Real Numbers vs Extended Real Numbers

In summary, the conversation discussed the concept of extended reals, which includes +infinity and -infinity, and whether it is common to use the symbol ℝ to represent it. It was noted that while some may use ℝ as shorthand, it is not a good idea as it can cause confusion. Additionally, it was mentioned that in the reals, there is no concept of infinity or a boundary, but in the extended reals, 1/infinity is defined as 0. However, in certain situations, such as limits, 1/infinity can be considered an indeterminate form and equal to 0.
  • #1
srfriggen
306
5
Today I watched a lecture that talked about the extended reals, ℝ U {+infinity, -infinity}.

Every course I've taken so far (differential calculus through multivariable calculus, linear algebra, a proofs writing course, etc) always defined the Reals with just they symbol ℝ and we always included infinity and negative infinity.

Were all of my teachers just using ℝ as shorthand for ℝ U {+infinity, -infinity} or is there some fundamental difference?

Actually, I shouldn't ask on this forum what my teachers were doing, but rather, is it commonplace to use ℝ as shorthand for the extended reals?
 
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  • #2
It is not common and a bad ideal to use R to denote extended reals as it might be confused with the reals better to use [tex]\mathbb{\overline{R}}[/tex] or something.
 
  • #3
So do the reals, R, have infinity? How would you define the boundaries?
 
  • #4
srfriggen said:
So do the reals, R, have infinity? How would you define the boundaries?

By its very definition, the reals do not have infinity. And the reals do not have a boundary (i.e. the boundary is empty).
 
  • #5
okay thanks.

one last question: is 1/∞ an indeterminate form or does it equal zero exactly. I know the limit as n approached infinity of 1/∞ = zero, but what about the equation.

I ask because I've read posts where people say it equals exactly zero and even two professors (one in my class, the other on a youtube lecture) differed in the answer.
 
  • #6
srfriggen said:
okay thanks.

one last question: is 1/∞ an indeterminate form or does it equal zero exactly. I know the limit as n approached infinity of 1/∞ = zero, but what about the equation.

I ask because I've read posts where people say it equals exactly zero and even two professors (one in my class, the other on a youtube lecture) differed in the answer.

This depends on the situation:

- In the real numbers, there is no thing as [itex]\infty[/itex]. So [itex]1/\infty[/itex] is nonsense.

- In the extended reals, we do define [itex]1/\infty =0[/itex]. But things like 1/0 are still undefined.

- When working with limits, if we encounter a limit of the type [itex]1/\infty[/itex], then the limit is 0. For example

[tex]\lim_{x\rightarrow +\infty}{\frac{1}{x}}=0[/tex]

Also see the following FAQ: https://www.physicsforums.com/showthread.php?t=507003
 
Last edited by a moderator:

1. What is the difference between real numbers and extended real numbers?

The main difference between real numbers and extended real numbers is that real numbers represent all numbers on the number line, including both positive and negative numbers, fractions, and irrational numbers like pi and square root of 2. On the other hand, extended real numbers include all real numbers as well as two additional numbers, positive infinity and negative infinity, which represent values that are infinitely large and infinitely small, respectively.

2. How are real numbers and extended real numbers used in mathematics?

Real numbers are used in mathematics to represent quantities and to perform operations such as addition, subtraction, multiplication, and division. They are also used to measure and compare quantities. Extended real numbers, on the other hand, are used in advanced mathematics, particularly in calculus and analysis, to represent limits and to extend the range of possible values in mathematical equations.

3. Can extended real numbers be graphed on a number line?

No, extended real numbers cannot be graphed on a number line because they represent values that are infinitely large or small, which cannot be accurately represented on a finite number line. However, they can be visualized and understood using concepts such as limits and infinity in advanced mathematics.

4. How do you determine if a number is a real number or an extended real number?

A number is a real number if it can be represented on the number line, including both positive and negative numbers, fractions, and irrational numbers. On the other hand, a number is an extended real number if it includes all real numbers as well as positive infinity and negative infinity. For example, 5 is a real number, while positive infinity is an extended real number.

5. Are extended real numbers used in real-world applications?

Extended real numbers are not commonly used in real-world applications, as they represent values that are infinitely large or small and are difficult to measure or calculate in practical situations. However, they are used in advanced mathematics and theoretical physics to describe abstract concepts and to solve complex equations.

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