Value of Irrational Number π (Part 2)

The edited version is as follows:In summary, the value of irrational number π, correct to ten decimal places (without rounding), is 3.1415926535. Archimedes showed that (223/71) < π < 22/7 and the use of the approximation (22/7) for π was introduced to the western world through the writings of Boethius. Among all fractions with numerators and denominators less than 100, the fraction (22/7) is the best approximation to π. However, it is possible to compare 22/7 with π without using a calculator by using long division. Choosing the best approximation with denominators less than 100 is trickier and may require a program to
  • #1
mathdad
1,283
1
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?
 
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  • #2
Comparing $22/7$ with \(\displaystyle \pi\) 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.
 
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  • #3
Evgeny.Makarov said:
Comparing $22/7$ with \(\displaystyle \pi\) 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?
 
  • #4
RTCNTC said:
How it is done without using a calculator?

Evgeny.Makarov said:
use long division.
...
 
  • #5
Evgeny.Makarov said:
...

The original question has been edited.
 

Question 1: What is the significance of irrational number π in mathematics?

The irrational number π, approximately equal to 3.14159, is significant in mathematics as it represents the ratio of a circle's circumference to its diameter. It is a fundamental constant that is used in many mathematical equations and formulas, making it an essential concept in geometry, trigonometry, and calculus.

Question 2: How is the value of π calculated?

The value of π is calculated by dividing the circumference of a circle by its diameter. This ratio is constant and is approximately equal to 3.14159. However, due to its irrational nature, the exact value of π cannot be expressed as a finite decimal or fraction.

Question 3: Can the value of π be expressed as a fraction?

No, the value of π cannot be expressed as a fraction. This is because it is an irrational number, meaning it cannot be written as a simple fraction of two integers. It has an infinite number of decimal places and does not follow a repeating pattern, making it impossible to express as a fraction.

Question 4: What is the significance of π in real-world applications?

The value of π has many real-world applications, particularly in fields such as engineering, physics, and architecture. It is used to calculate the circumference and area of a circle, which is essential in designing structures such as bridges, buildings, and roads. It is also used in calculating the volume of cylinders, spheres, and other curved objects.

Question 5: Are there any other ways to represent the value of π?

Yes, the value of π can also be represented using the Greek letter π or the symbol π, which was introduced by the mathematician William Jones in the early 18th century. In addition, there are various approximations and numerical methods that can be used to estimate the value of π, such as the Monte Carlo method and the Chudnovsky algorithm.

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