MHB Value of Irrational Number π (Part 2)

[mathdad]

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?

 

[Evgeny.Makarov]

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.

 

[mathdad]

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?

 

[Evgeny.Makarov]

How it is done without using a calculator?

use long division.

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[mathdad]

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The original question has been edited.