Problem 95 - Elementary Algebra

Question

Do you think that \(2 \sqrt{2}\) is a rational or an irrational number? Defend your answer.

Step-by-Step Solution

Answer

The number \(2 \sqrt{2}\) is irrational because \(\sqrt{2}\) is irrational.

Step 1: Understanding Rational and Irrational Numbers

A rational number is any number that can be expressed as the quotient or fraction of two integers, with a non-zero denominator. An irrational number cannot be expressed as such a fraction.

Step 2: Identifying the Form of the Given Number

The problem asks about the number \(2 \sqrt{2}\). We know that \(\sqrt{2}\), the square root of 2, cannot be expressed as a fraction of two integers, thus it is an irrational number.

Step 3: Solving the Expression

When you multiply 2 (a rational number) by \(\sqrt{2}\) (an irrational number), the result \(2 \sqrt{2}\) inherits the irrationality from \(\sqrt{2}\).

Step 4: Concluding the Nature of the Number

Since \(2 \sqrt{2}\) cannot be expressed as a fraction of two integers, it is an irrational number.

Key Concepts

Rational Numbers

Rational numbers are a fundamental part of mathematics. They are numbers that can be written as the ratio of two integers. Let's break this down: the numerator is the integer above the fraction line, and the denominator is the integer below it. Importantly, the denominator cannot be zero, because division by zero is undefined.
For example, the number \( \frac{3}{4} \) is rational because it is expressed as the fraction of two integers: 3 and 4. Rational numbers include both whole numbers (like 5, which can be written as \( \frac{5}{1} \)) and fractions (like \( \frac{1}{2} \)).

• Rational numbers can be either positive or negative.
• They have terminating or repeating decimal expansions.
• The easy way to remember: Can you write the number as a simple fraction? If yes, it's rational!

Square Root

The square root is a special mathematical operation. It finds a number which, when multiplied by itself, gives the original value. For instance, the square root of 9 is 3, because 3 multiplied by itself (\(3 \times 3\)) equals 9.
However, not all numbers have square roots that are neat and tidy integers. For example, \( \sqrt{2} \) is not a whole number. It's approximately 1.414, but it doesn't end or repeat, making it an irrational number.

• Perfect squares yield integer square roots, like \( \sqrt{16} = 4 \).
• Non-perfect squares, like 2, have irrational square roots.
• Understanding square roots helps in comprehending why certain numbers are not rational.

Expressing Numbers as Fractions

Expressing a number as a fraction is a key skill in math. When asked if a number is rational, we essentially ask if it can be transformed into a fraction of two integers.
Rational numbers fit neatly into this definition. You can always express a number like 0.75 as \( \frac{3}{4} \).

• Whole numbers as fractions look like \( n/1 \), where \( n \) is the whole number.
• Decimals can often be converted into fractions by recognizing repeating or terminating patterns. For example, 0.333... equals \( \frac{1}{3} \).
• Irrational numbers, like \( \sqrt{2} \), don't fit this description, hence they're not expressible as fractions of integers.

Expressing numbers as fractions serves as a touchstone for distinguishing between rational and irrational numbers.