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Egyptian fractions

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a. Show that any rational number a/b , between 0 and 1, can be written as an Egyptian fraction.

b. Can an irrational number between 0 and 1 ever be expressed as an Egyptian fraction? Why?

*c.* Show that any positive rational number a/b can be written as an Egyptian fraction.

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Solution Summary

Steps are given in this solution to show how Egyptian fractions can be ascertained.

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a. Show that any rational number a⁄b, between 0 and 1, can be written as an Egyptian fraction.
b. Can an irrational number between 0 and 1 ever be expressed as an Egyptian fraction? Why?
c. Show that any positive rational number a⁄b can be written as an Egyptian fraction.

Solution:
a) Show that any rational number a⁄b, between 0 and 1, can be written as an Egyptian fraction.

Proof: Let 0<a⁄b<1 be a rational number between 0 and 1. If a=1, we're done. If a≠1, let n_1 be the smallest integer such that
n_1≥b/a "so that " 1/n_1 ≤a/b "and" 1/(n_1-1)>a/b
Now set
a_1/b_1 =a/b-1/n_1 =(an_1-b)/(bn_1 ), where a_1/b_1 "is in lowest terms."
Now since 1⁄((n_1-1)>a⁄b,) we ...

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