Irrational + irrational = rational

  • Context: Undergrad 
  • Thread starter Thread starter tonebone10
  • Start date Start date
  • Tags Tags
    Irrational Rational
Click For Summary

Discussion Overview

The discussion revolves around the question of whether it is possible for the sum of two irrational numbers to be rational. Participants explore various examples, counterexamples, and theoretical implications related to this concept, touching on mathematical reasoning and properties of irrational and rational numbers.

Discussion Character

  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants propose examples such as \(\frac{\pi}{4} + \frac{3\pi}{4}\) to illustrate the claim that two irrational numbers can sum to a rational number, although others challenge this by asserting that the result is still irrational.
  • Jameson suggests that the only way to achieve a rational sum is through specific cases like \(\sqrt{5} - \sqrt{5} = 0\), indicating skepticism about the general case.
  • Another participant mentions that if \(x\) is irrational, then \(y\) must also be irrational to maintain the condition of summing to a rational number.
  • One participant proposes a general case involving the difference between a rational number and an irrational number being irrational, suggesting this leads to the conclusion that the sum of two irrational numbers can be rational.
  • There are discussions about the implications of specific examples, such as \((1 - \pi) + \pi = 1\), with some participants questioning whether this truly demonstrates the general case.
  • Several participants engage in a back-and-forth about the validity of statements regarding the sum of irrational numbers, with some asserting that the sum can never be rational while others provide counterexamples.
  • One participant mentions the concept of non-repeating decimals as a way to construct examples of irrational numbers that can sum to a rational number.
  • There is a debate about the probability of randomly chosen real numbers summing to a rational number, with some asserting that the chance is negligible.

Areas of Agreement / Disagreement

Participants do not reach a consensus on whether the sum of two irrational numbers can be rational. Multiple competing views remain, with some asserting it is possible under certain conditions while others maintain it is not.

Contextual Notes

Some arguments rely on specific definitions of irrational numbers and properties of real numbers, which may not be universally accepted. The discussion also includes unresolved mathematical steps and assumptions about the nature of irrational numbers.

  • #31
Zurtex said:
If you take any 2 random real numbers, a and b, the chance of their sum being rational is 0.


What do you mean by random?
 
Physics news on Phys.org
  • #32
It is possible to have the sum of two random numbers equal a rational one, although the chances of it happening are extremely minute.

If you are choosing two random numbers, then it is possible for the random number to be any number. This means the two random numbers could be \frac{\ln6}{\ln3} and \frac{\ln1.5}{\ln3}.

As tongos pointed out, these two numbers add together to get the rational number 2. I understand that the chances of a random number being infinitely equivalent to the examples I gave are so small it wouldn't happen, but it's possible.


Jameson
 
  • #33
It's possible, certainly. The probability is 0 though.
 
  • #34
Jameson said:
It is possible to have the sum of two random numbers equal a rational one, although the chances of it happening are extremely minute.

That depends on what you (they/whomever) mean by random. When you say random I need to know what probability measure you are talking about. The real numbers have infinite support so you can't have a uniform distribution.
 
  • #35
I disagree. Probability is \frac{desired outcome}{all possible outcomes}. You are looking for 2 outcomes in the instance, out of a possible outcomes of infinity.

The \lim_{x\rightarrow\infty}\frac{2}{x} = 0, yes I agree, but using the definition of probability, there is a chance it could happen in the possible number of outcomes, therefore it has a probability that is greater than 0.
 
  • #36
When I say a random number, I mean one that can be inifinitely small to infinitely large (including number of digits) for both positive and negative numbers.
 
  • #37
Jameson said:
When I say a random number, I mean one that can be inifinitely small to infinitely large (including number of digits) for both positive and negative numbers.

That doesn't tell me what you mean by random. Are all the numbers equally likely? What is the distribution?
 
  • #38
Sorry. Yes, I am saying all numbers are equally likely... I don't know what you are asking by distribution though.
 
  • #39
Jameson said:
Sorry. Yes, I am saying all numbers are equally likely... I don't know what you are asking by distribution though.

http://mathworld.wolfram.com/DistributionFunction.html

That is not possible. When you say that the numbers are equally likely then you are saying that you are using a uniform distribution. You can’t have a uniform distribution on an unbounded set
 

Similar threads

Undergrad Questions about this video on Taylor series
  • · Replies 11 ·
Replies
11
Views
2K
Irrational Number Raised To Irrational Number
  • · Replies 30 ·
2
Replies
30
Views
3K
MHB Piecewise function - rational and irrational
  • · Replies 4 ·
Replies
4
Views
8K
Undergrad Is the Ratio of Circumference to Radius of a Circle Always Irrational?
  • · Replies 19 ·
Replies
19
Views
3K
Sum of rational and irrational is irrational
  • · Replies 6 ·
Replies
6
Views
3K
Undergrad Calc I: Limits at Infinity: Solving r Irrational
  • · Replies 4 ·
Replies
4
Views
1K
Irrational Numbers a and b used in various expressions
  • · Replies 13 ·
Replies
13
Views
2K
Undergrad Are Real Numbers Essential in Scientific Measurements and Models?
  • · Replies 85 ·
3
Replies
85
Views
9K
Graduate Spivak Thomae's Function proof explanation
  • · Replies 35 ·
2
Replies
35
Views
8K
Undergrad Basic measurement theory question
  • · Replies 5 ·
Replies
5
Views
2K