Problem 5 - Book of Proof

Question

Use the method of proof by contradiction to prove the following statements. (In each case, you should also think about how a direct or contrapositive proof would work. You will find in most cases that proof by contradiction is easier.) Prove that \(\sqrt{3}\) is irrational.

Step-by-Step Solution

Answer

The assumption that \( \sqrt{3} \) is rational leads to a contradiction. Hence, \( \sqrt{3} \) is irrational.

Step 1: Rational Number Definition

Assume to the contrary that \( \sqrt{3} \) is rational. That means it can be expressed as a ratio of two integers. Let's put \( \sqrt{3} = \frac{a}{b} \) where \( a \) and \( b \) are co-prime integers and \( b \neq 0 \), and then simplify the equation to get \( 3 = \frac{a^2}{b^2} \) or \( a^2 = 3b^2 \).

Step 2: Formulate the Contradiction

Notice that since \( a^2 = 3b^2 \), the square of \( a \) is a multiple of 3. This implies that \( a \) is also a multiple of 3, because the square of a non-multiple of 3 is also not a multiple of 3. So we let \( a = 3c \), where \( c \) is an integer. Substituting \( a \) with 3c in the equation yields \( 9c^2 = 3b^2 \) or \( 3c^2 = b^2 \). The similar reasoning from before tells us that \( b \) is also a multiple of 3.

Step 3: Contradiction and Conclusion

However, in Step 1, we stated that \( a \) and \( b \) are co-prime, meaning they have no common factors other than 1, but in Step 2, we derived that both \( a \) and \( b \) are multiples of 3, which means they have at least one common factor, 3, which contradicts our original assumption that \( a \) and \( b \) are co-prime. Therefore, due to contradiction, our initial assumption that \( \sqrt{3} \) is rational is false. The proof concludes that \( \sqrt{3} \) is irrational.

Key Concepts

Irrational Numbers

Irrational numbers are numbers that cannot be written as a fraction, or ratio, of two integers. The decimal representation of an irrational number goes on forever without repeating. Examples of irrational numbers include \( \pi \), \( e \), and numbers like \( \sqrt{2} \) and \( \sqrt{3} \).
Irrational numbers cannot be expressed as a precise fraction like rational numbers can. This is because they have endless, non-repeating decimals. For instance, while \( \sqrt{2} \) starts as 1.4142135..., it carries on infinitely without forming a predictable pattern.
Understanding irrational numbers helps us comprehend the vastness of numbers, revealing that not all numbers can be neatly expressed with fractions. These are just a few insights into the complexity and beauty of irrational numbers.

Rational Numbers

Rational numbers are numbers that can be written as a fraction where both the numerator and denominator are integers, and the denominator is not zero. Examples are \( \frac{1}{2} \), \( -3 \), and \( 4.75 \) (equivalent to \( \frac{19}{4} \)).
Key properties of rational numbers include:

• They can have a finite or repeating decimal expansion.
• Negative numbers and whole numbers are also rational, as they can be expressed as fractions (e.g., \( -3 = \frac{-3}{1} \)).
• They can be easily added, subtracted, multiplied, and divided, except by zero.

Understanding rational numbers builds the foundation for dealing with fractions and helps distinguish them from non-terminating, non-repeating decimals, which are irrational numbers.

Mathematical Proof Methods

Proof methods are essential in mathematics for confirming whether a statement is true. One of these methods is proof by contradiction, which is used when proving statements that directly show the logical inconsistency of the opposite assumption.
Here’s how proof by contradiction works:

• First, assume the opposite of what you want to prove.
• Then, logically deduce consequences from this assumption.
• If these consequences lead to a contradiction, the initial assumption must be false, thus proving your statement true.

For example, the proof that \( \sqrt{3} \) is irrational uses this method. By assuming it was rational (expressible as a fraction of integers), we discovered a contradiction in the initial assumptions about the integers’ properties. Therefore, \( \sqrt{3} \) is confirmed as irrational.
Proof by contradiction is powerful for cases where direct proof is difficult. Besides contradiction, other proof methods include direct proof, contrapositive proof, and induction proof. Each serves different scenarios, helping mathematicians validate theories systematically.