Defining a transcendental number and countability

In summary, the conversation discusses the concept of transcendental numbers that cannot be specified by a finite amount of data. The participants also explore the possibility of creating an uncountable set of these numbers through instructions, but conclude that it is not possible. They also mention the existence of computable numbers, which are countable, and the fact that almost every real number is not computable.
  • #1
BWV
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Is there a term for transcendental numbers that cannot be specified by an operation with a finite amount of data?

for example pi or e have various finite definitions and one could generate other transcendental numbers with operations on these.

On the other hand if n= some randomly chosen number on the real number line, there is no real way to express or differentiate that number from some other arbitrarily close transcendental number

The transcendental numbers are uncountable, but is the set of transcendental numbers that can be defined by some finite operation countable?
 
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  • #2
I wonder how you define "define". When you say that ##\pi## has a finite definition, I think that you are allowing a wide definition of "define". If that is true, just identifying a transfinite number that has no finite definition might define it. It hurts my head.
 
  • #3
FactChecker said:
I wonder how you define "define". When you say that ##\pi## has a finite definition, I think that you are allowing a wide definition of "define". If that is true, just identifying a transfinite number that has no finite definition might define it. It hurts my head.
Yes, struggling with the terminology, by define I mean uniquely identify the number - with pi or e one could (mathematically, perhaps not physically) compute any finite sequence of digits within it and differentiate it from some other transcendental number arbitrarily close to it
 
  • #4
##\pi## is the ratio of a circle's circumference to its diameter. Is that a definition?
 
  • #5
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  • #6
Office_Shredder said:
I think you're looking for something like this
https://en.m.wikipedia.org/wiki/Computable_number

A fun fact is that the computable numbers are countable, so almost every real number is not computable.
yes, that is what I was trying to ask, thanks
 
  • #7
Office_Shredder said:
I think you're looking for something like this
https://en.m.wikipedia.org/wiki/Computable_number

A fun fact is that the computable numbers are countable, so almost every real number is not computable.
Note that there are numbers that can be defined but cannot be computed. e.g Chaitin's omega.
 
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  • #8
So is if the number ‘take pi and replace decimal places 601-610 with my phone number’ is computable, you could generate an infinite amount of arbitrary real numbers with instructions like this - simply manipulating digits of another computable irrational number. But you could not create an uncountable set of these?
 
  • #9
BWV said:
So is if the number ‘take pi and replace decimal places 601-610 with my phone number’ is computable, you could generate an infinite amount of arbitrary real numbers with instructions like this - simply manipulating digits of another computable irrational number. But you could not create an uncountable set of these?
Correct. You could not. Every set of instructions that you could generate would amount to a finite string. And the set of finite strings [over a countable alphabet] is countable.
 
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1. What is a transcendental number?

A transcendental number is a real number that is not the root of any non-zero polynomial equation with integer coefficients. In other words, it is a number that cannot be expressed as a fraction or a root of a whole number.

2. How is a transcendental number different from an algebraic number?

Algebraic numbers are numbers that can be expressed as the root of a polynomial equation with integer coefficients, while transcendental numbers cannot be expressed in this way.

3. Can you give an example of a transcendental number?

One of the most famous examples of a transcendental number is pi (π). It is a non-repeating, non-terminating decimal that cannot be expressed as a fraction or the root of a whole number.

4. What is the significance of transcendental numbers in mathematics?

Transcendental numbers play a crucial role in mathematics, as they provide a bridge between the two major branches of mathematics: algebra and analysis. They also have applications in fields such as physics and engineering.

5. Is it possible to count the number of transcendental numbers?

No, it is not possible to count the number of transcendental numbers. In fact, there are infinitely many transcendental numbers between any two real numbers. This is because the set of transcendental numbers is uncountable, meaning it cannot be put into a one-to-one correspondence with the set of natural numbers.

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