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Chapter 2: Problem 7

Is \(.1234567891011121314151617181920212223242526 \cdots\) rational?

Short Answer

Expert verified
No, the number \(0.1234567891011121314151617181920212223242526 \ldots\) is not rational.

Step by step solution

01

Understand the Definition

Recall that a rational number is any number that can be expressed as the quotient or fraction \(\frac{p}{q}\) where \(p\) and \(q\) are integers and \(q\) is non-zero. Also, rational numbers have either terminating decimals or repeating decimal patterns.
02

Analyze the Given Number

Examine the number \(0.1234567891011121314151617181920212223242526 \ldots\) closely. Notice that it consists of an increasing sequence of digits. It does not have a clear repeating pattern nor does it terminate.
03

Conclude from Analysis

Since the decimal expansion of the given number does not terminate and does not have a repeating pattern, it does not fit 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.

Definition of Rational Numbers
Rational numbers are numbers that can be written as a fraction \(\frac{p}{q}\), where \(p\) and \(q\) are integers, and \(q eq 0\). This means they can be expressed exactly without any approximation. Think of fractions like \(1/2\) or \(3/4\), or whole numbers like \(5\), which can be seen as \(5/1\).
In simple terms, if you can write a number as a simple division of two integers, it's rational.
But there's more to it! When you convert these fractions to decimals, their decimal expansions either terminate or repeat.
For example, \(1/2 = 0.5\) (terminating) and \(1/3 = 0.333\bar{3}\) (repeating).
Keep this in mind as we explore decimal expansions next.
Decimal Expansions
When rational numbers are expressed as decimals, their digits follow specific patterns.
These expansions are either terminating or repeating.
  • Terminating Decimal: A terminating decimal ends after a finite number of digits, e.g., \(0.75\) (which is \(3/4\)).
  • Repeating Decimal: A repeating decimal has digits that keep repeating infinitely, e.g., \(0.333\bar{3}\) (which is \(1/3\)).
Understanding these patterns helps in recognizing whether a number is rational or not.
The exercise given involves a long sequence of numbers after the decimal point.
Does it terminate? Does it repeat? We'll explore that further.
Non-Terminating Decimals
Non-terminating decimals go on forever without ending.
But for a number to be rational, even if it doesn't terminate, it must have a repeating pattern.
For instance, \(0.666\bar{6}\) doesn't terminate but repeats.
The given number, \(0.1234567891011121314151617181920212223242526\text{...}\), does not stop.
By looking at it, you can see it strings together an ever-increasing sequence of numbers.
Since there's no point where the digits come to a stop, it's non-terminating.
But does it have a repeating pattern? Let's find out next.
Non-Repeating Patterns
For a number to be rational and non-terminating, it must have a repeating pattern.
Otherwise, it's irrational.
The number \(0.1234567891011121314151617181920212223242526\text{...}\) shows an interesting sequence.
It starts from \(1\) and keeps adding digits in an increasing order, like \(1, 2, 3, 4, \text{...}\frac{}{}\).
This means there is no repeating pattern, as each segment differs from the previous ones.
Without a repeating segment, the number can't be written as a fraction of integers.
Hence, such a number is not rational.
This important observation is crucial in identifying rational and irrational numbers.

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Most popular questions from this chapter

We have seen that it is often a lot harder to find the value of an infinite sum than to show that it exists. Here are some sums that can be handled. $$\text { (a) Calculate } \sum_{n=1}^{\infty}\left(\frac{2}{3}\right)^{n} \text { and } \sum_{n=1}^{\infty}\left(-\frac{2}{3}\right)^{n}$$ $$\begin{aligned} &\text { (b) Prove } \sum_{n=1}^{\infty} \frac{1}{n(n+1)}=1 . \text { Hint: Note that } \sum_{k=1}^{n} \frac{1}{k(k+1)}=\\\ &\sum_{k=1}^{n}\left[\frac{1}{k}-\frac{1}{k+1}\right] \end{aligned}$$ $$\text { (c) Prove that } \sum_{n=1}^{\infty} \frac{n-1}{2^{n+1}}=\frac{1}{2} . \text { Hint: Note that } \frac{k-1}{2^{k+1}}=\frac{k}{2^{k}}-\frac{k+1}{2^{k+1}}$$ (d) Use (c) to calculate \(\sum_{n=1}^{\infty} \frac{n}{2^{n}}\)

Let \(t_{1}=1\) and \(t_{n+1}=\frac{t_{n}^{2}+2}{2 t_{n}}\) for \(n \geq 1 .\) Assume that \(\left(t_{n}\right)\) converges and find the limit.

For points \(x, y\) in \(\mathbb{R}^{k},\) let $$ d_{1}(x, y)=\max \left\\{\left|x_{j}-y_{j}\right|: j=1,2, \ldots, k\right\\} $$ and$$d_{2}(x, y)=\sum_{j=1}^{k}\left|x_{j}-y_{j}\right| $$ (a) Show that \(d_{1}\) and \(d_{2}\) are metrics for \(\mathbb{R}^{k}\). (b) Show that \(d_{1}\) and \(d_{2}\) are complete.

Let \(\left(s_{n}\right)\) be a nondecreasing sequence of positive numbers and define \(\sigma_{n}=\frac{1}{n}\left(s_{1}+s_{2}+\cdots+s_{n}\right) .\) Prove that \(\left(\sigma_{n}\right)\) is a nondecreasing sequence.

Let \(\left(s_{n}\right)\) be a sequence of nonnegative numbers, and for each \(n\) define \(\sigma_{n}=\frac{1}{n}\left(s_{1}+s_{2}+\cdots+s_{n}\right)\). (a) Show that $$\lim \inf s_{n} \leq \lim \inf \sigma_{n} \leq \lim \sup \sigma_{n} \leq \lim \sup s_{n}$$ Hint: For the last inequality, show first that \(M>N\) implies \(\sup \left\\{\sigma_{n}: n>M\right\\} \leq \frac{1}{M}\left(s_{1}+s_{2}+\cdots+s_{N}\right)+\sup \left\\{s_{n}: n>N\right\\}\). (b) Show that if \(\lim s_{n}\) exists, then \(\lim \sigma_{n}\) exists and \(\lim \sigma_{n}=\) \(\lim s_{n}\).
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