Problem 5 - A Gentle Introduction to the Art of Mathematics

Question

Write a proof that \(\sqrt{3}\) is irrational.

Step-by-Step Solution

Answer

Assuming \(\sqrt{3}\) is rational leads to a contradiction, so \(\sqrt{3}\) is irrational.

Step 1 - Assume the Opposite

Assume, for the sake of contradiction, that \(\sqrt{3}\) is rational. This means it can be written as the fraction of two integers \(\frac{a}{b}\), where \(a\) and \(b\) have no common factors (i.e., the fraction is in its simplest form).

Step 2 - Express the Assumption Algebraically

Under this assumption, we can write: \(\sqrt{3} = \frac{a}{b}\). Squaring both sides of the equation gives us: \(3 = \frac{a^2}{b^2}\). This simplifies to: \(a^2 = 3b^2\).

Step 3 - Analyze the Equation

The equation \(a^2 = 3b^2\) implies that \(a^2\) is a multiple of 3. Hence, \(a\) must also be a multiple of 3 (since if the square of a number is a multiple of 3, the number itself must be as well). We can write \(a = 3k\) for some integer \(k\).

Step 4 - Substitute Back

Substitute \(a = 3k\) back into the equation \(a^2 = 3b^2\): \((3k)^2 = 3b^2\). This simplifies to: \(9k^2 = 3b^2\), which can be further simplified to: \(3k^2 = b^2\).

Step 5 - Derive the Contradiction

The equation \(3k^2 = b^2\) implies that \(b^2\) is a multiple of 3. Hence, \(b\) must also be a multiple of 3. Therefore, both \(a\) and \(b\) are multiples of 3, which contradicts our initial assumption that \(a\) and \(b\) have no common factors.

Step 6 - Conclude the Proof

The contradiction arises from our initial assumption that \(\sqrt{3}\) is rational. Therefore, we must conclude that \(\sqrt{3}\) is irrational.

Key Concepts

sqrt 3

The symbol \( \sqrt{3} \) represents the square root of 3. This is a number which, when multiplied by itself, results in 3. Understanding this concept is key to following the proof we'll discuss. Since 3 is not a perfect square (such as 1, 4, 9, etc.), \( \sqrt{3} \) cannot be simplified to an exact integer or simple fraction.

To determine if \( \sqrt{3} \) is a rational number, we'll explore its properties using a logical process. The key is understanding how to handle expressions involving \( \sqrt{3} \).

assume the opposite

In the proof that \( \sqrt{3} \) is irrational, we start by assuming the opposite of what we want to prove. This is an important mathematical strategy called *proof by contradiction*. By assuming that \( \sqrt{3} \) is rational, we look to find an inconsistency, or contradiction.

We say that if \( \sqrt{3} \) were rational, it could be written as a fraction \(\frac{a}{b}\), where \( a \) and \( b \) are integers with no common factors (i.e., the fraction is in its simplest form). This assumption sets up the framework we need to uncover a contradiction.

contradiction

A contradiction in mathematical proofs happens when an assumption leads to a conclusion that is impossible or false. For example, if \( \sqrt{3} \) were rational, the subsequent steps lead to a situation where both \( a \) and \( b \) need to be divisible by 3.

This situation contradicts our initial assumption that \( a \) and \( b \) have no common factors. Recognizing this contradiction means our original assumption (that \( \sqrt{3} \) is rational) must be false. Thus, we conclude that \( \sqrt{3} \) is irrational.

rational numbers

Rational numbers are numbers that can be written as the division of two integers (i.e., \( 1.5 =\frac{3}{2} \) or \( 0.25 = \frac{1}{4} \)). In essence, any number that can be expressed as \( \frac{a}{b} \), where both \( a \) and \( b \) are integers, and \( b \) is not zero, is a rational number.

To check if \( \sqrt{3} \) is rational, we try to express it in such a fraction \(\frac{a}{b}\). However, as we process through the proof, we observe that no such fraction can exist without breaking the rules for rational numbers.

integer properties

Understanding integers and their properties is crucial for this proof. An integer is a whole number that can be positive, negative, or zero (such as -2, 0, 5). Important integer properties here include:

• Multiples: If a number is a multiple of another number, it can be divided exactly by that number without leaving a remainder.
• Prime Factors: The prime factors of a number are the prime numbers that multiply together to give the original number.

When we assume \( a^2 = 3b^2 \), it tells us that \( a \) must be a multiple of 3. Rewriting \( a \) as \( 3k \) and substituting back leads us to the contradiction that both \( a \) and \( b \) must share 3 as a common factor, invalidating our assumption.