1/1, 2/1, 1/2, 1/3, 2/2, 3/1, 4/1, 3/2, 2/3, 1/4, 1/5, 2/4, 3/3, 4/2, 5/1 I t skip i skip 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 Using this matching, what natural number is matched with the rational number
1/1, 2/1, 1/2, 1/3, 2/2, 3/1, 4/1, 3/2, 2/3, 1/4, 1/5, 2/4, 3/3, 4/2, 5/1 I t skip i skip 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 Using this matching, what natural number is matched with the rational number
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
Arrange the positive rational numbers in a table such that the first row of the arrangement has all positive rational numbers with denominator 1, the second row has all denominators 2, and so on. Then trace through the arrangement following the line shown on the right, skipping over numbers that have already been encountered. This path results in the matching with the natural numbers shown.
Transcribed Image Text:The image presents a matrix-like arrangement of fractions, followed by a series of blue diagonal lines. These fractions represent pairs of numbers in the form of \( \frac{m}{n} \), where \(m\) is the numerator, and \(n\) is the denominator. The diagram suggests a systematic arrangement and traversal of these pairs. Below is the detailed transcription and explanation of the content in the image:
### Transcription:
1/1, 2/1, 3/1, 4/1, 5/1, ...
1/2, 2/2, 3/2, 4/2, 5/2, ...
1/3, 2/3, 3/3, 4/3, 5/3, ...
1/4, 2/4, 3/4, 4/4, 5/4, ...
1/5, 2/5, 3/5, 4/5, 5/5, ...
### Explanation of Diagonal Lines:
The diagonal lines in blue denote a method for traversing the matrix. Starting from the top-left element of the matrix:
1. The first line begins at 1/1.
2. The second line starts at 2/1 and then moves to 1/2.
3. The third line includes 3/1, 2/2, and 1/3.
4. The fourth line covers 4/1, 3/2, 2/3, and 1/4.
5. The fifth line spans 5/1, 4/2, 3/3, 2/4, and 1/5.
This systematic diagonal traversal can be used for various mathematical applications or algorithms where systematically visiting each pair based on their position in the grid is necessary.
This illustration and traversal technique could be particularly useful in advanced topics such as Farey sequences, Cantor's diagonal argument, or continued fractions.
![**Title: Rational Numbers and Their Natural Number Matches**
**Content:**
In this educational lesson, we explore a unique way of matching rational numbers with natural numbers. This approach systematically matches each rational number with a distinct natural number without repetition.
**Instructions:**
Below is a sequence demonstrating this matching method:
\[
\begin{array}{ccccccccccc}
\frac{1}{1} & \frac{2}{1} & \frac{1}{2} & \frac{1}{3} & \frac{2}{2} & \frac{3}{1} & \frac{4}{1} & \frac{2}{3} & \frac{3}{2} & \frac{4}{3} & \frac{3}{3} & \frac{4}{2} & \frac{5}{1} & \cdots \\
\downarrow & \downarrow & \downarrow & \downarrow & \text{skip} & \downarrow & \downarrow & \downarrow & \downarrow & \downarrow & \text{skip} & \downarrow & \downarrow & \cdots \\
1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & \cdots \\
\end{array}
\]
**Explanation:**
1. The first row lists pairs of integers as fractions representing rational numbers.
2. The second row uses arrows to point to corresponding positions in the natural numbers sequence (1, 2, 3, ...).
3. The "skip" indicates that certain rational numbers, which are equivalent to earlier ones, are omitted to avoid duplication (e.g., 2/2 = 1/1, 4/2 = 2/1, etc.).
**Question:**
Using this matching method, what natural number is matched with the rational number \(\frac{5}{2}\)?
**Answer:**
The natural number that is matched with the rational number \(\frac{5}{2}\) is \(\boxed{11}\).
\( (Simplify your answer.)\)
This example demonstrates the matching process and helps in understanding the systematic way in which every unique rational number corresponds to a natural number. This type of exercise can deepen one’s understanding of the correspondence between different sets of numbers in mathematics.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd645a57e-8fec-4f58-b186-7cff590ec41c%2Fe7293dc6-13f1-42f5-9406-5bd0c3d93d18%2F0ais4n4_processed.png&w=3840&q=75)
Transcribed Image Text:**Title: Rational Numbers and Their Natural Number Matches**
**Content:**
In this educational lesson, we explore a unique way of matching rational numbers with natural numbers. This approach systematically matches each rational number with a distinct natural number without repetition.
**Instructions:**
Below is a sequence demonstrating this matching method:
\[
\begin{array}{ccccccccccc}
\frac{1}{1} & \frac{2}{1} & \frac{1}{2} & \frac{1}{3} & \frac{2}{2} & \frac{3}{1} & \frac{4}{1} & \frac{2}{3} & \frac{3}{2} & \frac{4}{3} & \frac{3}{3} & \frac{4}{2} & \frac{5}{1} & \cdots \\
\downarrow & \downarrow & \downarrow & \downarrow & \text{skip} & \downarrow & \downarrow & \downarrow & \downarrow & \downarrow & \text{skip} & \downarrow & \downarrow & \cdots \\
1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & \cdots \\
\end{array}
\]
**Explanation:**
1. The first row lists pairs of integers as fractions representing rational numbers.
2. The second row uses arrows to point to corresponding positions in the natural numbers sequence (1, 2, 3, ...).
3. The "skip" indicates that certain rational numbers, which are equivalent to earlier ones, are omitted to avoid duplication (e.g., 2/2 = 1/1, 4/2 = 2/1, etc.).
**Question:**
Using this matching method, what natural number is matched with the rational number \(\frac{5}{2}\)?
**Answer:**
The natural number that is matched with the rational number \(\frac{5}{2}\) is \(\boxed{11}\).
\( (Simplify your answer.)\)
This example demonstrates the matching process and helps in understanding the systematic way in which every unique rational number corresponds to a natural number. This type of exercise can deepen one’s understanding of the correspondence between different sets of numbers in mathematics.
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