Now consider the product of a nonzero rational number and an irrational number. Again, assume [tex]x=\frac{a}{b}[/tex], where [tex]a[/tex] and [tex]b[/tex] are integers and [tex]b \neq 0[/tex]. This time let [tex]y[/tex] be an irrational number. If we assume the product [tex]x \cdot y[/tex] is rational, we can set the product equal to [tex]\frac{m}{n}[/tex], where [tex]m[/tex] and [tex]n[/tex] are integers and [tex]n \neq 0[/tex]. The steps for solving this equation for [tex]y[/tex] are shown.
Based on what we established about the classification of [tex]y[/tex] and using the closure of integers, what does the equation tell you about the type of number [tex]y[/tex] must be for the product to be rational? What conclusion can you now make about the result of multiplying a rational and an irrational number?
Answer (1)
Assume x is a nonzero rational number and y is an irrational number, and their product x o b re ak ⋅ y is rational.
Express x ⋅ y = n m , where m and n are integers, and solve for y to get y = na mb .
Since m , b , n , a are integers, mb and na are also integers, making y a rational number.
This contradicts the initial assumption that y is irrational, therefore, the product of a nonzero rational number and an irrational number must be irrational: irrational .
Explanation
Problem Setup and Initial Assumptions We are given that x is a nonzero rational number, which can be expressed as x = b a , where a and b are integers and b = 0 . We also know that y is an irrational number. We assume that the product x ⋅ y is rational, so we can write x ⋅ y = n m , where m and n are integers and n = 0 . Our goal is to determine what type of number y must be for the product to be rational and to draw a conclusion about the result of multiplying a rational and an irrational number.
Analyzing the Equation for y We are given the equation y = na mb . Since m , b , n , and a are all integers, and integers are closed under multiplication, it follows that mb and na are also integers. Therefore, the fraction na mb represents a rational number, because it is a ratio of two integers.
Identifying the Contradiction The equation y = na mb implies that y is a rational number. However, we initially stated that y is an irrational number. This creates a contradiction. If we assume that the product of a nonzero rational number x and an irrational number y is rational, then y must be rational, which contradicts our initial condition that y is irrational.
Conclusion Since our assumption that the product x ⋅ y is rational leads to a contradiction, that assumption must be false. Therefore, the product of a nonzero rational number and an irrational number must be irrational.
Examples
Consider a scenario where you want to calculate the total cost of buying 2 acres of land, and the price per acre is a rational number, say 2 3 million dollars. Since 2 is irrational and 2 3 is rational, their product, 2 3 ⋅ 2 , will be irrational. This means the total cost cannot be expressed as a simple fraction, and you'll need to work with irrational numbers or approximations to determine the exact cost. This principle is crucial in various fields, including engineering and economics, where irrational numbers frequently appear in calculations.
