Problem 57 - Calculus Single Variable

Question

About 2400 years ago, the followers of Pythagoras discovered that if \(x\) is a positive number such that \(x^{2}=2\), then \(x\) is irrational. Complete the following outline to obtain a proof of this fact. a. Suppose that \(x=a / b\) where \(a\) and \(b\) are integers with no common factors. b. Conclude that \(2=x^{2}=a^{2} / b^{2}\). c. Conclude that \(2 b^{2}=a^{2}\). d. Conclude that 2 divides \(a\) evenly, with no remainder. Therefore \(a=2 \alpha\) for some integer \(\alpha\) e. Conclude that \(b^{2}=2 \alpha^{2}\). f. Conclude that 2 divides \(b\) evenly, with no remainder. g. Notice that parts \(\mathrm{d}\) and \(\mathrm{f}\) contradict part a.

Step-by-Step Solution

Answer

Assume x is rational, derive a contradiction, proving x is irrational.

Step 1: Express x as a fraction

Assume that the number \(x\) can be expressed as a fraction \(x = \frac{a}{b}\), where \(a\) and \(b\) are integers with no common factors (i.e., the fraction is in its simplest form).

Step 2: Relate x² to the fraction

From the assumption given that \(x^2 = 2\), we set up the equation \(2 = x^2 = \frac{a^2}{b^2}\).

Step 3: Clear the fraction

Multiply both sides by \(b^2\) to eliminate the fraction: \(2b^2 = a^2\).

Step 4: Analyze a² being even

Since \(a^2 = 2b^2\), \(a^2\) is even because it is 2 times an integer \(b^2\). Thus, \(a\) must be even because the square of an odd number is odd, leading to \(a = 2\alpha\) for some integer \(\alpha\).

Step 5: Substitute for a in the equation

Substitute \(a = 2\alpha\) in \(2b^2 = a^2\), resulting in \(2b^2 = (2\alpha)^2\), simplifying to \(2b^2 = 4\alpha^2\).

Step 6: Simplify the equation

Divide the entire equation \(2b^2 = 4\alpha^2\) by 2 to obtain \(b^2 = 2\alpha^2\).

Step 7: Analyze b² being even

Since \(b^2 = 2\alpha^2\), it follows that \(b^2\) is even, which implies \(b\) must also be even.

Step 8: Identify the contradiction

Both \(a\) and \(b\) are even, meaning they have a common factor of 2, which contradicts the initial statement that \(a\) and \(b\) have no common factors, proving that \(x\) is irrational as it cannot be expressed as \(\frac{a}{b}\) with coprime integers.

Key Concepts

Pythagorean Theorem

The Pythagorean Theorem is one of the most famous mathematical principles, named after the ancient Greek mathematician Pythagoras. The theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This can be expressed with the formula:\[ c^2 = a^2 + b^2 \]where \(c\) is the length of the hypotenuse, and \(a\) and \(b\) are the lengths of the triangle's other two sides. This theorem is foundational in geometry and is used not only for solving problems related directly to right-angled triangles but also in many areas of mathematics and science. It provides a simple and effective way to determine the length of any side of a right triangle as long as the other two sides are known.

Proof by Contradiction

Proof by contradiction is a logical method used to establish the truth of a statement by assuming the opposite is true and then showing that this assumption leads to a contradiction. Here's how it works:- Assume the opposite of what you want to prove (let's call the statement you want to prove "P" and the opposite "not P").- Show that assuming "not P" leads to a logical contradiction or an impossibility.- Conclude that since the assumption "not P" leads to a contradiction, "P" must be true.This technique is powerful in mathematics, especially in proving that numbers are irrational or that certain properties must hold. In the exercise, we used this method by assuming a supposedly rational expression for \(x\) and finding that it logically led to the contradiction that both numerator \(a\) and denominator \(b\) of the fraction had to be even, contradicting the initial assumption that they have no common factors.

Rational Numbers

Rational numbers are numbers that can be expressed as a fraction of two integers, where the numerator and the denominator have no common factors other than 1. That is, they can be written as \( \frac{a}{b} \) where \(a\) and \(b\) are integers and \(b eq 0\). Rational numbers include integers, fractions, and terminating or repeating decimals.Characteristics of rational numbers:

• They can be positive, negative, or zero.
• They have a finite or repeating decimal representation.
• They can always be exactly located on the number line.

In the problem, when we tried to express \(x\) as a rational number \(\frac{a}{b}\) and reached a contradiction, it provided a compelling argument that \(x\) could not be rational.

Integer Factorization

Integer factorization involves expressing an integer as a product of other integers. When numbers are decomposed into factors, a special case of interest is expressing them as a product of prime numbers, also known as prime factorization. Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves.Key points about integer factorization:

• Every integer greater than 1 can be uniquely factored into primes, according to the Fundamental Theorem of Arithmetic.
• Finding common factors is crucial in simplifying fractions.
• Having only 1 as a common factor implies that two numbers are coprime or relatively prime.

In our exercise, we assumed \(a\) and \(b\) to be coprime, which set the foundation for finding a contradiction when both were found to be even, leading to the conclusion that the initial assumption was false.