• Real Analysis
• 1. Sets and Relations
• 2. Infinity and Induction
• 2.1. Countable Infinity
• 2.2. Uncountable Infinity
• 2.3. The Principle of Induction
• 2.4. The Real Number System
• 2.5. Pi and e are irrational
• 3. Sequences of Numbers
• 4. Series of Numbers
• 5. Topology
• 6. Limits, Continuity, and Differentiation
• 7. The Integral
• 8. Sequences of Functions
• 9. Historical Tidbits
• Java Tools

2.4. The Real Number System

Theorem 2.4.1: No Square Roots in Q

Proof:

Suppose there was such an x. Being a rational number, we can write it as

• x = a / b (with no common divisors)

Since x² = x * x = 2 we have

• a² = 2 b²

In other words, a² is even, and therefore a must be even as well. (Can you prove this ?). Hence,

• a = 2 c for some integer c.

But then we have that

• 4 c² = 2 b², or 2 c² = b²

As before, this means that b is even.. But then both a and b are divisible by 2. That's a contradiction, because a and b were supposed to have no common divisors.

Interactive Real Analysis, ver. 2.0.1(c)
(c) 1994-2018, Bert G. Wachsmuth
Page last modified: Jan 26, 2023