Mathematical Awareness #1

Number Theory Level 3

$\large \color{skyblue}{\boxed{0.1234567891011...}} \quad \color{#BA33D6}{\boxed{\sqrt{1729}}} \quad \color{#20A900}{\boxed{\sqrt{2}}} \quad \color{#3D99F6}{\boxed{e} } \quad \color{#D61F06}{\boxed{\pi} }$

How many of the above numbers are transcendental numbers?

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0 1 2 3 4 5 Can't be predicted

1 solution

Rohit Ner
May 27, 2015

After going through the definition , we can conclude that from the given numbers, only $0.1234567891011...$ , $e$ and $\pi$ are transcendental numbers. Square root of any positive real may become irrational however it can be a solution of a single-variable polynomial equation with all integral coeffecients. Thus the answer is $\huge\boxed { 3 }$

@rohit How can we prove that 0.123456... is a transcendental number???

Jaka Ong - 6 years ago

@Jaka Ong , $0.1234567891011...$ is an irrational no. Moreover, it cannot be expressed as ${ n }^{ th }$ root of any positive real number and hence satisfies the condition of transcendental numbers.

Rohit Ner - 6 years ago

You didn't answer the question. How can you PROVE that it's transcendental? What you did is you simply regurgitate the definition of a transcendental number.

This constant is called Champernowne constant . The proof of it is hard to find and most likely very hard to understand.

Pi Han Goh - 6 years ago

@Pi Han Goh – So you expect me to explain the Champernowne constant for a level 3 problem? That's a bit fussy. There is no need to prove the thing. We just have to refine all the numbers through the given conditions. I hope my solution is quite clear to give the idea about my approach.

Rohit Ner - 6 years ago

@Rohit Ner – I didn't expect you to explain it. I'm just pointing out the fact that you didn't prove that the first number is transcendental. What do you mean by "We just have to refine all the numbers through the given conditions."?

Pi Han Goh - 6 years ago

@Pi Han Goh – I would like to know about your approach.

Rohit Ner - 6 years ago

@Rohit Ner – Didn't you read what I said earlier? And you didn't answer my question.

Pi Han Goh - 6 years ago

@Pi Han Goh – Here is the proof . Be happy now! :)

Rohit Ner - 6 years ago

@Rohit Ner –

There is no need to prove the thing.

You do realise the initial commenter asked you to prove that $0.123...$ is transcendental.

Siddhartha Srivastava - 6 years ago

Thank you @rohit ner

Jaka Ong - 6 years ago

Argh! I thought that the first number was the all-powerful $\dfrac{1}{(\displaystyle \sum_{n=0}^a 10^{n} \times 9)^2}$ where $a$ is a certain integer.

Sharky Kesa - 6 years ago

This question is ridiculous. Unless you are familiar with Champernowne’s Number, and the proof involved (which the author THEMSELVES claims is tedious and "fussy" for a problem of this number), you do not know that the decimal (0.1234567891011...) is transcendental. The definition of a transcendental number is simple, but unless you are familiar with that unique proof you have NO way of knowing that the number isn't the nth root of some positive real number.

Malekai Zagorski - 6 years ago

Exactly, "..." could mean that it repeats or that it doesn't repeat. If they had written "Chamernowne's Number", that would be a totally different story. It's just an attempt at making a trick question that ends up making the problem fuzzy.

Alex Li - 5 years, 11 months ago

What would be the point in writing out that it's the Champernowne constant? It's not a trick and you have no excuse on this.

Kenneth Choo - 4 years, 10 months ago

No one is asking you to prove it. I'm not sure but the proof is possibly still difficult even for a level 5 problem. All you have to do is just to know the Champernowne constant. If the problem is asking about transcendental numbers, then you have to have knowledge on them, including knowing the numbers that are proven to be transcendental. You have no excuse here.

Kenneth Choo - 4 years, 10 months ago

sorry ...but what are transcendental no. ???

Harshal Choudhary - 6 years ago

transcendental numbers are numbers that cannot be expressed in ax^n + bx^(n-1) + cx^(n-2) etcetera. So for example it is not an answer (root) of the equation 4x^3 + 7x^2 - 5x + 3, or any other equation. As I say in my comment I do not know of a way of showing that a number either is or is not transcendental. Regards, David

David Fairer - 3 years, 10 months ago

I DO NOT KNOW THE ANSWER! It is quite trivial that any square root is not transcendental. Because if x=squroot (y) (I don't know how to write this correctly) then x^2 - y = 0. So squroot 1729 and squroot 2 are not transcendental. I knew that e is irrational, though I didn't know it is transcendental before I was told in this article. I know that pi (I do not know of a way to type this either) is transcendental. So the question becomes is 0.12345678910111213.... transcendental? I do not even know if it is irrational, though I suspect that it is?!?! And I do not know of a way to show either that it is transcendental or not. So I tried to 'circle' the answer 'Can't be predicted' though I do not think that I did this correctly. Regards, David

David Fairer - 3 years, 10 months ago

Solution of a single-variable polynomial equation with all integral coefficients. @Rohit Ner

Sandeep Bhardwaj - 6 years ago

Thank you for correcting me. :)

Rohit Ner - 6 years ago

Mathematical Awareness #1

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