Problem 7 Explain why the product of a non... [FREE SOLUTION]

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Precalculus A Prelude to Calculus

Sheldon Axler

$Math Studyset 91影视 Explanations$ Math

1 Edition

Chapter 0: Problem 7

Explain why the product of a nonzero rational number and an irrational number is an irrational number.

Short Answer

Expert verified

Assuming the product of a nonzero rational number, $a = \frac{p}{q}$, and an irrational number, $x$, is rational, we denote their product as $ax = \frac{m}{n}$ where $p, q, m,$ and $n$ are integers, and $q$ and $n$ are nonzero. Finding an expression for $x$, we get $x = \frac{(ax)}{a} = \frac{\frac{m}{n}}{\frac{p}{q}} = \frac{mq}{np}$. However, since $p, q, m,$ and $n$ are all integers, their ratio, $\frac{mq}{np}$, becomes a rational number, contradicting our initial definition of $x$ being irrational. Thus, the product of a nonzero rational number and an irrational number must be irrational.

Step by step solution

Understanding the definitions of rational and irrational numbers

A rational number is any number that can be expressed as the quotient or fraction p/q of two integers, where p and q are integers, and q is not equal to 0. Irrational numbers, on the other hand, are numbers that cannot be expressed as a fraction p/q, where p and q are integers.

Using contradiction and assuming that the product is rational

Let's assume, for the purpose of contradiction, that the product of a nonzero rational number and an irrational number is rational. Let's denote the nonzero rational number as $a$ and the irrational number as $x$. So, according to our assumption, their product, $ax$, is rational.

Writing the rational numbers in fraction form

Since $a$ is a rational number and $ax$ is assumed to be a rational number, we can represent them in the form of a fraction, as $a = \frac{p}{q}$ and $ax = \frac{m}{n}$, where $p, q, m$, and $n$ are integers and $q$ and $n$ are nonzero.

Calculating x using the expressions for a and ax

Now, we want to find an expression for $x$, which we know to be irrational. We can obtain this by dividing the expressions for $ax$ and $a$, which can be expressed as follows: $x = \frac{(ax)}{a} = \frac{\frac{m}{n}}{\frac{p}{q}} = \frac{mq}{np}$

Realizing the contradiction

Since $p, q, m,$ and $n$ are all integers and both $q$ and $n$ are nonzero, then $mq$ and $np$ are also integers. Their ratio, $\frac{mq}{np}$, is a rational number, which contradicts our initial definition of $x$ being an irrational number. Thus, our assumption that the product of a nonzero rational number and an irrational number is rational must be false.

Conclusion

After showing the contradiction in assuming that the product of a nonzero rational number and an irrational number is rational, we can conclude that the product of a nonzero rational number and an irrational number is indeed an irrational number.

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91影视

Explain why the product of a nonzero rational number and an irrational number is an irrational number.

Short Answer

Step by step solution

Understanding the definitions of rational and irrational numbers

Using contradiction and assuming that the product is rational

Writing the rational numbers in fraction form

Calculating x using the expressions for a and ax

Realizing the contradiction

Conclusion

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