Question

c) how many rational numbers are there from 1 through 10?
d) how many irrational numbers are there from 1 through 10?
e) how many real numbers are there from 1 through 10?

1. explain what it means for a number set to have the density property.
2. which of the following sets have the density property?

• natural numbers
• whole numbers
• integers
• rational numbers
• irrational numbers
• real numbers

1. does the set of even integers have the density property? explain.
2. the following sequences of numbers each have a limit. that is, they gradually get closer and closer to a specific number, called the limit. identify the limit of each of the following sequences.

a) 7.1, 7.01, 7.001, 7.0001,...
b) 1/2, 1/3, 1/4, 1/5, 1/6,...
c) 5/1, 5/4, 5/9, 5/16, 5/25,...

Explanation:

Step1: Define density property

The density property of a number - set means that between any two distinct elements of the set, there exists another element of the set. For example, in the set of rational numbers, if \(a\) and \(b\) are two rational numbers with \(a < b\), then \(\frac{a + b}{2}\) is also a rational number and \(a<\frac{a + b}{2}

Step2: Analyze sets for density property

• Natural numbers (\(\mathbb{N}=\{1,2,3,\cdots\}\)): No. For example, between \(1\) and \(2\), there is no other natural number.
• Whole numbers (\(\mathbb{W}=\{0,1,2,\cdots\}\)): No. For example, between \(0\) and \(1\), there is no other whole - number.
• Integers (\(\mathbb{Z}=\{\cdots,- 2,-1,0,1,2,\cdots\}\)): No. For example, between \(1\) and \(2\), there is no other integer.
• Rational numbers (\(\mathbb{Q}\)): Yes. As shown in the definition example above.
• Irrational numbers (\(\mathbb{I}\)): Yes. Between any two distinct irrational numbers \(x\) and \(y\) (\(x < y\)), there is another irrational number. For example, \(x<\frac{x + y}{2}\sqrt{2}/\sqrt{2}
• Real numbers (\(\mathbb{R}\)): Yes. Since \(\mathbb{R}=\mathbb{Q}\cup\mathbb{I}\), and both rational and irrational numbers have the density property, real numbers have the density property.

Step3: Analyze even integers

The set of even integers (\(\{ \cdots,-4,-2,0,2,4,\cdots\}\)) does not have the density property. For example, between \(2\) and \(4\), there is no other even integer.

Step4: Find limits of sequences

a)

The sequence \(7.1,7.01,7.001,7.0001,\cdots\) can be written as \(a_n = 7+\frac{1}{10^n}\). As \(n\to\infty\), \(\frac{1}{10^n}\to0\). So the limit is \(7\).

b)

The sequence \(\frac{1}{2},\frac{1}{3},\frac{1}{4},\frac{1}{5},\frac{1}{6},\cdots\) can be written as \(a_n=\frac{1}{n + 1}\). As \(n\to\infty\), \(\frac{1}{n+1}\to0\). So the limit is \(0\).

c)

The sequence \(\frac{5}{1},\frac{5}{4},\frac{5}{9},\frac{5}{16},\frac{5}{25},\cdots\) can be written as \(a_n=\frac{5}{n^2}\). As \(n\to\infty\), \(\frac{5}{n^2}\to0\). So the limit is \(0\).

Answer:

1. A number - set has the density property if between any two distinct elements of the set, there exists another element of the set.
2. Rational numbers, Irrational numbers, Real numbers.
3. No. Because between two consecutive even integers, there is no other even integer.
4. a) \(7\); b) \(0\); c) \(0\)