Problem 2 Show that $5-\sqrt{2}$ is an i... [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 2

Show that $5-\sqrt{2}$ is an irrational number.

Short Answer

Expert verified

Assuming that $5-\sqrt{2}$ is rational and can be expressed as $\frac{p}{q}$, we derive a contradiction when trying to solve for $p$, as we obtain $p=5q - q\sqrt{2}$, in which $q\sqrt{2}$ must not be an integer. This contradicts our assumption, therefore proving that $5 - \sqrt{2}$ is an irrational number.

Step by step solution

Assume $5-\sqrt{2}$ is rational

Let's assume, for the sake of contradiction, that $5-\sqrt{2}$ is a rational number. This means that it can be expressed as a fraction. Let's express it in the simplest form as a fraction: $5 - \sqrt{2} = \frac{p}{q}$ Where $p$ and $q$ are integers with no common factors other than 1 (i.e., they are co-prime), and $q \neq 0$.

Express $\sqrt{2}$ in terms of $p$ and $q$

Next, express $\sqrt{2}$ in terms of $p$ and $q$: $\sqrt{2} = 5 - \frac{p}{q}$

Rearrange the equation for $p$

Now rearrange the equation to solve for $p$: $p = 5q - q\sqrt{2}$

Observe the properties of $p$

If $q$ is an integer, then $5q$ is clearly an integer. Also, if $q\sqrt{2}$ is an integer, then $(5q - q\sqrt{2})$ must be an integer as well. However, we know that $\sqrt{2}$ is irrational. Therefore, we can conclude that $q\sqrt{2}$ is not an integer.

Derive a contradiction

Since we have assumed that $5-\sqrt{2}$ is rational, it can be expressed as $\frac{p}{q}$ where both $p$ and $q$ are integers. But now we have found that for the equation to hold true, $p = 5q - q\sqrt{2}$, and $q\sqrt{2}$ must not be an integer. This leads to a contradiction as $p$ cannot be an integer.

Conclude that $5-\sqrt{2}$ is irrational

Since our assumption that $5-\sqrt{2}$ is rational leads to a contradiction, we can conclude that $5-\sqrt{2}$ is an irrational number.

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91影视

Show that \(5-\sqrt{2}\) is an irrational number.

Short Answer

Step by step solution

Assume \(5-\sqrt{2}\) is rational

Express \(\sqrt{2}\) in terms of \(p\) and \(q\)

Rearrange the equation for \(p\)

Observe the properties of \(p\)

Derive a contradiction

Conclude that \(5-\sqrt{2}\) is irrational

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