Value of Irrational Number π (Part 2)

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Discussion Overview

The discussion revolves around the value of the irrational number π and its approximation using the fraction 22/7. Participants explore how closely 22/7 approximates π, the historical context of this approximation, and methods to compare the two values, including the possibility of doing so without a calculator.

Discussion Character

  • Exploratory, Technical explanation, Conceptual clarification, Debate/contested, Homework-related

Main Points Raised

  • One participant states that the value of π to ten decimal places is 3.1415926535 and questions how many decimal places 22/7 agrees with π.
  • Another participant suggests that comparing 22/7 with π is straightforward with a calculator or through long division, but notes that finding the best approximation with denominators less than 100 is more complex.
  • A participant expresses interest in how to compare 22/7 with π without using a calculator, prompting further discussion on methods.
  • There is a mention of Archimedes' historical contribution to the understanding of π and the introduction of 22/7 as an approximation through Boethius' writings.

Areas of Agreement / Disagreement

Participants appear to agree that comparing 22/7 with π can be done through various methods, but there is no consensus on the best approach or the feasibility of doing so without a calculator. The discussion remains unresolved regarding the best approximation with denominators less than 100.

Contextual Notes

Some participants mention the use of calculators and long division, but the specifics of how to perform these comparisons without a calculator are not fully explored. There is also a lack of clarity on the criteria for determining the "best" approximation.

mathdad
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The value of irrational number π, correct to ten decimal places (without rounding), is 3.1415926535. By using your calculator, determine to how many decimal places the following quantity (22/7) agrees with π.

Extra notes from textbook:

Archimedes (287-212 B.C.) showed that
(223/71) < π < 22/7. The use of the approximation (22/7) for π was introduced to the western world through the writings of Boethius (ca 480-520), a Roman philosopher, mathematician, and statesman. Among all fractions with numerators and denominators less than 100, the fraction (22/7) is the best appriximation to π. Do you agree?

I was wondering if this question can be answered without a calculator. Can we show that (22/7) in terms of decimal places agrees with pi?
 
Last edited:
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Comparing $22/7$ with $$\pi$$ is simple if you have a calculator or use long division. Choosing the best approximation with denominators less than 100 is trickier. I would write a program for this.
 
Last edited:
Evgeny.Makarov said:
Comparing $22/7$ with $$\pi$$ is simple if you have a calculator or use along division. Choosing the best approximation with denominators less than 100 is trickier. I would write a program for this.

How it is done without using a calculator?
 
RTCNTC said:
How it is done without using a calculator?

Evgeny.Makarov said:
use long division.
...
 
Evgeny.Makarov said:
...

The original question has been edited.
 

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