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Chapter 5: Problem 50

For Exercises \(47-58,\) identify the number as rational or irrational. $$3$$

Short Answer

Expert verified
The number 3 is rational.

Step by step solution

01

Define Rational and Irrational Numbers

A rational number can be expressed as the quotient or fraction \(\frac{p}{q}\), where \(p\) and \(q\) are integers and \(q eq 0\). An irrational number cannot be expressed as a fraction of two integers.
02

Express the Given Number as a Fraction

Identify if the number 3 can be expressed as a fraction where the numerator and the denominator are integers. The number 3 can be expressed as \( \frac{3}{1} \).
03

Conclusion

Since the number 3 can be expressed as the fraction \( \frac{3}{1} \), it meets the definition of a rational number.

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Key Concepts

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

Rational Numbers
Rational numbers are a central concept in mathematics. A rational number can be written as a fraction, where both the numerator and the denominator are integers, and the denominator is not zero. For example, the number 3 can be expressed as the fraction \(\frac{3}{1}\). This ability to be written as a fraction means 3 is a rational number.
Rational numbers include positive and negative whole numbers, fractions, and mixed numbers. They can also be represented as terminating or repeating decimals. Here's a quick example:
  • \(\frac{1}{2}\) is a rational number because it is a fraction with integers in the numerator and denominator.
  • 0.75 is rational because it can be expressed as \(\frac{3}{4}\).
Being able to convert numbers into fractions and identify rational numbers is a skill that will help you understand more complex mathematical concepts in the future.
Integers
Integers are the simplest form of rational numbers. They include all positive whole numbers, negative whole numbers, and zero. In mathematical terms, integers are represented by the set \(\boldsymbol{Z}\) which includes:
  • ... -3, -2, -1, 0, 1, 2, 3, ...

One key point to remember is that every integer is a rational number, but not every rational number is an integer. For instance, 3 is an integer and a rational number because it can be expressed as \(\frac{3}{1}\), but \(\frac{1}{2}\) is a rational number that is not an integer. Understanding integers helps lay a strong foundation for working with all kinds of rational numbers.
Fractions
Fractions represent parts of a whole and are a critical part of rational numbers. A fraction is written as \(\frac{p}{q}\), where \(\boldsymbol{p}\) represents the numerator and \(\boldsymbol{q}\) represents the denominator. For example, the rational number 3 can be written as the fraction \(\frac{3}{1}\).
Fractions can be:
  • Proper fractions where the numerator is less than the denominator, like \(\frac{3}{4} \).
  • Improper fractions where the numerator is greater than or equal to the denominator, like \(\frac{5}{3}\).
  • Mixed numbers which are a combination of a whole number and a fraction, like 1 \(\frac{1}{2}\).
Understanding fractions is crucial, as they can sometimes be simplified or converted to and from decimals, and are frequently used in various applications like measurements, probabilities, and ratios.

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Most popular questions from this chapter

The prices of 10 single-family, three-bedroom homes for sale in Boston, Massachusetts, are listed for a recent year. Find the mean, median, and mode (if one exists).$$\begin{aligned}&\$ 300,000, \$ 2,495,000, \$ 2,120,000, \$ 220,000\\\&\$ 194,000, \$ 391,000, \quad \$ 315,000, \quad \$ 330,000\\\ &\$ 435,000, \$ 250,000\end{aligned}$$

Jennifer earns \(\$ 720.00\) per week. Alex earns \(\frac{3}{4}\) of what Jennifer earns per week. a. How much does Alex earn per week? b. If they both work 40 hr per week, how much do they each make per hour?

For Exercises \(85-90,\) refer to the table. The table gives the closing stock prices (in dollars per share) for the first day of trading for the given month. $$\begin{array}{l|c|c|c|c|c} \text { Stock } & \text { January } & \text { February } & \text { March } & \text { April } & \text { May } \\ \hline \text { IBM } & \$ 97.27 & \$ 99.00 & \$ 92.27 & \$ 95.21 & \$ 103.17 \\\ \hline \text { FedEx } & \$ 109.77 & \$ 111.89 & \$ 114.16 & \$ 106.07 & \$ 106.11 \\ \hline \end{array}$$ Between which two consecutive months did the IBM stock decrease the most? What was the amount of decrease?

Which number is between 3.12 and 3.13? Circle all that apply. a. 3.127 b. 3.129 c. 3.134 d. 3.139

The table shows the thickness of four U.S. coins. If you stacked three quarters and a dime in one pile and two nickels and two pennies in another pile, which pile would be higher? $$\begin{array}{l|l} \text { Coin } & \text { Thickness } \\ \hline \text { Quarter } & 1.75 \mathrm{mm} \\ \hline \text { Dime } & 1.35 \mathrm{mm} \\ \hline \text { Nickel } & 1.95 \mathrm{mm} \\ \hline \text { Penny } & 1.55 \mathrm{mm} \\ \hline \end{array}$$
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