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Chapter 2: Problem 46

Is \(x\) rational? Suppose that \(I\) is a fixed but unknown irrational number. Consider the equation \(4 x-I=2 / 3\). Is it possible to determine if the value for \(x\) that satisfies the equation is rational or irrational? Explain your answer.

Short Answer

Expert verified
The solution for \( x \) is irrational.

Step by step solution

01

Set Up the Equation

We start with the given equation: \( 4x - I = \frac{2}{3} \). Our goal is to isolate \( x \) to determine its nature whether rational or irrational.
02

Isolate x

To isolate \( x \), we'll first add \( I \) to both sides of the equation, which gives us: \( 4x = \frac{2}{3} + I \).
03

Solve for x

Next, divide both sides of the equation by 4 to solve for \( x \): \( x = \frac{1}{4}(\frac{2}{3} + I) \). This simplifies to \( x = \frac{1}{4} \times \frac{2}{3} + \frac{I}{4} \).
04

Analyze the Expression

The expression for \( x \) is \( \frac{1}{6} + \frac{I}{4} \). Since \( I \) is an irrational number, \( \frac{I}{4} \) is also irrational. The sum of a rational number \( \frac{1}{6} \) and an irrational number \( \frac{I}{4} \) is irrational.
05

Conclusion

Since \( \frac{1}{6} + \frac{I}{4} \) is an irrational number, \( x \), as expressed in our equation, is irrational. Therefore, given that \( I \) is irrational, \( x \) must also be irrational.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Equation Solving
Equation solving is all about finding the unknowns in an equation. In this particular exercise, we're working with the equation \( 4x - I = \frac{2}{3} \). Our mission is to determine whether \( x \) is rational or irrational. We start by trying to isolate \( x \) on one side of the equation so we can clearly see its value.
Here's how we solve it step-by-step:
  • First, add \( I \) to both sides, turning the equation into \( 4x = \frac{2}{3} + I \).
  • Then, divide everything by 4, giving us \( x = \frac{1}{4}(\frac{2}{3} + I) \).
  • Simplify the multiplication to get \( x = \frac{1}{6} + \frac{I}{4} \).
Equation solving doesn't just stop at isolating the variable; analyzing the nature of the solution is crucial.
In this case, the expression suggests that \( x \) can be irrational due to its components.
Irrational Numbers
Irrational numbers are numbers that cannot be expressed as a simple fraction. They're a bit like the mystery in math because they don't seem to end or repeat in decimal form. They're crucial in this exercise because we've got \( I \), a fixed but unknown irrational number that plays a starring role in our equation.
When we break it down:
  • \( I \) cannot be made into a fraction like \( \frac{2}{3} \), so it's not aligning neatly in the world of rational numbers.
  • Upon simplifying \( x = \frac{1}{6} + \frac{I}{4} \), we see \( \frac{I}{4} \) stays irrational. Multiplying or dividing an irrational number still leaves us with an irrational piece.
  • The addition of a rational (\( \frac{1}{6} \)) and an irrational (\( \frac{I}{4} \)) gives us an irrational overall value.
Understanding these friendly irrational numbers makes the identification of \( x \) as irrational simpler and more logical.
Mathematical Reasoning
Mathematical reasoning is like tying up the loose ends to make sense of all the math puzzles we've encountered. We need to use logical thinking to draw conclusions based on what we know. Let's put this into practice for our equation.
Here's our logical breakdown:
  • First, identify all known components like \( \frac{2}{3} \) (rational) and \( I \) (irrational).
  • Since \( x = \frac{1}{6} + \frac{I}{4} \), rational \( \frac{1}{6} \) and irrational \( \frac{I}{4} \) combine to form an irrational number for \( x \).
  • Therefore, rational numbers can't absorb the 'irrationality' by just simple addition.
  • Even though calculations simplified the expression, logical reasoning using known properties of rational and irrational numbers confirms \( x \) is irrational.
Drawing these conclusions helps students realize why math isn't just about operations but making sense out of patterns and properties encountered.

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Most popular questions from this chapter

Irrational with 1 's and some 2 's. Is it possible to build an irrational number whose decimal digits are just 1 's and 2 's and only finitely many 2 's appear? If so, describe such a number and show why it's irrational. If not, explain why.

With a group of folks. In a small group, discuss and work through the arguments that the number \(0.12345678910 \ldots\) is irrational and that \(0.99999 \ldots=1\). After your discussion, write a brief narrative describing the arguments in your own words.

A proof for \(\pi\). Suppose we look at the first one billion decimal digits of \(\pi\). Those digits do not repeat. Does that prove that \(\pi\) is irrational? Why or why not? What if we examined the first trillion digits?

Is \(y\) irrational? You decide to create the digits of a decimal number \(y\) between 0 and 1 using the function \(f(n)=3 n+1\). Here's your system. Compute \(f(1)\) and put the result as the first digit of \(y\) to the right of the decimal point. Compute \(f(2)\) and put the result as the second digit of \(y\). Compute \(f(3)\) and put the result as the third and fourth digits of \(y\). Compute \(f(4)\) and put the result as the fifth and sixth digits of \(y\). And so on. What's the tenth digit of \(y\) ? Do you think \(y\) is irrational? Why or why not?

Irrationals and zero. Is there an irrational number that is closer to zero than any other irrational? If so, describe it. If not, give a sequence of irrational numbers that get closer and closer to zero. (Hint: Start by considering \(\sqrt{2} / 2\) and \(\sqrt{2} / 3\).
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