Egyptian fraction

Long ago, people in Egypt used math. They shared food in fair ways. They split one thing into small parts. Each part had a one on top.

This is a special way to write numbers. It can help you share pizza with friends. You could give someone half and then an eighth.

Old scrolls show these math ideas. One famous scroll is the Rhind Papyrus. It has many math problems in it.

People used these ways to measure grain. They also used them to measure bread. It was a very useful tool.

Math helps us solve puzzles every day. Can you find a way to share?

Ancient Egyptians had a special way to write fractions. They used what we call Egyptian fractions. These are sums of unit fractions. A unit fraction is a part where the top number is one. For example, one-half or one-eighth are unit fractions. In this system, every part must be a different size.

This method was very useful for sharing. Imagine you have five pizzas for eight people. You could give each person one-half of a pizza. Then, you could give them one-eighth of a pizza too. This makes a fair share for everyone.

We know about this math from old scrolls. The Rhind Mathematical Papyrus is a famous one. A man named Ahmes wrote it. It contains 84 math problems. The scroll also has a table of fractions. Egyptians also used these parts to measure grain and bread. They even used special symbols called Horus-Eye fractions. These helped them measure small amounts of food. Even though we use decimals now, mathematicians still study these old ways.

Imagine you are sharing five pizzas among eight hungry friends. Instead of saying everyone gets five-eighths, you could say each person gets one-half of a pizza plus one-eighth. This way of breaking a number into different parts is called an Egyptian fraction. In this system, every part is a unit fraction, which means the top number is always one. Every single part in the sum must also be a different size. This math is very useful for dividing things like food into fair shares.

This way of working with numbers is more than just a way to write things down. It can solve tricky puzzles, like the rope-burning puzzle. Imagine you have ropes that burn at different speeds to measure time. You can use these fractions to decide exactly when to light the ropes. You can also use them to measure volume for things like grain or bread. Even though we use decimals today, these fractions still help us understand how to split things up.

We know about these methods from very old writings from the Middle Kingdom of Egypt. One famous scroll is the Rhind Mathematical Papyrus. A scribe named Ahmes wrote this important text during the Second Intermediate Period. It contains a special table of fractions and 84 different word problems. Other old texts include the Moscow Mathematical Papyrus and the Kahun Papyrus. Some even used special symbols called Horus-Eye fractions to measure small amounts of food.

Writing these fractions in ancient times was a careful job. In hieroglyphs, scribes placed a special symbol above a number to show it was a unit fraction. They also had special symbols for certain numbers to make big calculations easier. In the Middle Ages, a famous mathematician named Fibonacci wrote about these fractions too. His book, the Liber Abaci, was written in 1202. He showed how to turn regular fractions into Egyptian fractions using different steps.

Today, mathematicians still find these old patterns very interesting. They study how to find the shortest way to write a fraction. They also look at how large the bottom numbers in the fractions can get. A famous mathematician named Paul Erdős spent much of his life studying these ideas. He proved things about how these fractions work with whole numbers. Even though the ancient Egyptians are gone, their way of thinking still helps us solve math puzzles today.

An Egyptian fraction is a finite sum of distinct unit fractions. A unit fraction is a fraction where the numerator is exactly 1 and the denominator is a positive integer. For these sums to be considered Egyptian fractions, every denominator must be different from the others. For example, the expression 1/2 + 1/3 + 1/6 is an Egyptian fraction because it sums to 1 using unique unit fractions. Every positive rational number can be represented this way. While modern math uses decimals or vulgar fractions, Egyptian fractions remain a vital subject in number theory and historical studies.

This notation offers practical advantages for fair division. If you must divide 5 pizzas among 8 diners, an Egyptian fraction allows you to give each person 1/2 of a pizza plus 1/8 of a pizza. This is achieved by splitting 4 pizzas into halves and the remaining pizza into eighths. Another application involves rope-burning puzzles to measure time. If a rope burns at a non-uniform rate, you can measure any rational fraction of a unit of time by lighting multiple ropes. For each unit fraction in the expansion, you light a rope so that there are simultaneously lit points burning. This method may require an infinite number of re-lighting steps.

Historical records show these methods developed during the Middle Kingdom of Egypt. Several early texts contain these fractions, including the Egyptian Mathematical Leather Roll and the Moscow Mathematical Papyrus. The Reisner Papyrus, the Kahun Papyrus, and the Akhmim Wooden Tablet also provide evidence. A later, highly significant text is the Rhind Mathematical Papyrus. Written by a scribe named Ahmes during the Second Intermediate Period, it includes a table of expansions for fractions with denominators from 2 to 100. It also contains 84 word problems, with all final answers expressed in Egyptian fraction notation.

Ancient scribes used specific symbols to denote these values. In hieroglyphic script, they placed the symbol D21, meaning "one among," above a number to show its reciprocal. In hieratic script, they simply drew a line over the number. The Egyptians also used "Horus-Eye fractions" to subdivide a hekat, which was a unit of volume for grain or bread. These were special fractions of the form 1/2^n. If a remainder existed after using these, it was written as multiples of a ro, which equals 1/320 of a hekat.

Historians study the Rhind papyrus to understand Egyptian calculation methods. For small odd prime denominators, the Egyptians used specific expansions. For larger primes, they used a method involving a number with many divisors between the prime and its double. They often chose expansions that kept the largest denominator as small as possible. For composite denominators, they sometimes used a method where the expansion for n/m was derived from the expansion for n/(m/k). This allowed them to scale known expansions to larger numbers.

Egyptian fraction notation persisted through Greek times and into the Middle Ages. In 1202, Leonardo of Pisa, known as Fibonacci, wrote the Liber Abaci. This text explores converting vulgar fractions into Egyptian fractions. Fibonacci suggested several methods, such as splitting a numerator into divisors of the denominator. If that failed, he proposed a "greedy" algorithm. This algorithm repeatedly chooses the smallest possible denominator that does not exceed the remaining fraction. While effective, Fibonacci noted that this greedy method often produced very long expansions with large denominators.

Modern number theory continues to investigate these mathematical structures. Mathematicians study the maximum possible length of an expansion and the size of the denominators used. The famous mathematician Paul Erdős contributed significantly to this field. He proved that a harmonic progression cannot form an Egyptian fraction representation of an integer. He also proved that every integer has a representation where all denominators are products of three primes. Additionally, the Erdős–Graham conjecture, proven by Ernest S. Croot III in 2003, relates to how integers can be partitioned into subsets with unit fraction sums.