## What does it mean to say that a number x is irrational?

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marty.frmn
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### What does it mean to say that a number x is irrational?

What does it mean to say that a number x is irrational?
Prove by contradiction statements A and B below, where p and q are real numbers.
A: If pq is irrational, then at least one of p and q is irrational.
B: If p + q is irrational, then at least one of p and q is irrational.
Disprove by means of a counterexample statement C below, where p and q are real numbers.
C: If p and q are irrational, then p + q is irrational.
If the numbers e, pi, pi^2, e^2, e*pi are irrational, prove that at most one of the numbers pi+e, pi-e, pi^2+e^2, pi^2-e^2 is rational.

stapel_eliz
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marty.frmn wrote:What does it mean to say that a number x is irrational?

To learn about the different number types, try here.

marty.frmn wrote:Prove by contradiction statements A and B below, where p and q are real numbers.
A: If pq is irrational, then at least one of p and q is irrational.
B: If p + q is irrational, then at least one of p and q is irrational.

Assume, for (A), that each of p and q is a fraction in lowest terms. Multiply them. What do you get?

Assume, for (B), that each of p and q is a fraction in lowest terms. Add them. What do you get?

marty.frmn wrote:Disprove by means of a counterexample statement C below, where p and q are real numbers.
C: If p and q are irrational, then p + q is irrational.

Take an irrational (like the square root of two), and add and subtract the same whole number from it. What are the results of these processes? What types of numbers are the answers? But what do you get when you add them back together?

marty.frmn wrote:D: If the numbers e, pi, pi^2, e^2, e*pi are irrational, prove that at most one of the numbers pi+e, pi-e, pi^2+e^2, pi^2-e^2 is rational.

Use the results from (A) and (B).