1/1, 21, 12, 1/3, 22, З/1, 4/1, 32, 23, 14, 1/5, 2/4, 33, 42, 5/1 1 1 1 1 skip 1 1 1 t t 1 1, 2, 3, 4, skip 5, 6, 7, 8, 9, 10, 11 Using this matching, what rational number corresponds to the natural number 15? The rational number that corresponds to the natural number 15 is (Simplify your answer.)
1/1, 21, 12, 1/3, 22, З/1, 4/1, 32, 23, 14, 1/5, 2/4, 33, 42, 5/1 1 1 1 1 skip 1 1 1 t t 1 1, 2, 3, 4, skip 5, 6, 7, 8, 9, 10, 11 Using this matching, what rational number corresponds to the natural number 15? The rational number that corresponds to the natural number 15 is (Simplify your answer.)
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.
Transcribed Image Text:The image shows a mathematical representation of rational numbers arranged in a matrix-like grid, with an illustration of a specific traversal path.
**Matrix of Rational Numbers:**
```
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 ...
...
```
Each cell in the grid represents a rational number where the numerator and denominator correspond to the row and column indices, respectively.
**Traversal Path Explanation:**
Within the grid, blue lines are drawn to indicate a specific traversal path through the matrix:
1. The path starts from the top-left cell, which is the rational number 1/1.
2. The path proceeds diagonally to the next terms: 2/1, 1/2.
3. It continues to the next set of terms: 1/3, 2/2, 3/1.
4. Then to the next set: 4/1, 3/2, 2/3, 1/4.
5. And further into: 1/5, 2/4, 3/3, 4/2, 5/1.
6. The pattern appears to proceed in a zig-zag diagonal manner, covering each rational number in the grid before moving on to the next level.
![### Matching Rational and Natural Numbers
In this exercise, we match rational numbers with natural numbers in a specific sequence to find which rational number corresponds to a given natural number. The set of rational numbers is arranged in a grid pattern, specifically:
\[
\begin{array}{ccccccccccc}
1/1, & 2/1, & 1/2, & 1/3, & 2/2, & 3/1, & 4/1, & 3/2, & 2/3, & 1/4, & 1/5, \\
2/4, & 3/3, & 4/2, & 5/1 & \ldots \\
\end{array}
\]
Below each rational number, there are corresponding natural numbers increasing sequentially from left to right. It's important to note that certain rational numbers are skipped if they are not in their simplest form (i.e., if they can be simplified further).
For example:
- \(2/2\) is skipped because it can be simplified to \(1/1\),
- \(3/3\) is also skipped because it simplifies to \(1/1\), and
- \(2/4\) is skipped since it can be simplified to \(1/2\).
To understand the mapping better, here’s the detailed sequence:
1. \(1/1 \) – 1
2. \(2/1 \) – 2
3. \(1/2 \) – 3
4. \(1/3 \) – 4
5. Skip \(2/2\)
6. \(3/1 \) – 5
7. \(4/1 \) – 6
8. \(3/2 \) – 7
9. \(2/3 \) – 8
10. \(1/4 \) – 9
11. \(1/5 \) – 10
12. Skip \(2/4\)
13. \(3/3 \) – skipped (simplifies to \(1/1\))
14. \(4/2 \) – 11
15. ...
### Exercise:
Using this matching, what rational number corresponds to the natural number 15?
The rational number that corresponds to the natural number 15 is \( \boxed{4/3} \).
(S](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd645a57e-8fec-4f58-b186-7cff590ec41c%2Fe4c56c2c-d217-4b10-903f-23006cf75970%2Fz19ye_processed.png&w=3840&q=75)
Transcribed Image Text:### Matching Rational and Natural Numbers
In this exercise, we match rational numbers with natural numbers in a specific sequence to find which rational number corresponds to a given natural number. The set of rational numbers is arranged in a grid pattern, specifically:
\[
\begin{array}{ccccccccccc}
1/1, & 2/1, & 1/2, & 1/3, & 2/2, & 3/1, & 4/1, & 3/2, & 2/3, & 1/4, & 1/5, \\
2/4, & 3/3, & 4/2, & 5/1 & \ldots \\
\end{array}
\]
Below each rational number, there are corresponding natural numbers increasing sequentially from left to right. It's important to note that certain rational numbers are skipped if they are not in their simplest form (i.e., if they can be simplified further).
For example:
- \(2/2\) is skipped because it can be simplified to \(1/1\),
- \(3/3\) is also skipped because it simplifies to \(1/1\), and
- \(2/4\) is skipped since it can be simplified to \(1/2\).
To understand the mapping better, here’s the detailed sequence:
1. \(1/1 \) – 1
2. \(2/1 \) – 2
3. \(1/2 \) – 3
4. \(1/3 \) – 4
5. Skip \(2/2\)
6. \(3/1 \) – 5
7. \(4/1 \) – 6
8. \(3/2 \) – 7
9. \(2/3 \) – 8
10. \(1/4 \) – 9
11. \(1/5 \) – 10
12. Skip \(2/4\)
13. \(3/3 \) – skipped (simplifies to \(1/1\))
14. \(4/2 \) – 11
15. ...
### Exercise:
Using this matching, what rational number corresponds to the natural number 15?
The rational number that corresponds to the natural number 15 is \( \boxed{4/3} \).
(S
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