Decimal Infinity
Calculus
Level
3
Is it possible to make a list of all the decimals?
Yes
No
It is impossible to tell
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1 solution
Cantor's diagonal proof: Imagine we have every decimal, we put them in rows. We take the first digit of the first number, second digit of the second number, and so on. We then make a new number using all the numbers we have collected with the 1st number's digit being the first digit of the new number, the 2nd number's digit being the second and so on. If any of these digits is a 2, we change it to a 3 and we change all the other digits to a 2. This new number is different to all the other decimals as the first digit is different to the first number's first digit, the second digit is different to the second number's second digit and so on. Therefore, we can never list every decimal.
Here is an identical idea but with a slightly different way to change the digits.