Problem 57 - Precalculus

Question

Give an example of irrational numbers a and b such that the indicated expression is (a) rational and (b) irrational. $$a b$$

Step-by-Step Solution

Answer

Use \( a = \sqrt{2} \) and \( b = \sqrt{2} \) for rational product; \( a = \sqrt{2} \) and \( b = \sqrt{3} \) for irrational product.

Step 1: Understanding Rational and Irrational Numbers

A rational number can be expressed as a fraction \( \frac{p}{q} \) where \( p \) and \( q \) are integers and \( q eq 0 \). An irrational number cannot be expressed as a simple fraction.

Step 2: Example for Rational Product \( ab \)

Consider \( a = \sqrt{2} \) and \( b = \sqrt{2} \). Although both \( \sqrt{2} \) and \( \sqrt{2} \) are irrational numbers, their product is \( ab = \sqrt{2} \times \sqrt{2} = 2 \), which is rational.

Step 3: Example for Irrational Product \( ab \)

Consider \( a = \sqrt{2} \) and \( b = \sqrt{3} \). Both \( \sqrt{2} \) and \( \sqrt{3} \) are irrational, and their product \( ab = \sqrt{2} \times \sqrt{3} = \sqrt{6} \) is also irrational because 6 is not a perfect square.

Key Concepts

Rational Numbers

Rational numbers are numbers that can be written as a quotient or fraction \( \frac{p}{q} \). Here, \( p \) and \( q \) are integers, and \( q eq 0 \). This means any number you can think of that fits nicely into a fraction is rational.
For example:

• \( \frac{1}{2} \)
• \( \frac{-3}{4} \)
• \( 5 \) (which can be written as \( \frac{5}{1} \))

These numbers terminate or repeat after a certain point if you were to write them as decimals. So, when you encounter a number that ends or has a repeating pattern in its decimal form, it’s likely rational.

Properties of Irrational Numbers

Irrational numbers, unlike rational numbers, cannot be expressed as a simple fraction. Their decimal forms are non-repeating and non-terminating, meaning they continue on forever without forming a predictable pattern.
Some important properties of irrational numbers include:

• They cannot be written as \( \frac{p}{q} \) where both \( p \) and \( q \) are integers.
• Their decimal expansion does not repeat or terminate.
• Their sums or products with specific conditions can result in either rational or irrational outcomes.

Understanding these properties helps in recognizing and working with irrational numbers effectively.

Examples of Irrational Numbers

There are many numbers that are considered irrational. These numbers are not as straightforward as rational ones.
Common examples of irrational numbers include:

• \( \pi \) (the ratio of the circumference of a circle to its diameter)
• \( e \) (the base of natural logarithms)
• \( \sqrt{2} \), \( \sqrt{3} \), \( \sqrt{5} \) (since these square roots are not perfect squares)

These numbers often appear in mathematical formulas and natural growth models, given their unique properties.

Irrational and Rational Multiplication

Multiplying irrational numbers can yield interesting results. Understanding the outcome of such multiplication requires attention to the properties of the numbers involved.

• An example of multiplying two identical irrational numbers is \( \sqrt{2} \times \sqrt{2} = 2 \), which results in a rational number even though both multiplicands were irrational.
• Multiplying two different irrational numbers, such as \( \sqrt{2} \times \sqrt{3} = \sqrt{6} \), results in another irrational number since 6 is not a perfect square.

Therefore, when dealing with irrational and rational multiplication, checking the conditions is crucial to predict the rationality of the product.