6 ms·

I want to know if there are more decimal numbers between 0 and 1 than there are integers between 0 and infinity.

warent 4y ago
My understanding is that this is true because there are infinite decimals between every decimal, infinitely. For example, there is infinity between 0.1 and 0.2, and infinity between 0.1 and 0.11, etc. i.e. infinite sets of infinity rather than one set of infinity. In the end it's all infinity, but their sets have higher cardinality described in Aleph terms ... (or something) https://en.m.wikipedia.org/wiki/Aleph_number https://en.m.wikipedia.org/wiki/Aleph_number
- daef 4y ago
  You can uniquely map all rationals onto the natural numbers, thus they are of the same quantity. That doesn't work for all real numbers thou.
  - warent 4y ago
    Oh right this is only true for irrational and transcendental numbers
  - aimor 4y ago
    Maybe this is misguided cheat, but couldn't you map any real number (between 0 and 1) to a natural number by mirroring the decimal digits across the decimal point. So 0.123 -> 321, but also sqrt(2)/2 -> ?601707 where ? is the rest of the decimal representation. This creates infinitely large numbers, but it's still a 1-to-1 mapping.
    - dangond 4y ago
      Unfortunately, numbers with infinitely many digits are not natural numbers. You cannot count to ?601707, even with an infinite amount of time.
- rcme 4y ago
  It’s not because there are infinite decimals between every two decimal numbers. That applies to the rational numbers too, e.g. there are infinite rational numbers between 1/2 and 3/4. Rather, the real numbers are more dense in a way that makes them fundamentally larger than the integers / rational numbers. “Larger” means not being able to pair up the two sets one by one so that each element of both sets is the member of a pair. No matter how you pair up the integers to the reals, you can prove that some real numbers will be unpaired.
contravariant 4y ago
Depends what you mean by decimal. Decimal is a system of notation, does it count as a decimal number if it cannot be written in decimal notation (in finite time)? if not then they are equal, if yes then there are more decimal numbers between 0 and 1 than integers.
- yamtaddle 4y ago
  I don't usually use that term, but I take it to mean "number you (may, and, if not using e.g. fractions, must) write using a decimal point" because that seems to always be what people intend by it. Everybody experienced writing irrational numbers using decimal notation in school, so those definitely count.
  - xigoi 4y ago
    The correct mathematical term is “real number”.
    - yamtaddle 4y ago
      Yes, but you'll see "decimal" more in the wild, and that's what people mean by it. "You write it with a decimal point", and they do usually mean to include the irrationals. So, yes, real numbers, but the reasoning behind their usage is "you write it with a decimal point". I'd bet more people understand "decimal number" used in that sense, than understand "real number".
      - bombolo 4y ago
        i've never seen an irrational number written in digits in my whole life. Have you????? I've seen them expressed as letters or formule
        yamtaddle 4y ago
        Never. Never in school? It's totally normal in the US, it's the reason as many people know that π starts "3.14" as do.
daef 4y ago
How do you define 'decimal numbers'? Do you only count rationals, or all real numbers?
pxx 4y ago
Assuming sqrt(2)/2 is one of the elements of your set between 0 and 1, there are! See https://en.m.wikipedia.org/wiki/Countable_set https://en.m.wikipedia.org/wiki/Countable_set
- jameshart 4y ago
  Sqrt(2)/2 can not be written as a decimal number. A decimal number is a rational whose denominator is an integer power of 10.
  - zorgmonkey 4y ago
    No, any rational number with a denominator that has only the prime factors of 2 and 5 will have a finite and exact decimal representation in base 10.
    - jameshart 4y ago
      That's exactly the same thing as I said, using different words. And anyway, is Sqrt(2)/2 such a number?
      - davrosthedalek 4y ago
        That's not quite what you have said. 3/2 does not have an integer power of 10 as the denominator, but can be written as a decimal. Of course, it can also be written as a fraction with a integer power, for example 15/10. (You are of course right on the 1/sqrt(2) issue)
eimrine 4y ago
I have an opinion that number of decimal numbers between 0 and e is equal to number of decimal numbers between e and +Infinity, because a parabola with a=e will grow in x with same speed as in y.
- quantified 4y ago
  Turn it into a proof? I need to revisit Cantor's proof, the argument I was taught left out key aspects of numbers, particularly how "number" and "string of digits you've printed out so far" aren't the same thing. It's really about creating a space-filling curve.
- dangond 4y ago
  The amount of real numbers between any two distinct real numbers (a,b) is the same as the amount of all real numbers. This is true for (0,e), (0,1), and any other combination.
tobiasSoftware 4y ago
There are, and it turns out that this is a significant mathematical concept. The integers between 0 and infinity are defined as "countably infinite". Other infinities are considered countably infinite, or the "same" infinity, if and only if you can arrange it in a list such that each item in the list pairs to an integer in our 0 to infinity list. So the set of even numbers is countably infinite because for every i that is an even number, it pairs with the number i/2. To demonstrate: 0 -> 0, 2 -> 1, 4 -> 2, 6 -> 3, ... The decimal (real) numbers between 0 and 1 are not countably infinite, and we know this from a concept called Cantor diagonalization. What Cantor did was a proof by contradiction: assume that the numbers are countably infinite, then you can arrange them in a list. However, he then builds a number by altering the first decimal place of the first number, the second decimal place of the second number, and so on. Finally, he shows that this built number is both a real number and is not on the list. Therefore, the real numbers between 0 and 1 cannot be ordered into a list, therefore they are not countably infinite, and there are more decimal numbers between 0 and 1 than integers between 0 and infinity.
- bombolo 4y ago
  does decimal numbers mean fractions or real numbers? If it means fractions only, they are countable.
- denton-scratch 4y ago
  > The decimal (real) numbers between 0 and 1 The way I parse "decimal number" in this context is a number expressible as a (finite?) string of decimal numerals. Those numbers are not reals, they are rationals.
- shidoshi 4y ago
  This account discrete maths. Bravo!
- dbtc 4y ago
  "Countably infinite" makes zero sense to me. Whatever method you use to generate your decimals, you can just slap an integer on each step of the way. You'll never run out of integers. I'll put Cantor and his proof in a box, tell him to give me his fancy decimals quick as he can, and I can match each one with an integer no problem. And pairing one infinite list with another infinite list doesn't make either one any more countable, because however high you count, they keep on going.
deleted 4y ago
[deleted]
ftxbro 4y ago
If you thought of this question from no real math training then that's pretty interesting. You should have been a mathematician. Your question is one of the most important and concisely stated questions about infinity that you can ask!
- Eisenstein 4y ago
  I was thinking if Pi never repeats itself and infinity of integers can only go up then it seems to make sense that there are more decimals between any two numbers than infinite integers. I can't describe the thought process behind it just seems intuitive.
School-Cotton 4y ago
If you consider non-repeating, non-terminal decimals (like say pi/4) then yes. Otherwise no.
jlokier 4y ago
The correct answer is: - No, even if you include all the decimals which can be individually described in any notation whatsoever. You can disregard all the arguments in the other comments about whether "decimals" includes fractions like 1/7 using decimal repeat notation, or irrationals described by a formula like sqrt(2), or transcendentals from mathematical definitions like pi and e. Those are interesting and deep rabbit holes, but they don't change the answer to your question, because it is still "no" with all of those. Even with all possible definitions which can be written in any symbolic language. This is because the set of all possible definitions which can be written can be enumerated systematically in a list, and mapped 1:1 to all the integers. - Yes, if you include all the other numbers in the range 0 to 1 which are not ones you can individually describe. Most real numbers in the range 0 to1 are actually these type of "individually undescribables". But I can't point out an individual one, of course. The "real numbers" contain these. They are present due to a consequence of logic that keeps regular math simpler and more consistent than it would be otherwise. (Aside: The question of whether 1/3 = 0.3(repeating), times 3 = 0.9(repeating), is equal to 1 is an example of choosing the simpler and more consistent logic. Of course 1/3 times 3 is 1 so 0.9(repeating) must be defined as equal to 1, or fractions wouldn't be consistent with decimals...) But the rationals (fractions), algebraic numbers (solutions to polynomials with integer coefficients, such as square roots), computables (numbers you can define by any algorithm), and even some types of uncomputables (such as all Chaitin's constants for all enumeration rules), and all mathematically individually definable transcendental numbers like pi/4 and e/3 do not contain these. It follows that all the "individually undescribables" in the real numbers can only, conceptually, be imagined as infinitely long decimals with no repeats and no pattern to the digits definable by a finite-length rule in any language. You obviously can't write one of them down, you can only conceptualise what one already written down might look like. (For example a spiral of digits of ever descreasing size would fit one in finite area.) And we can only reason about them as a set by logical construction. If you were to pick a random real number uniformly (ie. fairly) from the range 0 to 1 by picking a sequence of random decimal digits, it would ɓe one of these infinitely long decimals with probability 1. Because simple random values from a continuous range are like this, perhaps this explains why they are actually a natural and not unreasonable concept. So the answer depends on whether your meaning of "decimal numbers" means the "real numbers" in the continuous range 0 to 1, or just certain ways of writing numbers. From the other comments, evidently some people include all sort of things in their idea of "decimals" including 1/3 = 0.333...(repeating) and the exact value of pi/4 for example, not just finite strings of digits. While other people think of "decimals" as being only strings of digits you can write down, so they would not include the exact value of pi/4 for example. These two meanings of "decimal numbers" give different answers to your question.
paulddraper 4y ago
There are more reals between 0 and 1 than integers. There are not more numbers with terminating decimals between 0 and 1 than integers.

My understanding is that this is true because there are infinite decimals between every decimal, infinitely. For example, there is infinity between 0.1 and 0.2, and infinity between 0.1 and 0.11, etc. i.e. infinite sets of infinity rather than one set of infinity. In the end it's all infinity, but their sets have higher cardinality described in Aleph terms ... (or something) https://en.m.wikipedia.org/wiki/Aleph_number https://en.m.wikipedia.org/wiki/Aleph_number

daef 4y ago
You can uniquely map all rationals onto the natural numbers, thus they are of the same quantity. That doesn't work for all real numbers thou.
- warent 4y ago
  Oh right this is only true for irrational and transcendental numbers
- aimor 4y ago
  Maybe this is misguided cheat, but couldn't you map any real number (between 0 and 1) to a natural number by mirroring the decimal digits across the decimal point. So 0.123 -> 321, but also sqrt(2)/2 -> ?601707 where ? is the rest of the decimal representation. This creates infinitely large numbers, but it's still a 1-to-1 mapping.
  - dangond 4y ago
    Unfortunately, numbers with infinitely many digits are not natural numbers. You cannot count to ?601707, even with an infinite amount of time.
rcme 4y ago
It’s not because there are infinite decimals between every two decimal numbers. That applies to the rational numbers too, e.g. there are infinite rational numbers between 1/2 and 3/4. Rather, the real numbers are more dense in a way that makes them fundamentally larger than the integers / rational numbers. “Larger” means not being able to pair up the two sets one by one so that each element of both sets is the member of a pair. No matter how you pair up the integers to the reals, you can prove that some real numbers will be unpaired.

contravariant 4y ago

Depends what you mean by decimal. Decimal is a system of notation, does it count as a decimal number if it cannot be written in decimal notation (in finite time)? if not then they are equal, if yes then there are more decimal numbers between 0 and 1 than integers.

yamtaddle 4y ago
I don't usually use that term, but I take it to mean "number you (may, and, if not using e.g. fractions, must) write using a decimal point" because that seems to always be what people intend by it. Everybody experienced writing irrational numbers using decimal notation in school, so those definitely count.
- xigoi 4y ago
  The correct mathematical term is “real number”.
  - yamtaddle 4y ago
    Yes, but you'll see "decimal" more in the wild, and that's what people mean by it. "You write it with a decimal point", and they do usually mean to include the irrationals. So, yes, real numbers, but the reasoning behind their usage is "you write it with a decimal point". I'd bet more people understand "decimal number" used in that sense, than understand "real number".
    - bombolo 4y ago
      i've never seen an irrational number written in digits in my whole life. Have you????? I've seen them expressed as letters or formule
      - yamtaddle 4y ago
        Never. Never in school? It's totally normal in the US, it's the reason as many people know that π starts "3.14" as do.

daef 4y ago

How do you define 'decimal numbers'? Do you only count rationals, or all real numbers?

pxx 4y ago

Assuming sqrt(2)/2 is one of the elements of your set between 0 and 1, there are! See https://en.m.wikipedia.org/wiki/Countable_set https://en.m.wikipedia.org/wiki/Countable_set

jameshart 4y ago
Sqrt(2)/2 can not be written as a decimal number. A decimal number is a rational whose denominator is an integer power of 10.
- zorgmonkey 4y ago
  No, any rational number with a denominator that has only the prime factors of 2 and 5 will have a finite and exact decimal representation in base 10.
  - jameshart 4y ago
    That's exactly the same thing as I said, using different words. And anyway, is Sqrt(2)/2 such a number?
    - davrosthedalek 4y ago
      That's not quite what you have said. 3/2 does not have an integer power of 10 as the denominator, but can be written as a decimal. Of course, it can also be written as a fraction with a integer power, for example 15/10. (You are of course right on the 1/sqrt(2) issue)

eimrine 4y ago

I have an opinion that number of decimal numbers between 0 and e is equal to number of decimal numbers between e and +Infinity, because a parabola with a=e will grow in x with same speed as in y.

quantified 4y ago
Turn it into a proof? I need to revisit Cantor's proof, the argument I was taught left out key aspects of numbers, particularly how "number" and "string of digits you've printed out so far" aren't the same thing. It's really about creating a space-filling curve.
dangond 4y ago
The amount of real numbers between any two distinct real numbers (a,b) is the same as the amount of all real numbers. This is true for (0,e), (0,1), and any other combination.

tobiasSoftware 4y ago

There are, and it turns out that this is a significant mathematical concept. The integers between 0 and infinity are defined as "countably infinite". Other infinities are considered countably infinite, or the "same" infinity, if and only if you can arrange it in a list such that each item in the list pairs to an integer in our 0 to infinity list. So the set of even numbers is countably infinite because for every i that is an even number, it pairs with the number i/2. To demonstrate: 0 -> 0, 2 -> 1, 4 -> 2, 6 -> 3, ... The decimal (real) numbers between 0 and 1 are not countably infinite, and we know this from a concept called Cantor diagonalization. What Cantor did was a proof by contradiction: assume that the numbers are countably infinite, then you can arrange them in a list. However, he then builds a number by altering the first decimal place of the first number, the second decimal place of the second number, and so on. Finally, he shows that this built number is both a real number and is not on the list. Therefore, the real numbers between 0 and 1 cannot be ordered into a list, therefore they are not countably infinite, and there are more decimal numbers between 0 and 1 than integers between 0 and infinity.

bombolo 4y ago
does decimal numbers mean fractions or real numbers? If it means fractions only, they are countable.
denton-scratch 4y ago
> The decimal (real) numbers between 0 and 1 The way I parse "decimal number" in this context is a number expressible as a (finite?) string of decimal numerals. Those numbers are not reals, they are rationals.
shidoshi 4y ago
This account discrete maths. Bravo!
dbtc 4y ago
"Countably infinite" makes zero sense to me. Whatever method you use to generate your decimals, you can just slap an integer on each step of the way. You'll never run out of integers. I'll put Cantor and his proof in a box, tell him to give me his fancy decimals quick as he can, and I can match each one with an integer no problem. And pairing one infinite list with another infinite list doesn't make either one any more countable, because however high you count, they keep on going.

deleted 4y ago

[deleted]

ftxbro 4y ago

If you thought of this question from no real math training then that's pretty interesting. You should have been a mathematician. Your question is one of the most important and concisely stated questions about infinity that you can ask!

Eisenstein 4y ago
I was thinking if Pi never repeats itself and infinity of integers can only go up then it seems to make sense that there are more decimals between any two numbers than infinite integers. I can't describe the thought process behind it just seems intuitive.

School-Cotton 4y ago

If you consider non-repeating, non-terminal decimals (like say pi/4) then yes. Otherwise no.

jlokier 4y ago

The correct answer is: - No, even if you include all the decimals which can be individually described in any notation whatsoever. You can disregard all the arguments in the other comments about whether "decimals" includes fractions like 1/7 using decimal repeat notation, or irrationals described by a formula like sqrt(2), or transcendentals from mathematical definitions like pi and e. Those are interesting and deep rabbit holes, but they don't change the answer to your question, because it is still "no" with all of those. Even with all possible definitions which can be written in any symbolic language. This is because the set of all possible definitions which can be written can be enumerated systematically in a list, and mapped 1:1 to all the integers. - Yes, if you include all the other numbers in the range 0 to 1 which are not ones you can individually describe. Most real numbers in the range 0 to1 are actually these type of "individually undescribables". But I can't point out an individual one, of course. The "real numbers" contain these. They are present due to a consequence of logic that keeps regular math simpler and more consistent than it would be otherwise. (Aside: The question of whether 1/3 = 0.3(repeating), times 3 = 0.9(repeating), is equal to 1 is an example of choosing the simpler and more consistent logic. Of course 1/3 times 3 is 1 so 0.9(repeating) must be defined as equal to 1, or fractions wouldn't be consistent with decimals...) But the rationals (fractions), algebraic numbers (solutions to polynomials with integer coefficients, such as square roots), computables (numbers you can define by any algorithm), and even some types of uncomputables (such as all Chaitin's constants for all enumeration rules), and all mathematically individually definable transcendental numbers like pi/4 and e/3 do not contain these. It follows that all the "individually undescribables" in the real numbers can only, conceptually, be imagined as infinitely long decimals with no repeats and no pattern to the digits definable by a finite-length rule in any language. You obviously can't write one of them down, you can only conceptualise what one already written down might look like. (For example a spiral of digits of ever descreasing size would fit one in finite area.) And we can only reason about them as a set by logical construction. If you were to pick a random real number uniformly (ie. fairly) from the range 0 to 1 by picking a sequence of random decimal digits, it would ɓe one of these infinitely long decimals with probability 1. Because simple random values from a continuous range are like this, perhaps this explains why they are actually a natural and not unreasonable concept. So the answer depends on whether your meaning of "decimal numbers" means the "real numbers" in the continuous range 0 to 1, or just certain ways of writing numbers. From the other comments, evidently some people include all sort of things in their idea of "decimals" including 1/3 = 0.333...(repeating) and the exact value of pi/4 for example, not just finite strings of digits. While other people think of "decimals" as being only strings of digits you can write down, so they would not include the exact value of pi/4 for example. These two meanings of "decimal numbers" give different answers to your question.

paulddraper 4y ago

There are more reals between 0 and 1 than integers. There are not more numbers with terminating decimals between 0 and 1 than integers.