Chapter I: Real Variables: Rational Numbers: Examples I

Examples I.

1) If r and s are rational numbers, then r+s, r-s, rs, and r/s are rational numbers, unless in the last case s=0 (when r/s is of course meaningless).

Proof:

Part i): Given r and s are rational numbers. Let r=a/b, s=c/d, where a, b, c and d are integers, and b and d are not zero; where a and b do not have any common factors, where c and d do not have any common factors, and c and d are positive integers.

Then, r+s=a/b+c/d=(ad+bc)/bd, which is clearly rational as both the numerator and denominator are new integers (closure in addition and multiplication).

Part ii) Similar to part (i).

Part iii) By closure in multiplication.

Part iv) By definition of division in fractions, and closure in multiplication.

2) If \lambda , m, n are positive rational numbers, and m > n, then prove that \lambda(m^{2}-n^{2}), 2\lambda mn, \lambda(m^{2}+n^{2}) are positive rational numbers. Hence, show how to determine any number of right-angled triangles the lengths of all of whose sides are rational.

Proof:

This follows from problem 1 where we proved that the addition, subtraction and multiplication of rational numbers is rational.

Also, Pythagoras’ theorem holds in the following manner:

\lambda^{2}(m^{2}-n^{2})^{2}+(2\lambda m n)^{2}=\lambda^{2}(m^{2}+n^{2})^{2}

3) Any terminated decimal represents a rational number whose denominator contains no factors other than 2 or 5. Conversely, any such rational number can be expressed, and in one way only, as a terminated decimal.

Proof Part 1:

This is obvious since the divisors other than 2 or 5, namely, 3,6,7,9, and other prime numbers do not divide 1 into a terminated decimal.

Proof Part 2:

Since the process of division produces a unique quotient.

4) The positive rational numbers may be arranged in the form of a simple series as follows:

1/1, 2/1, 1/2, 3/1, 2/2, 1/3, 4/1, 3/2, 2/3, 1/4, \ldots

Show that p/q is the [\frac{p}{q}(p+q-1)(p+q-2)+q]th term of the series.

Proof:

Suggested idea. Try by mathematical induction.

More later,

Nalin Pithwa