Numbers Not Expressed with 0-9

Question

I hope this question is okay, and makes some sense. I am by no means a mathematician. My 12 year old son says that there are numbers that cannot be expressed using 0-9. He says it isn’t about their size; being too big or too small. Maybe it’s about their complexity. That’s kind of what I get from him. Anyway, can anyone explain this to me? Is the statement that numbers exist which cannot be expressed with (or solely by) 0-9 true?

Answer 1

My take on what might be meant... is that what it means to be a "number" is incredibly arbitrary. Maybe we might imagine a scenario where we have nonstandard "numbers" who can be added or subtracted... like for instance a scenario with the "numbers" True and False. We can do "math" with them just fine despite them not being represented with digits like traditional numbers from natural numbers or real numbers etc... Heck, there's nothing stopping it from being more exotic, like the "numbers" being dog names or flavors, etc... so long as the operations are well defined for them. The example of complex numbers might fall under this interpretation.

The other take I might have on this, is that certain natural numbers are not fully known or understood by mortals (and might never be) like Graham's number or the Ramsey number $R(7,7)$. We know they should exist and will be finite natural numbers, but we do not know many more details about them. An omniscient hyperintelligent being like God might, but we can not with our current tools and language begin to try to write these numbers down, though we can unambiguously refer to them. We know that $R(7,7)$ must be some natural number between $205$ and $497$ but we do not know which one.

"Erdős asks us to imagine an alien force, vastly more powerful than us, landing on Earth and demanding the value of R(5, 5) or they will destroy our planet. In that case, he claims, we should marshal all our computers and all our mathematicians and attempt to find the value. But suppose, instead, that they ask for R(6, 6). In that case, he believes, we should attempt to destroy the aliens" - Joel Spencer

Answer 2

Of course there are many generalizations in which numbers cannot be written solely by $0,1,\ldots,9$. The "simplest" generalization are complex numbers but one could go further in this direction by adding other "units" such as the $i$ in $\mathbb{C}$.

Answer 3

One insight is the world of complex analysis. As one commenter has pointed above, intuitively $\sqrt-1$ does not exist, that is why $i$ is introduced, to define that number and all those even roots of negative numbers. However, this world becomes even more interesting since there are generalizations such as quaternions, octonions, sedenions and so on. Here is a link: https://en.wikipedia.org/wiki/Quaternion

Answer 4

I like this type of question; this site is for anyone studying math at any level, so I think it should be allowed.

Firstly, there is a question about what a number should be, and this is not a trivial one. Some people might reasonably argue that complex numbers are all the "numbers", but one could also talk about quarternions, octonions and so forth.

This family of examples all work by taking the real numbers, which can be seen as all numbers that have a decimal expansion (though it may be infinite and non-repeating), and adding new "numbers" with interesting and useful properties, such as $i$ with the property that $i^2=-1$. It is not hard to convince yourself that no number with a decimal expansion could satisfy this property.

One could also talk about ordinals, which is another generalization of what one typically would call a number. I won't go into detail as to what they are, but there are many ressources online to find.