The Egyptians of anti quity expressed a fraction as a sum of fractions whose numerators we...

The Egyptians of anti quity expressed a fraction as a sum of fractions whose numerators were 1. For example, 5/6 might be expressed as

 =  +

We say that a fraction p/q, where p and q are positive integers is in Egyptian form if

where n1, n2, … nk are positive integers satisfying n­1 < n2 < … <nk.

By completing the following steps, give a proof by induction on p to show that every fraction p/q with 0 <p/q < 1 may be expressed in Egyptian form.

(a) Verify the Basis Step (p = 1).

(b) Suppose that 0 <p/q< 1 and that all fractions i/q' with 1 ≤ i<p and q' arbitrary, can be expressed m Egyptian form. Choose the smallest positive integer n with 1/n ≤ p/q. Show that

 

(c) show that if p/q = 1/n, the prool is complete.

(d) Assume that 1/n < p/q. Let

p1= np – q and q1 = nq

Show that

Conclude that

with n1, n2, …, nk distinct.

(e) Show that p1/q1< 1/n.

(f) Show that

and n1, n2, …, nk are distinct.