Irrational number approximation by a rational number

In summary, there is a theorem that states that for a given irrational number, there is always a rational number that can approximate it to a desired number of decimal places. This can be achieved by writing out the decimal expansion of the irrational number to the required number of digits and rounding it off. This may result in larger denominators, but it is the most common method used. Alternatively, the command "Rationalize" in Mathematica uses continued fractions to find the best rational approximation. For multiple irrational numbers, the LLL algorithm can be used to efficiently find a relation between them using rational numbers.
  • #1
n.karthick
245
0
Is there a way ( a theorem ) to find a rational number for a given irrational number such that it is an approximation to it to the required decimal places of accuracy. For example 22/7 is an approximate for pi for 2 decimal places.
 
Physics news on Phys.org
  • #2
best rational approximation (via continued fractions)
 
  • #3
There is always the trivial solution. Write out the decimal expansion for the irrational to the required number (plus one) of places. Then truncate (or round). This rational number will satisfy the criterion.
 
  • #4
^That tends to give big denominators, how distasteful.
 
  • #5
lurflurf said:
^That tends to give big denominators, how distasteful.

Most of the time you can't do much else.
 
  • #6
  • #7
Write down the decimal expansion to the accurate number of digits required+1 and make the penultimate digit to round figure and then convert it to fraction by dividing it by 10 to power number of decimal digits.
 

1. What is an irrational number?

An irrational number is a number that cannot be expressed as a ratio of two integers. This means that it cannot be written in the form of a fraction. Common examples of irrational numbers include pi (π) and the square root of 2 (√2).

2. Why is it important to approximate irrational numbers?

Approximating irrational numbers allows us to better understand and work with these numbers in practical applications. It also helps us to represent these numbers in a more manageable and understandable form.

3. How do you approximate an irrational number by a rational number?

One method to approximate an irrational number by a rational number is through decimal approximations. For example, to approximate pi (π), we can use the fraction 22/7 or the decimal approximation 3.14. Another method is through continued fractions, which involves repeatedly taking the integer part of the irrational number and using the remaining fractional part to create a new fraction.

4. What are the limitations of rational number approximation for irrational numbers?

Rational number approximation can only provide an estimate of the irrational number, and it may not always be an accurate representation. This is because irrational numbers have infinitely long and non-repeating decimal expansions, making it impossible to represent them exactly as a rational number.

5. Are there any real-life applications for irrational number approximation?

Yes, irrational number approximation has many practical applications. For example, it is used in engineering and construction to make precise measurements and calculations. It is also used in computer algorithms and programming to improve the accuracy of numerical calculations.

Similar threads

  • Linear and Abstract Algebra
Replies
33
Views
3K
  • Precalculus Mathematics Homework Help
Replies
30
Views
2K
  • General Math
Replies
5
Views
1K
  • Precalculus Mathematics Homework Help
Replies
8
Views
888
  • Linear and Abstract Algebra
Replies
6
Views
963
  • Classical Physics
3
Replies
85
Views
4K
  • Precalculus Mathematics Homework Help
Replies
13
Views
1K
  • Set Theory, Logic, Probability, Statistics
Replies
25
Views
2K
  • Precalculus Mathematics Homework Help
Replies
2
Views
2K
  • Precalculus Mathematics Homework Help
Replies
1
Views
983
Back
Top