What formula to use when finding patterns of sum of unit fractions #egyptian fractions
To find patterns in the sum of unit fractions, such as Egyptian fractions, we can use a formula called the Greedy Algorithm. This algorithm helps us express any rational number as a sum of unit fractions.
The Greedy Algorithm starts with the largest possible unit fraction, which is 1 divided by the integer just greater than or equal to the given rational number. Here's how it works step by step:
1. Start with the given rational number or fraction that you want to express as a sum of unit fractions.
2. Take the denominator of the given fraction and divide it by the numerator. The result will be the largest integer possible (let's call it n) such that 1/n will be a unit fraction in the sum.
3. Subtract 1/n from the given rational number, and write down 1/n as one of the unit fractions in the sum.
4. Repeat the process with the remaining fraction until the whole number part becomes zero.
5. If there are remaining fractions after making the whole number part zero, go back to step 2 and continue until there are no remaining fractions left.
For example, let's say we want to express the fraction 7/8 as a sum of unit fractions using the Greedy Algorithm:
1. Start with the fraction 7/8.
2. Find the largest possible unit fraction: 8/7 = 1 + 1/7.
3. Subtract 1/7 from 7/8: 7/8 - 1/7 = 49/56.
4. Write down 1/7 as the first unit fraction in the sum.
5. Repeat steps 2-4 with 49/56: 56/49 = 1 + 7/49. Subtract 7/49 from 49/56: 49/56 - 7/49 = 35/56.
6. Write down 1/7 and 1/49 as unit fractions in the sum.
7. Repeat steps 2-4 with 35/56: 56/35 = 1 + 21/35. Subtract 21/35 from 35/56: 35/56 - 21/35 = 14/56.
8. Write down 1/7, 1/49, and 1/35 as unit fractions in the sum.
9. The remaining fraction 14/56 can be simplified to 1/4.
10. Write down 1/7, 1/49, 1/35, and 1/4 as the final unit fractions in the sum.
By following the Greedy Algorithm, we have expressed the fraction 7/8 as the sum of the unit fractions 1/7, 1/49, 1/35, and 1/4.