Problem 44 - Algebra for College Students

Question

Use the rational root theorem to argue that \(\sqrt{12}\) is not a rational number.

Step-by-Step Solution

Answer

\( \sqrt{12} \) is not rational because it implies contradictory prime factorization assumptions.

Step 1: Identify the expression

We are given the expression \( \sqrt{12} \) and need to determine if it is a rational number. A rational number can be expressed as the fraction \( \frac{p}{q} \), where \( p \) and \( q \) are integers, \( q eq 0 \).

Step 2: Express the equation

Assume \( \sqrt{12} = \frac{p}{q} \) where \( p \) and \( q \) are integers with no common factors other than 1. Thus, we have \( 12 = \frac{p^2}{q^2} \) leading to \( p^2 = 12q^2 \).

Step 3: Examine prime factorization

The equation \( p^2 = 12q^2 \) implies that \( p^2 \) is divisible by 12. The prime factorization of 12 is \( 2^2 \times 3^1 \). This implies \( p^2 \) and hence \( p \) itself must be divisible by 2 and 3.

Step 4: Simplify conditions for \( q \)

If \( p \) is divisible by 2 and 3, it can be expressed as \( p = 6k \) (since \( 2 \times 3 = 6 \)). Substituting \( p = 6k \) into \( p^2 = 12q^2 \) giving us \( 36k^2 = 12q^2 \). Simplifying, we get \( 3k^2 = q^2 \).

Step 5: Assess divisibility

The equation \( 3k^2 = q^2 \) implies \( q^2 \) is divisible by 3, hence \( q \) must be divisible by 3. However, if both \( p \) and \( q \) are divisible by 3, they have a common factor greater than 1, contradicting our assumption that \( \frac{p}{q} \) is in simplest form.

Step 6: Conclude non-rationality

Since assuming \( \sqrt{12} \) equals a fraction leads to a contradiction (both \( p \) and \( q \) sharing a prime factor), \( \sqrt{12} \) cannot be expressed as a rational number.

Key Concepts

Rational Numbers

Rational numbers are numbers that can be expressed as the quotient or fraction of two integers. That means if you can represent a number in the form \( \frac{p}{q} \), where \( p \) and \( q \) are integers and \( q \) is not zero, then it’s a rational number. Think of rational numbers as any number that can have a clean fraction form.

• Examples include fractions like \( \frac{3}{4} \), \( -\frac{2}{5} \), and whole numbers like 5 (which can be written as \( \frac{5}{1} \)).
• Even recurring decimals such as 0.3333... (repeating) count as rational because they can be rewritten as \( \frac{1}{3} \).

All this means is that rational numbers are just numbers we can comfortably put on the number line with fractions. They are easy to recognize because they don't go on forever without repeating, and they make arithmetic neat and orderly.

Prime Factorization

Prime factorization is a process of breaking down a number into its basic building blocks, which are prime numbers. A prime number is one that has no divisors other than 1 and itself.

• For example, the prime factorization of 12 is \( 2^2 \times 3 \), which means 12 is the result of multiplying two 2s and one 3 together.
• This is a useful tool because it helps us see the number's original components for simplification or solving divisibility issues.

When analyzing whether a number like \( p^2 = 12q^2 \) could stem from rational roots, prime factorization tells us the kind of numbers \( p \) must be composed of. It reveals incompatibilities when seeking rational solutions and is a fundamental step in proving that certain roots, like \( \sqrt{12} \), are indeed not rational.

Irrational Numbers

Irrational numbers are numbers that cannot be expressed as a simple fraction, meaning they cannot be written as \( \frac{p}{q} \) where \( p \) and \( q \) are integers. They extend infinitely without repeating, making them quite different from rational numbers.

• Famous examples include \( \sqrt{2} \), \( \sqrt{3} \), and the number \( \pi \).
• Decimals that go on indefinitely without forming any repetition are classic signs of irrational numbers.

The Rational Root Theorem is useful for asserting that certain roots are irrational. In the case of \( \sqrt{12} \), through logical deduction and examination of divisibility, we discover that assuming it's rational leads to a contradiction. Therefore, despite looking like a clean square root, because it can't break into a neat fraction, \( \sqrt{12} \) is classified as an irrational number. This group's uniqueness adds richness to our number system by ensuring even predictable, common numbers can spring up surprises.