MthT 430 Chapter 2b Projects

In class September 12, 2007, Turn in September 19, 2007

Rational and Irrational Numbers

1.

Prove that Ö3 is irrational.

2.

Let the set of numbers Q_Ö3 consist all the real numbers, x, of the form

x = p + q Ö3,

where p and q are rational numbers. .

·

Prove that if x = p + q Ö3, where p and q are rational numbers, then

x-1 = 1 p + q Ö3 = a + b Ö3,

for some rational a and b.

·

Prove that if x = p + q Ö3 where p and q are rational numbers, and m is a natural number, then x^m = a + b Ö3 for some rational a and b.

Remark: One can show that Q_Ö3 satisfies P1 - P12; more briefly: Q_Ö3 is an ordered field.

Cauchy - Schwartz Inequality

3.

Prove by mathematical induction or otherwise:

æ è m å j=1  xj yj ö ø 2   £ æ è m å j=1  xj2 ö ø · æ è m å j=1  yj2 ö ø .

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File translated from T_EX by T_TH, version 3.77.
On 12 Sep 2007, 10:51.