I will begin this with a disclaimer that I barely understand the p-adic analysis beyond very general ideas of what the p-adic numbers are and how they are constructed. But this is a question I've been pondering for a while now; is there something special about the real and complex numbers that make them a natural first attempt at creating algebraic closures of integers? Might an alternate chain of human development stumble upon a p-adic field prior to the reals? If this is possible how significant would this be for scientific development? Are the p-adic numbers in anyway better for approaching question in specific disciplines?

  • $\begingroup$ Comments are not for extended discussion; this conversation has been moved to chat. $\endgroup$
    – HDE 226868
    Commented Apr 9, 2021 at 16:01
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    $\begingroup$ This and this might be helpful. I also saw the phrase "adelic physics" when I searched "p-adic numbers and science." $\endgroup$ Commented Apr 9, 2021 at 23:16
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    $\begingroup$ Here is some more on that. $\endgroup$ Commented Apr 9, 2021 at 23:19
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    $\begingroup$ wow. I did not understand the question, so I went a-surfing and studied up on this field.. and.... Now I do no undertand the question, and I have a headache. $\endgroup$
    – PcMan
    Commented Apr 10, 2021 at 19:41

4 Answers 4


The difference between real and $p$-adic numbers is in how they measure size

In the reals, a number becomes big when you add it to itself many times. For instance,

$$1 + 1 + 1 + \cdots (\text{many times}) \cdots + 1 = \text{big number}$$

For $p$-adics, a number never grows beyond a fixed size no matter how many times you add it to itself. More importantly, it becomes small if $p$ divides it a lot.

For instance (taking $p = 2$), let's call:

an even number something like $10$,

a "very even number" something like $24$ (i.e. divisible by $8$ rather than just $2$)

an "extremely even number" something like $1024$ (divisible by $2^{10}$),

and so on.

The more even a number is, the smaller it is. So $1024$ is very small. But $1025 = 1024 + 1$ is not that small at all; it has the same size at $1$.

So what?

Why might a $p$-adic number system be invented? Well, it pinpoints numbers that are highly divisible by $p$.

So if you had some alien culture with, say, a religious obsession with a particular prime $p = 7$, then they might have a strong motivation to devise a way to systematically keep track of which integers are divisible by lots and lots of $7$s. This is what the $p$-adic integers do.


It is quite possible - after all, we (for certain definition of "we") are using other fields. It happens $\mathbb{R}$ seems most intuitive, simple and "natural" to a layman (though it isn't, in fact), so that it is the first stop (as something that has a direct interpretation in geometry). The second stop is $\mathbb{C}$, less intuitive, but still somewhat explainable without heavy math, and very useful in physics. (Then there are vectors and matrices, but usually we do not call them "numbers").

What is however very feasible and somewhat surprising it did not happen on Earth is a widespread adoption of hyperreals. They also form a closed field, have almost all the properties of reals, tackle infinitesimals and infinities in a somewhat intuitive way, and make grasping calculus much easier.

  • $\begingroup$ Interesting. On reflection, I do agree that it's surprising that a hyperreal field wasn't more widely accepted. Perhaps the sheer amount of work to make the hyperreals rigorous creates a boundary? While the transfer principle makes sense intuitively, it takes some leg work to do all of the model theory and such to construct a satisfactory field. $\endgroup$
    – tox123
    Commented Apr 9, 2021 at 23:37

Taking a different approach to the problem, lets posit the question as 'What would our math system be without fractions?'

If we had never 'invented' the fraction, the only numbers we would have would be 'counting numbers'. There would be no 'real number system', methinks, because all of the issues with real numbers begins and ends with fractions (decimals are just an extension of fractions).

Consider if the entire evolved math system and the concept of 'unity' had been totally different? Would we even have a need for anything past 'counting numbers' and 'integers'?

Take for instance the process of taking a 'whole' pizza and divided it up into 8 pieces. Our 'math' creates a concept of a fraction of a whole - each piece is one-eighth of the whole, or 0.125 of the whole. But suppose this concept had never been created or even entertained by humans? Suppose, instead, we stopped at 'there are now eight 'wholes'? Without the concept of fractions, there would be no one-third fractions to become 0.333333333.... and thus no 'real' numbers.

Consider the digitization of our accounting system. We have the concept of the 'dollar', and of 'pennies'. One hundred pennies make a dollar. So we use the 'fraction' (decimal) annotation of $3.20 to represent three dollars and twenty pennies. The British, with pounds and shillings and pence, could not ascribe to such notation. Every transaction was done with 'whole numbers' of counting individual pounds, shillings, and pence, not 'fractions of a pound' and 'decimals of a dollar'. Imagine if the concept had ended there?

When the computer was invented, and thanks to a road trip to some quaint country-side pub, we use binary instead of decimal as the numerical representation in computer accounting, and the problem with digital notation of money soon became evident. In our accounting systems (spreadsheets and such) decimals - cents - were converted, in the computer circuitry, to bicemals. The problem is, amounts such as 13 cents, or 0.13, created an infinite bicemal when converted to binary. The digital computer could not handle it properly. Rounding occurred. Over thousands of transactions per day, the small differences added up. It is rumored that one bank almost went under, when their 'daily interest' calculations on the computer did not match with daily interest calculations computed manually. The books done manually did not balance with he computer numbers. The balance sheet was out by thousands of dollars. Fraud was suspected. It was a very common problem with early spreadsheets. The solution was to enter $8.13 as 813 pennies, not dollars, and revert back to using only counting numbers, not fractions and decimals.

So, imagine a society that had never gotten their heads around fractions (like so many students in the early grades) and used only whole numbers (or, eventually, integers). This society would never have fractions, and thus decimals, and thus real numbers.

Consider our concept of time. Sixty seconds form a minute. Each second is a unitary concept, and then we create a new unitary concept 'minute' out of the sum of the parts. But somewhere along the line, humans went in reverse. They thought only of the minute as being the primary unitary concept (instead of a 'bunch' of unitary concepts 'seconds') and created the system of 'A second is 1/60th of a minute'. Given the absolute chaotic confusion this presents when primary school teachers attempt to get it across to their students, it is definitely not a 'natural' concept, but a very 'contrived' concept. Suppose a society had never even conceived of fractions? Suppose they had developed a system of mathematical notation where the 'second' was always considered a 'unitary concept', not a 'fraction of the whole'? And 'one minute 30 seconds' was always referred to as 'one minute thirty seconds', not as 'one and a half minutes'?

For division, 7 divided by 3 is 2 with one remainder, and it stops there. No 'one-third remainder'. That last remaining animal is not cut up into 'thirds', it remains a 'whole'.

That is, this society wold END at integers, not develop any alternative system of representing fractions. No p-adic necessary.

Elementary students would love it. It certainly makes their life easier. But that is the issue. We would not HAVE science as we now know it. Instead of dollars and cents, we might still be using the British system of pounds shillings, and pence.

What would our science and math look like? Perhaps this answer could come from children who have never experienced the horror of 'fractions'. Methinks it would be far more based on biology and the environment, rather than physics and engineering. Whatever it looked like, it would be very different from our world. A true 'world-building' exercise.

But one thing for sure, pies would always be round, and cake are square.


Mathematics is the science1 of developing formal models and applying them to get insights into the real world.

Some of those models are more useful than others. Consider euclidean and non-euclidean geometry. Euclidean geometry is taught in most schools because it is applicable to the real world at human-sized scales. Non-euclidean geometry is taught later, mostly because thinking about in comparison with euclidean geometry teaches thinking about models.

When it comes to real numbers, studying them gives insight in some real-world physical processes and also teaches thinking about models. The former is a reason why sufficiently advanced aliens would probably know real numbers. To the core of your question, decades ago I went through the traditional human progression of mathematical education, and it seems natural to progress from integers to real numbers. That could be the bias I picked up from my teachers.

1 It isn't exactly like other hard sciences, but close enough for me to apply that label.

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    $\begingroup$ the science of developing formal models and applying them to get insights into the real world is called physics. Mathematics "doesn't care" of the applicability into real world of its models and theories. $\endgroup$
    – L.Dutch
    Commented Apr 9, 2021 at 6:25
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    $\begingroup$ @L.Dutch-ReinstateMonica, I disagree. There is applied mathematics, there is non-applied mathematics funded in the hope that it will some day benefit applied mathematics (if only by teaching well-rounded students), and there is non-applied mathematics which "doesn't care." The last often has to struggle for funding, so jobs go into the first two. $\endgroup$
    – o.m.
    Commented Apr 9, 2021 at 10:38
  • $\begingroup$ Mathematics is the science of figuring out answers to questions we won't realize are actual problems for another 500 years. $\endgroup$
    – John O
    Commented Apr 9, 2021 at 16:06
  • $\begingroup$ @JohnO, nice quip, but people like Black and Scholes would disagree, and so do the bankers who use their work. $\endgroup$
    – o.m.
    Commented Apr 9, 2021 at 17:16
  • $\begingroup$ This answer not only ignores the facts that a lot of early math was in fact motivated by mysticism and that other fields and rings are very useful in describing real-world phenomena but also ignores the meat of the question. "It seems natural to progress from integers to real numbers." Why? $\endgroup$
    – tox123
    Commented Apr 9, 2021 at 23:29

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