# Egyptian fractions

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 This solution is **FREE** courtesy of BrainMass!

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 have

b/(n_1-1)>a

b>a(n_1-1)

b>an_1-a

a>an_1-b

Thus we have

a/b=1/n_1 +a_1/b_1 , "where" a_1<a.

If a_1=1, we're done. If a_1≠1, we continue the process with a_1⁄b_1 to obtain

a_1/b_1 =1/n_2 +a_2/b_2 , "where" a_2<a_1.

Then we have

a/b=1/n_1 +1/n_2 +a_2/b_2

Since the numerator a_k of a_k⁄b_k is decreasing, the process will stop at some point when a_k=1 and n_k=b_k.

Then we obtain

a/b=1/n_1 +1/n_2 +⋯+1/n_(k-1) +1/n_k ,

where n_1<n_2<⋯<n_(k-1)<n_k.

Thus every 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?

Solution: Since every Egyptian fraction is a sum of distinct unit fractions, it follows that every Egyptian fraction is a rational number, i.e. can be written in the form m⁄(n,) where m and n are integers. By definition, an irrational number cannot be written in the form m⁄(n,) where m and n are integers. So no irrational number between 0 and 1 can ever be expressed as an Egyptian fraction.

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

Proof: We've shown that any rational number a⁄b, between 0 and 1 can be represented as an Egyptian fraction, i.e.

a/b=1/n_1 +1/n_2 +⋯+1/n_k ,

where n_1,n_2,...,n_k. Note that this representation is never unique. For example,

1/n=1/(n+1)+1/(n(n+1))

This gives us

1=1/2+1/2=1/2+1/3+1/6

1=1/3+1/6+1/4+1/12+1/7+1/42

1=1/4+1/12+1/7+1/42+1/5+1/20+1/13+1/156+1/8+1/56+1/43+1/1806

So we obtain two Egyptian fraction representations of 1

1=1/2+1/3+1/6

and

1=1/4+1/12+1/7+1/42+1/5+1/20+1/13+1/156+1/8+1/56+1/43+1/1806

that have no unit fractions in common.

So by rewriting the terms as sums of unit fractions with larger denominators, we can obtain infinitely many Egyptian fraction representations of 1 that have no unit fractions in common.

Then if a⁄b is a rational number greater than 1, we can write it as

a/b=q+r/b, where q and r are integers and 0≤r/b<1.

Thus we have

a/b=⏟(1+1+⋯+1)┬(q times)+r/b.

We've already shown that 1 and r⁄b can be expressed as Egyptian fractions and rewriting the unit fractions as sums of smaller unit fractions with larger denominators, we can write a/b as the sum of distinct unit fractions.

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