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Chapter 5: Problem 153

What does it mean when we say that the set of rational numbers is dense?

Short Answer

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In mathematics, the set of rational numbers is said to be dense because for any two rational numbers, one can always find another rational number in between them. For example, \( \frac{1}{2} \) and \( \frac{2}{3} \) are two rational numbers and \( \frac{3}{5} \) is a rational number that lies between them. Thus, the term 'dense' as applied to the set of rational numbers, refers to the property where there is no gap in the set.

Step by step solution

01

Understanding Rational Numbers

Rational numbers are any numbers that can be expressed as a ratio of two integers where the denominator is not zero, i.e., it can be written in the form \( \frac{a}{b} \) where \( a \) and \( b \) are integers and \( b 鈮� 0 \). Examples of rational numbers include \( \frac{1}{2}, 2, -\frac{3}{4} \) etc.
02

Defining Dense Sets

A set is said to be dense if between any two numbers within the set, there is another number from the same set. The key point here is that there should be no gap between any two numbers.
03

Density of Rational Numbers

The set of rational numbers is dense because between any two rational numbers, one can always find another rational number. For example, between \( \frac{1}{2} \) and \( \frac{2}{3} \) one can find \( \frac{3}{5} \) or \( \frac{7}{12} \), thus proving that there is no gap in the set of rational numbers

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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 special numbers for a few reasons. They are numbers that you can write as a fraction, like \( \frac{a}{b} \), where \( a \) and \( b \) are whole numbers, and \( b \) is not zero. This means any number you can think of that fits this form is a rational number. Here鈥檚 a look at some examples:
  • Whole numbers like 2 or 5 are rational because they can be written as \( \frac{2}{1} \) or \( \frac{5}{1} \).
  • Fractions such as \( \frac{1}{2} \) or \( -\frac{3}{4} \) are rational.
  • Even some decimals, like 0.5 (equivalent to \( \frac{1}{2} \)), can be rational.
Essentially, if you can express a number as a fraction with the above rules, it鈥檚 rational. This makes rational numbers versatile and abundant in the world of mathematics. They include negative numbers, fractions, and whole numbers as long as they can take the form \( \frac{a}{b} \).
Dense Sets
The concept of a dense set might seem tricky, but it鈥檚 quite intriguing. Imagine you have a number line, which goes on forever in both directions. A set is called dense if, no matter how close you look between two numbers in the set, you can always find another number from that same set. The idea of a dense set is like this:
  • There are no gaps between numbers.
  • If you pick any two numbers, you can find more numbers in between.
  • The numbers just keep on coming, no matter how close you zoom in on the line.
This definition highlights that a dense set isn鈥檛 just packed full of numbers; it鈥檚 filled with infinite possibilities between any two given members. This property is crucial for understanding how some number sets behave differently than others.
Mathematical Concepts
It's important to integrate the mathematical concepts that underpin the idea of dense sets and rational numbers. When mathematicians talk about density in numbers, they are often looking at the structure and distribution of these numbers along a number line. Here's how it connects:
  • The rational numbers being dense means there's no spot on the number line where you can't find a rational number close by.
  • This makes the rational numbers 'crowd' the number line, filling in all sorts of gaps you might assume were there.
  • The concept of density is not about the total number of rational numbers but the infinitesimal spaces they can fill.
This understanding shows how rational numbers are applicable in measuring, calculating, and modeling in mathematics, as they can precisely describe many quantities. These concepts tie into broader mathematical ideas, like continuity and approximation, illuminating how useful rational numbers are in real-life applications and theoretical mathematics alike.

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Most popular questions from this chapter

Enough curiosities involving the Fibonacci sequence exist to warrant a flourishing Fibonacci Association. It publishes a quarterly journal. Do some research on the Fibonacci sequence by consulting the research department of your library or the Internet, and find one property that interests you. After doing this research, get together with your group to share these intriguing properties.

Company A pays \(\$ 44,000\) yearly with raises of \(\$ 1600\) per year. Company B pays \(\$ 48,000\) yearly with raises of \(\$ 1000\) per year. Which company will pay more in year 10 ? How much more?

Write the first six terms of the geometric sequence with the first term, \(a_{1}\), and common ratio, \(r\). \(a_{1}=10, r=-4\)

Write the first six terms of the geometric sequence with the first term, \(a_{1}\), and common ratio, \(r\). \(a_{1}=20, r=-4\)

You will develop geometric sequences that model the population growth for California and Texas, the two most populated U.S. states. The table shows the population of Texas for 2000 and 2010 , with estimates given by the U.S. Census Bureau for 2001 through \(2009 .\) $$ \begin{aligned} &\begin{array}{|l|c|c|c|c|c|c|} \hline \text { Year } & \mathbf{2 0 0 0} & \mathbf{2 0 0 1} & \mathbf{2 0 0 2} & \mathbf{2 0 0 3} & \mathbf{2 0 0 4} & \mathbf{2 0 0 5} \\ \hline \begin{array}{l} \text { Population } \\ \text { in millions } \end{array} & 20.85 & 21.27 & 21.70 & 22.13 & 22.57 & 23.02 \\ \hline \end{array}\\\ &\begin{array}{|l|c|c|c|c|c|} \hline \text { Year } & \mathbf{2 0 0 6} & \mathbf{2 0 0 7} & \mathbf{2 0 0 8} & \mathbf{2 0 0 9} & \mathbf{2 0 1 0} \\ \hline \begin{array}{l} \text { Population } \\ \text { in millions } \end{array} & 23.48 & 23.95 & 24.43 & 24.92 & 25.15 \\ \hline \end{array} \end{aligned} $$ a. Divide the population for each year by the population in the preceding year. Round to two decimal places and show that Texas has a population increase that is approximately geometric. b. Write the general term of the geometric sequence modeling Texas's population, in millions, \(n\) years after \(1999 .\) c. Use your model from part (b) to project Texas's population, in millions, for the year 2020 . Round to two decimal places.
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