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Help: Understanding infinity

Hello, i have a few questions regarding infinity, which i'd like to discuss with you all.

  • Is \( \frac{1}{0} = \infty \)
  • Is \( \frac{1}{\infty} = 0 \)
  • Is \( \tan90^\circ = \pm\infty \)
  • Is \( \log_x0 = -\infty \)

All my teachers tell me that the above expressions are undefined, and so is \(\infty \) (undefined), and we cannot compare the two.

I understand that we cannot determine the exact value of the above expressions, but what is the problem of terming them as \(\infty \)?

On graphing these, we do notice that they approach infinity, so why say that they are undefined? You may say that infinity is a concept, and not an actual number, but then isn't everything a concept?

If the number 1 is defined, then what is its definition? 1? (circular proof) Every number is a concept, and I don't see how certain operations on them cannot lead to infinity, (another concept).

Note by Harshit Kapur
4 years, 11 months ago

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Harshit,

There are numerous questions that you raise in this single discussion. In general, when you're asking for clarification, keeping things to a single question / query (and opening several discussions) could help ensure that you get proper replies to all of them.

I can classify the numerous questions that you raised in the following 4 broad categories:

1) What is infinity? Can we compare infinities? Can we do any arithmetic with infinities? What do those equations mean, if anything?

2) What concepts can be represented by another concept? For example, can the concept of 1 be represented by the number line? Can the concept of infinity be represented by the number line? Can the concept of pigeonhole principle be represented by the number line?

3) What is the definition of 1? How about 2, 3? How about 0? Is there a way to define it that isn't circular?

4) Why are some operations defined on some numbers, but not on others?

For the purposes of this discussion, let's keep it mainly to 1. If you're interested in the others, please open up another discussion topic for each of them.

Calvin Lin Staff - 4 years, 11 months ago

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1) I will (partially) answer the first part. The is significant theory involved, that I'm not going into detail about. If you are interested, you can read up on set theory.

The concept of infinity is best expressed using set theory. Without having to assume the existence of 0 or 1, we know that there is the empty set \( \emptyset\), Furthermore, we can build more sets by taking the union of their elements, or taking the set of the set. We know that there is the set of the empty set \( \{ \emptyset \} \). We know that there is the set of the union of them empty set and the previous set. \( \{ \emptyset, \{ \emptyset \} \} \). So on and so forth.

We say that 2 representations are equal, if there is a bijection between them. For example, \( \{ \emptyset \} \) and \( \{ \{ \emptyset \} \} \)are equivalent. (Prove the following statement using the transitive property of bijections.) These representations form an equivalence relation, which means that it doesn't matter which choice of representations we use.

This allows us to build up a concept of counting. We can let the a set of \(n\) elements represent the integer \(n\). In the first paragraph, we gave representations of 0, 1 and 2. Verify for yourself that taking the union of 2 sets is equal to taking the sum of the integers that they represent. What does taking the set of a set do to the integer? I.e. What is \( \{ (any set) \} \) equal to?

Any set that can be built up in this way (taking unions, and sets), is called a finite set. Any (nonnegative) integer that is represented in this way is finite.

An infinite set is a set that is not finite. An infinite number is a number represented by an infinite set. It is not obvious that an infinite set exists, since it is defined "not something".

You should show the following:

  1. The set of natural numbers is an infinite set. Hint: To show this, show that it is not equivalent to any finite set. Hence, it is "not finite set".

  2. The set of natural numbers is equivalent to the set of rational numbers

  3. The set of natural numbers is not equivalent to the set of real numbers.

Calvin Lin Staff - 4 years, 10 months ago

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how to solve dis problem? title" solving a second-order IVP" Y"-3y'+2y=e^-4t ;y(0)=1 ; y'(0)=5?

Ayes Bañanola - 5 days, 11 hours ago

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need to answer to correct.

Ayes Bañanola - 5 days, 11 hours ago

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Water at 100°c is taken off the stove . In 5 minutes off the burner the temperature decreased to 85°c Given that the room temperature is 25°c and newtons law of cooling is followed find the temperature of the water in degrees Celsius after an additional 5 minutes

Else Amofah - 1 week, 3 days ago

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How can direction of motion affect time ( i mean some particle is in future or past light cone ,something like that) If a clock is stationary and other is moving towards it and continues to pass by ,the stationery clock perceives the time differently ,plz explain ,as am getting same results in every situation

Vivek Bhal - 1 year, 5 months ago

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square ABCD divided into 18 smaller square. 17 squares of the sides of length 1. ABCD square area Find.

Mahla Salarmohammadi - 1 year, 7 months ago

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I have a doubt

0^n=0 Where n is any real number n^0=1

Where n is any real number

Then what is 0^0 ?
is it 0 or 1?

Bharath Raj - 1 year, 10 months ago

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There's actually a wiki on it: What is \(0^0\).You can check that out.

Abdur Rehman Zahid - 1 year, 10 months ago

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I have a comment on one over zero: \(\frac{1}{0}=\infty\) Multiply both sides by zero to remove the fraction bit. You get 1=0

Daniel Leong - 4 years, 3 months ago

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We can show that any number is equal to any other number by setting \(\frac{1}{0}=\infty\)

Abdur Rehman Zahid - 3 years ago

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hi we dont no wht infinity is so we say that till now it is not defined. whn we divide any no. by very small no. surely quotient will be>>> tht means infinity\not defined.is answer

Piyush Kumar Rai - 4 years, 11 months ago

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IT IS UNMEASURABLE

Narender Palugula - 1 year, 8 months ago

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Integration of [e^(ax).ln(x)]

Abhishek Mehta - 1 year, 9 months ago

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Post this as a note. It will reach to more people that way.

Abdur Rehman Zahid - 1 year, 9 months ago

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What is the probability that exactly one player gets all four aces if a randomly shuffled deck is dealt to four players?

Vivek S - 1 year, 9 months ago

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