Undefined For Different Reasons

[mathdad]

Both a/0 (a cannot be 0) and 0/0 are undefined but for different reasons. Explain the two reasons.

Let me see.

For a/0, where a cannot be 0, any number divided by zero is not possible. For example, it is not possible to divide 5 candy bars equally among 5 friends if I have no candy bars to begin with.

For 0/0, I learned long ago that the correct term is indeterminate. I have never taken calculus. I know that this idea of 0/0 is further explored in Calculus ll according to friends who have completed the Calculus l, ll, and lll sequence.

What do you say?

Thank you.

 

[R.M.]

a/0 ("a" being non-zero) means that there is a unique number (let's call it "n") that satisfies the equation n * 0 = a. Is there a non-zero real number that, when multiplied by 0, gives a non-zero result?

For 0/0, you can do the same thing. n * 0 = 0. What is the UNIQUE number "n"?

 

[fresh_42]

My favorite answer (among many) to this question is: Why do you even bother? Say we speak about rational numbers. They consist of a group for addition [imath] (\mathbb{Q},+) [/imath] with neutral element [imath] 0 [/imath] and a group for multiplication [imath] (\mathbb{Q}^*,\cdot) [/imath] with the neutral element [imath] 1 [/imath] that does not even has zero as one of its elements! No zero, no need for an inversion! It is an artificial question.

Now, you might say, that we do have [imath] 0\cdot 1 = 0[/imath] although [imath] 0\not\in \mathbb{Q}^* .[/imath] But this is no valid argument. All we have to connect the two groups is the distributive law. That's it. Nothing else.

The distributive law [imath] a\cdot(b+c)=a\cdot b+a\cdot c [/imath] makes [imath] 0\cdot 1 = 0[/imath] by the following argument:
[math]0\cdot 1 =(-1+1)\cdot 1=(-1)\cdot 1 +1\cdot 1=-1 + 1 =0[/math]
However, there is no way, no chance to use the distributive law to attach whatever meaning to [imath] \dfrac{1}{0} [/imath] without immediately running into contradictions.

One could as well ask why there isn't a (real) solution to [imath] e^x=-1. [/imath] But I never read that question.

 

[mathdad]

a/0 ("a" being non-zero) means that there is a unique number (let's call it "n") that satisfies the equation n * 0 = a. Is there a non-zero real number that, when multiplied by 0, gives a non-zero result?

For 0/0, you can do the same thing. n * 0 = 0. What is the UNIQUE number "n"?

1. There is no non-zero number that when multiplied by 0 yields a non-zero result.

2. For 0/0, n • 0 = 0. Unique number n? I assume that you mean n must = 0 to get 0 • 0 = 0 and to get the indeterminate form 0/0.

 

[fresh_42]

1. There is no non-zero number that when multiplied by 0 yields a non-zero result.

2. For 0/0, n • 0 = 0. Unique number n? I assume that you mean n must = 0 to get 0 • 0 = 0 and to get the indeterminate form 0/0.

Right. One of the other reasons. If [imath] \dfrac{0}{0}=x [/imath] then [imath] x\cdot 0 =0 [/imath] but [imath] x [/imath] isn't unique anymore. Which value should we use? We simply cannot do arithmetic anymore if we have infinitely many answers to such a simple algebraic equation.

You cannot give treats to a group of children if there are no children at all.

 

[mathdad]

Right. One of the other reasons. If [imath] \dfrac{0}{0}=x [/imath] then [imath] x\cdot 0 =0 [/imath] but [imath] x [/imath] isn't unique anymore. Which value should we use? We simply cannot do arithmetic anymore if we have infinitely many answers to such a simple algebraic equation.

You cannot give treats to a group of children if there are no children at all.

Well-stated. I concur. Real numbers requires taking a course in the subject. What course or textbook expands the concept of real numbers?

 

[fresh_42]

Real numbers requires taking a course in the subject. What course or textbook expands the concept of real numbers?

Well, that depends on the level at which you want to study them. I usually recommend OpenStax (https://openstax.org/subjects) for books (free pdf, low cost in print) that are meant to bridge the gap between different high school levels and college. It is a project of Rice University (Houston, TX).

These books are not what I would call a scientific textbook, but they are for free so it's cheap to check whether they are of help or not.

A standard textbook for calculus is https://www.amazon.de/Calculus-Michael-Spivak/dp/0914098918/

I'm not sure what to recommend if you are only interested in the number system. The real numbers as an object of abstract algebra are not as trivial as they seem. There are basically two different approaches. The classical approach is by set theory (Dedekind cuts) and a more modern approach is by a topological completion (Cauchy sequences). You see, it's not that easy. Even the Greeks had problems with non-rational numbers. They could draw a diagonal in a square of length one and they knew that its length, [imath] \sqrt{2}, [/imath] isn't a quotient anymore and called such numbers irrational (not rational). Now imagine numbers like [imath] \pi [/imath] or [imath] \mathrm{e} [/imath] or [imath] \sqrt{2}^{\sqrt{2}} . [/imath] They obviously exist, but what makes them unique so that we can speak about them and still mean the same number and not something that differs at the millionth digit? And there are literally uncountably many real numbers that don't even have a name or an algebraic expression to describe them.

 

[mathdad]

Well, that depends on the level at which you want to study them. I usually recommend OpenStax (https://openstax.org/subjects) for books (free pdf, low cost in print) that are meant to bridge the gap between different high school levels and college. It is a project of Rice University (Houston, TX).

These books are not what I would call a scientific textbook, but they are for free so it's cheap to check whether they are of help or not.

A standard textbook for calculus is https://www.amazon.de/Calculus-Michael-Spivak/dp/0914098918/

I'm not sure what to recommend if you are only interested in the number system. The real numbers as an object of abstract algebra are not as trivial as they seem. There are basically two different approaches. The classical approach is by set theory (Dedekind cuts) and a more modern approach is by a topological completion (Cauchy sequences). You see, it's not that easy. Even the Greeks had problems with non-rational numbers. They could draw a diagonal in a square of length one and they knew that its length, [imath] \sqrt{2}, [/imath] isn't a quotient anymore and called such numbers irrational (not rational). Now imagine numbers like [imath] \pi [/imath] or [imath] \mathrm{e} [/imath] or [imath] \sqrt{2}^{\sqrt{2}} . [/imath] They obviously exist, but what makes them unique so that we can speak about them and still mean the same number and not something that differs at the millionth digit? And there are literally uncountably many real numbers that don't even have a name or an algebraic expression to describe them.

1. I am not interested in a deep study of real numbers. For me, the basics is more than enough.

2. I have three math textbooks. All questions that have been posted thus far (with the exception of probability) come from Michael Sullivan's College Algebra Edition 9.

3. I also have a Precalculus textbook by Michael Sullivan and a Calculus textbook by the late James Stewart. These two books are for further studies.

 

[fresh_42]

1. I am not interested in a deep study of real numbers. For me, the basics is more than enough.

2. I have three math textbooks. All questions that have been posted thus far (with the exception of probability) come from Michael Sullivan's College Algebra Edition 9.

3. I also have a Precalculus textbook by Michael Sullivan and a Calculus textbook by the late James Stewart. These two books are for further studies.

Skipping all technical details, it is quite simple. Consider sequences of rational numbers, e.g. [imath] 1,\dfrac{1}{2},\dfrac{1}{4},\dfrac{1}{8},\dfrac{1}{16},\ldots [/imath] such that any two sequence numbers get closer and closer to each other if you only get far enough in the sequence, then the sequence converges to a specific number, [imath] 0 [/imath] in my example, and then add these numbers to the rationals. That's it, up to technical nuances like uniqueness or different sequences that converge to the same number.

E.g. if we set [imath] a_1=b_1=1 [/imath] and [imath] a_{n+1}=a_n+b_n\, , \,b_{n+1}=a_{n+1}+a_n [/imath] then the sequence of its quotients [imath] \dfrac{b_n}{a_n} [/imath] tends to [imath] \sqrt{2}. [/imath] The quotients are all rational numbers, their limit is not, so we add it to the rational numbers. That's the idea. If we do this for any such sequences (with this closer and closer condition), we arrive at the real numbers.

Real numbers have some important properties:

a) Regardless of how many ones we add, the sum will never be zero (other than your light switch where on + on = off, i.e. [imath] 1+1=0 [/imath]).
b) All sequences with the closer and closer condition converge, i.e. have a limit (per construction).
c) They are totally ordered, i.e. for any two numbers [imath] a,b [/imath] there always holds one of the cases: [imath] a=b\, , \,a<b [/imath] or [imath] a>b. [/imath]
d) For any two numbers [imath] a,b [/imath] such that [imath] n\cdot a\leq b [/imath] for all natural numbers [imath] n [/imath] it follows that [imath] a\leq 0. [/imath]
e) Squares are always positive, except for zero.

 

[mathdad]

Skipping all technical details, it is quite simple. Consider sequences of rational numbers, e.g. [imath] 1,\dfrac{1}{2},\dfrac{1}{4},\dfrac{1}{8},\dfrac{1}{16},\ldots [/imath] such that any two sequence numbers get closer and closer to each other if you only get far enough in the sequence, then the sequence converges to a specific number, [imath] 0 [/imath] in my example, and then add these numbers to the rationals. That's it, up to technical nuances like uniqueness or different sequences that converge to the same number.

E.g. if we set [imath] a_1=b_1=1 [/imath] and [imath] a_{n+1}=a_n+b_n\, , \,b_{n+1}=a_{n+1}+a_n [/imath] then the sequence of its quotients [imath] \dfrac{b_n}{a_n} [/imath] tends to [imath] \sqrt{2}. [/imath] The quotients are all rational numbers, their limit is not, so we add it to the rational numbers. That's the idea. If we do this for any such sequences (with this closer and closer condition), we arrive at the real numbers.

Real numbers have some important properties:

a) Regardless of how many ones we add, the sum will never be zero (other than your light switch where on + on = off, i.e. [imath] 1+1=0 [/imath]).
b) All sequences with the closer and closer condition converge, i.e. have a limit (per construction).
c) They are totally ordered, i.e. for any two numbers [imath] a,b [/imath] there always holds one of the cases: [imath] a=b\, , \,a<b [/imath] or [imath] a>b. [/imath]
d) For any two numbers [imath] a,b [/imath] such that [imath] n\cdot a\leq b [/imath] for all natural numbers [imath] n [/imath] it follows that [imath] a\leq 0. [/imath]
e) Squares are always positive, except for zero.

Very informative.