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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 Preview

    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 ...

    Solution Summary

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

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