraction that fits. The next example finds one.
Worked example: 2/5 with unit fractions
How do you write 2/5 with no numerator larger than 1?
- Take the largest unit fraction that fits. 1/2 is too big, because 1/2 > 2/5. The next one, 1/3, fits.
- Subtract. 2/5 − 1/3 = 6/15 − 5/15 = 1/15, which is already a unit fraction.
- Write the sum. The result is 1/3 + 1/15.
So 2/5 is written as 1/3 + 1/15.
Check. Adding, 5/15 + 1/15 = 6/15, and dividing top and bottom by 3 gives the fraction you started with.
The Egyptian fractions tool applies that same method to any positive fraction and shows each subtraction.
Mesopotamia: a base where 1/3 does end
The Babylonians wrote fractions with place value, just as we write decimals today, but with powers of 60 instead of powers of 10. MacTutor gives an example: 0;7 30 stands for 7/60 + 30/3600, which is 1/8. There was no sign to separate the whole part from the fractional part, so context decided.
The advantage of that base is that 60 divides evenly by 2, 3 and 5: a fraction in lowest terms has a finite expansion in base 60 if its denominator has only those prime factors. One third is 20/60 and is written with no remainder, while in base 10 it becomes 0.333… without end. Why 60 was chosen is not known, and MacTutor reviews several hypotheses without any being conclusive.
The legacy we live with today is certain. An hour has 60 minutes and a minute has 60 seconds, and the Babylonians divided the circle into 360 parts, the degrees. The 45 minutes in three quarters of an hour come from that legacy. The article on why an hour has 60 minutes develops how divisible 60 is compared with 100.
The horizontal bar: from al-Ḥaṣṣār to Fibonacci
In India and in the Islamic world, fractions were written with the numerator above the denominator and nothing in between. English Wikipedia records this for Indian mathematicians such as Brahmagupta (around 628), and Spanish Wikipedia cites it for al-Khwarizmi (9th century) and al-Uqlidisi (10th century). According to English Wikipedia, the horizontal bar first appears in the work of al-Ḥaṣṣār, a mathematician from Fez active around 1200 who specialized in inheritance. This is an attribution from secondary literature, not a quotation from a text by al-Ḥaṣṣār.
Fibonacci published the Liber Abaci in 1202 after returning to Italy. MacTutor describes it as a book mostly about Arabic numerals, which brought the Hindu-Arabic positional system to Europe. Spanish Wikipedia adds that it also spread the notation with a bar between numerator and denominator. By about 1700, the bar, horizontal or slanted, was in general use.
From Stevin’s decimals to the ÷ sign
Simon Stevin published La Thiende in 1585, a 29-page pamphlet aimed at astronomers, surveyors, merchants and mint officials. MacTutor makes clear that he did not invent decimals, which Arabs and Chinese had used long before, but introduced them into European mathematics. His notation was turned, through Clavius and Napier, into the one we use now. With decimals, 3/4 is written 0.75; the decimal to fraction tool goes the other way.
The ÷ sign appears as a division symbol in Johann Rahn’s Teutsche Algebra of 1659. According to Spanish Wikipedia, it may have been John Pell, the book’s editor, who introduced it. Earlier it was an editing mark, which Aristarchus of Samothrace used to flag doubtful passages in Homer. Today ÷ and the slanted bar in 3/4 are read the same way, but they are not the same sign: the bar is part of the number, while ÷ indicates an operation.
Three quarters in four notations
Each era wrote three quarters in its own way, and all four forms are exact. The fraction, decimal and percent table helps you move from one to another.
| Notation | Written form | How to check it |
|---|---|---|
| Unit fractions (Egypt) | 1/2 + 1/4 | 2/4 + 1/4 = 3/4 |
| Base 60 | 45/60 (0;45) | 45/60 = 3/4 after dividing by 15 |
| Horizontal bar | 3 over 4 | It is the fraction itself |
| Decimal (since Stevin) | 0.75 | 75/100 simplifies by 25 |
All four rows have the same value, and only which operation is convenient changes. Adding parts is easy with unit fractions, dividing time is easy in base 60, and comparing quantities is easy with decimals. To see when a decimal ends and when it repeats, use fraction to decimal; to understand what each part means, the guides what are fractions and reading and writing fractions start from zero.
Practice
Three checks
- Check that 1/4 + 1/6 + 1/12 equals one half.
- Write 1/8 of an hour in minutes.
- Write 7/10 as a sum of two distinct unit fractions.
Show the answers
- With common denominator 12, the sum is 3/12 + 2/12 + 1/12 = 6/12, which is 1/2.
- It is 7.5 minutes, that is, 7 minutes and 30 seconds, which matches the Babylonian 0;7 30.
- 1/2 + 1/5, because 5/10 + 2/10 = 7/10.
Sources
- MacTutor (University of St Andrews), Egyptian mathematical papyri: unit fractions and the table of doubled unit fractions (accessed October 16, 2026).
- Spanish Wikipedia, Rhind papyrus article, in Spanish: date, the scribe Ahmes and the older texts (accessed October 16, 2026).
- MacTutor, Babylonian numerals: fractions in base 60 and hypotheses about its origin (accessed October 16, 2026).
- Spanish Wikipedia, sexagesimal system article, in Spanish: use in time and angles (accessed October 16, 2026).
- Spanish Wikipedia, fraction article, in Spanish: notation in the Islamic world, Fibonacci and the bar (accessed October 16, 2026).
- English Wikipedia, Fraction: Indian notation, al-Ḥaṣṣār and the horizontal bar (accessed October 16, 2026).
- MacTutor, Fibonacci: Liber Abaci, 1202 (accessed October 16, 2026).
- MacTutor, Stevin: La Thiende, 1585 (accessed October 16, 2026).
- Spanish Wikipedia, obelus article, in Spanish: ÷ in the Teutsche Algebra, 1659 (accessed October 16, 2026).