The set of numbers that can be written with a finite number of 0-9 digits is countable. That set is called the decimal numbers, and is a subset of the rational numbers which itself is countable too.
If you allow infinitely long descriptions, then I don't know what's the score. In the case of "descriptions" that amount to (potentially infinite) decimal expansions, I think an argument akin to Cantor's diagonal will show that you can't possibly describe all real numbers (or all numbers between 0 and 1), but I'm not confident I can assert this to be true.
As fro your further edit: assuming √2 is rational, then the set of ~~rational~~ integers {q: q√2 ∈ ℤ} does not admit a minimum, indeed it contains arbitrarily small numbers, so we can't pick k the smallest member of that set.
I have heard of an anecdote that in some company, can't remember which, when meeting to take a decision, they would have people speak up in order of seniority, from the most junior to the most senior in the room. That way they could avoid the problem you're describing: you don't risk contradicting someone preceived as more experienced, or worse, your boss.
I found that an elegant solution, at least in abstract, I've never actually see it in practical use.
I can see how this sounds good, but my first reaction is that I think it is the kind of thing that would work great in a really good team that wouldn't even need it, and it would be awful for juniors in the kind of team that would get really excited about implementing it. Who wants to speak up when you don't know the ropes, you haven't gotten used to the social dynamic, etc? Seems to select for people with excessive confidence, who already have no difficulty advancing themselves.
This model seems insufficient and exclusionist to people who are socially withdrawn, or have a social disability, or come from a marginalized background, or any of the myriad reasons some developers already don't speak up. But it's a good starting point. I just would not implement it without additional frameworking to guide and encourage useful input from the juniors. In which case, if you're putting in the effort to understand why people aren't speaking more freely, this framework may be superfluous (or it could still be a valuable part of your practices).
I'll repeat myself: it's very obvious in context that they meant the positive solution of xˆ2 = 2.
My definition of the square root is as follows: the square root of a positive real number x is the positive number, noted √x, such that (√x) ^ 2 = x. To make this a useable definition, we need to prove that equation has a solution (using the fact the function t -> t^2 is zero for t=0, diverges to +inf when t -> +inf, and is continuous between the two) and that solution is unique (using the fact the same function is strictly increasing).
The most satisfying answer I have for the nature of the square root is to consider complex numbers.
For an arbitrary complex number, a + bi, we can plot this as a vector on a two axis scale (x axis real and y axis imaginary). We can also convert any complex number to the form z = r e^(i θ). In other words, draw the complex number vector as an angle and a magnitude, in polar form.
So any number can be drawn as a vector onto the complex plane. Multiplying two vectors together causes the angles to add and the magnitude to multiply — creating a rotated and extended (or shrunken) new vector as the product.
So what’s the square root of a complex number z? It’s the vector a that, when multiplied by itself, winds up with z.
When z has no imaginary component, it lies flat on the x axis; its magnitude is z and the angle is 0. What’s its square root?
There are two. |a|e^(i 0) and |a|e^(i 180). 180 degrees multiplied together rotates the vector back to 0. And the e^(i 180), in radians, is e^(i pi), or negative one.
So putting it together: there are only two solutions, with the same magnitude, and different signs.
The initial definition of a square root of a number X (from where the name actually comes from) is "the length of the sides of a square whose area is X". There are some straightedge and compass constructions for this even in Euclid's Elements. That's why the square root is always a positive number; bringing in complex numbers only serves to confuse the issue, and is anachronistic.
using the square root to convert area to perimeter, for example, is an application of square roots, one where the negative root is kinda useless. That’s not the definition of the operation. The definition of the operation is the solution to y = xx. And sure for many applications the negative root is useless, but you can’t argue against that both -2 -2 and 2*2 = 4.
You may not think I'm right, but that is the actual history. This operation was invented at a time where geometry was the main way mathematics was done. The square root is a much older concept than negative numbers (edit: at least in the Hellenistic world; Chinese and Indian mathematics may have had different histories).
So, by definition, the square root of 4 is +2. x * x = 4 has two solutions, which we dub +sqrt(2) and -sqrt(2).
Edit: to discuss just how much older, the concept of a square root appears in Euclid's Elements, c. 300 BC; on the other hand, Diophantus, in Arithmetica, c. 280 AD, was claiming that the equation 4x + 20 = 4 doesn't have any solutions/is absurd. So, the square root is more than 600 years older than negative numbers in the Hellenistic tradition.
> Diophantus, in Arithmetica, c. 280 AD, was claiming that the equation 4x + 20 = 4 doesn't have any solutions/is absurd. So, the square root is more than 600 years older than negative numbers in the Hellenistic tradition.
The gap is much wider than that; double-entry bookkeeping speaks of "credit" and "debit" (defined to be exactly the same concept as positive and negative numbers) because negative numbers were still nearly unknown to Europeans in the 15th century.
It's very obvious in context that they meant the positive solution of xˆ2 = 2.
My definition of the square root is as follows: the square root of a positive real number x is the positive number, noted √x, such that (√x) ^ 2 = x. To make this a useable definition, we need to prove that equation has a solution (using the fact the function t -> t^2 is zero for t=0, diverges to +inf when t -> +inf, and is continuous between the two) and that solution is unique (using the fact the same function is strictly increasing).
