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Since the decimal expansion of Pi is infinite and unpredictable, the number just to its left and just to its right would have to conform to Pi’s ever changing value if we are to suppose there are no gaps between successive numbers.

If that is so, then the two numbers flanking Pi, must also be flanked by such numbers that do not allow gaps. Logically, this process must continue for all numbers to the left and to the right of Pi.

If this is the case, then no other kind of number could ever lie to the left or right of Pi without creating a gap. How does Pi 'fit' on the number line?

Jandon Gropp
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    Well couldn’t you apply this argument to any number? – ArthD21 Oct 04 '22 at 01:35
  • The number “right next to” another number is irrational, because it would keep going on forever. For example, the number “right next to” 1 is irrational. It would be 1.0000000000000000000000000000..., and it can never end in a 1-digit, because there would always be another power of ten to add another digit 0. – ArthD21 Oct 04 '22 at 01:37
  • In other words, the real number-line continuous, not discrete. But natural numbers, integers, or whole numbers, for example, are. For example, there is a NATURAL NUMBER “right next to” 3, 4 or 2. – ArthD21 Oct 04 '22 at 01:40
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    The digits of $\pi$ are perfectly predictable, in the sense that we have very simple algorithms that can compute as many of them as we want. There are numbers with much less predictable digits than this, such as Chaitin's constant, where no such algorithm exists. – Qiaochu Yuan Oct 04 '22 at 01:53
  • It's also very important to note that $\pi$'s value is not "ever-changing", it has one specific value that stays the same, it just can't be expressed exactly as a rational number. – ConMan Oct 04 '22 at 04:44
  • There is no "next" or "previous" number in the reals , neither a rational one nor an irrational one. Between two real numbers , there are always infinite many other real numbers , no matter how close they are. – Peter Oct 04 '22 at 10:15

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The problem is that you could apply this type of reasoning to any number. On the real number line, it is impossible to define exactly which number is after another, as you have shown. We could get closer and closer rational approximations of $\pi$, but never $\pi$ exactly. The answer to your question is that numbers don't really 'fit' on the real number line as it is hard to even say what numbers like $\sqrt 2, e,\pi$ etc. really are, much less define the numbers that are perfectly adjacent to them.

Sidenote: as a result, the set of all real numbers is "uncountable".

bobeyt6
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