Question

Prove by contradiction that \(\sqrt{2}\) is not a rational number.

Short answer

By contradiction, \(\sqrt{2}\) is not a rational number.

Step-by-step solution

- Assume the Contradiction

To prove that \(\sqrt{2}\) is not a rational number, start by assuming the opposite: that \(\sqrt{2}\) is a rational number. This means \(\sqrt{2}\) can be expressed as a fraction \(\frac{a}{b}\) where \(a\) and \(b\) are integers with no common factors other than 1 (i.e., \(\frac{a}{b}\) is in its simplest form, with \(b eq 0\)).

- Derive the Equation

From the assumption, we have \(\sqrt{2} = \frac{a}{b}\). By squaring both sides of the equation, we obtain: \(2 = \frac{a^2}{b^2}\). This can be rearranged to \(a^2 = 2b^2\).

- Identify Properties of the Equation

The equation \(a^2 = 2b^2\) implies that \(a^2\) is an even number (since it is double another integer). If \(a^2\) is even, then \(a\) must also be even (since the square of an odd number is odd). Let \(a = 2k\) for some integer \(k\).

- Substitute and Analyze

Substitute \(a = 2k\) into the equation \(a^2 = 2b^2\): \((2k)^2 = 2b^2\), which simplifies to \(4k^2 = 2b^2\). Dividing both sides by 2 gives \(2k^2 = b^2\), showing that \(b^2\) is even, and hence \(b\) must also be even.

- Reach the Contradiction

Since both \(a\) and \(b\) are even, they have a common factor of 2. This contradicts our initial assumption that \(\frac{a}{b}\) is in its simplest form. Therefore, the initial assumption that \(\sqrt{2}\) is rational must be false.

Key concepts

Irrational Numbers

Irrational numbers are numbers that cannot be expressed as a simple fraction of two integers. Unlike rational numbers, they have non-terminating and non-repeating decimal expansions. For example, \( \pi \) and \(\sqrt{2}\)\ are classic examples of irrational numbers.

• They represent numbers that can't be written in the form \(\frac{a}{b}\)\, where \a\ and \b\ are integers and \b eq\ 0\.
• The decimal form of an irrational number goes on forever without repeating patterns.
• Irrational numbers can be placed on the real number line, filling in gaps left by rational numbers.

Understanding irrational numbers is crucial because they help complete the real number system. This prevents gaps in calculations, especially in geometry and calculus, where exact values are needed.

Rational Numbers

Rational numbers are any numbers that can be expressed as the quotient of two integers, \(rac{a}{b}\), where \b eq\ 0.\

• Every integer is a rational number, for example, the number \(-3\)\ can be represented as \(-\frac{3}{1}\).
• Rational numbers have decimal expansions that are either terminating or repeating, such as \(0.25\) (terminating) or \(0.333...\) (repeating).
• They are dense in the sense that between any two rational numbers, there's another rational number, thus densely populating the number line.

To express a rational number in its simplest form, its numerator and denominator should share no common factors besides 1. Recognizing rational numbers is essential in understanding fractions, decimals, and all arithmetic operations in mathematics.

Mathematical Proofs

Mathematical proofs are logical arguments that demonstrate the truth or falsity of a mathematical statement. They are foundational for establishing reliable mathematical truths. Proof by contradiction is a specific type of proof technique.

• Proof by contradiction involves assuming that the opposite of the statement to be proven is true.
• If this assumption leads to a contradiction, then the original statement must be true.
• This technique is powerful for negating claims, especially useful when direct proof is difficult.

In proving that \(\sqrt{2}\) is irrational, the proof by contradiction method shows that assuming \sqrt{2} as rational leads to inconsistencies, effectively demonstrating that \(\sqrt{2}\) cannot actually be expressed as a fraction of two integers in its simplest form.