Is $\sqrt2$ a tricky notation?

Question

When someone asked me how to solve $x^2=9$,I can easily say, $x=3$ or $-3$. But what about $x^2=2$? There is NOT any "ordinary" number to solve this question. It's an irrational number. So we say helplessly, the answer is $\pm\sqrt2$, but what does $\sqrt2$ mean? It's a number, when squared, equals $2$.

This is a cycle define, just like "what's grandfather mean?father's father - what's father mean、grandfather‘s son." It does NOT tell anything more. And if we can define notations ‎arbitrarily, we can give any questions answer tricky. For example, what's $123456789\times987654321$? We need not calculate, just define $f(x)=123456789x$, the answer is $f(987654321)$.

Answer 1

Very good, you understand what is going on here!

It does NOT tell anything more

Right, it is just a useful notation. Would you prefer to write

• $$ \sqrt 2 $$
• the unique number $\sigma \geq 0$ that has the property $\sigma^2=2$

every time you need that number in a computation? There is nothing more to the notation, it is a shortcut, that's all. The interesting world is somewhere else: In order to use such a notation, you have to prove that such a number exists and that it is unique.

And in order to do that you have to know a lot of things about the set $\mathbb R$: a very interesting story.

You may not believe it, but if a mathematician has to compute $123456789 \times x$ very often, he will define $f(x)=123456789x$, and his answer to the question: What is $123456789 \times 987654321$? will be:

You mean $f(987654321)$? Do you want to know the digits?

Answer 2

Yes, $\sqrt{2}$ only tells you that it is a number which, when squared, yields $2$. It's a whole lot more informative than any other thing you might write for the same number. However, the fact that $\sqrt{2}^2=2$ is really important in certain contexts. For instance, in higher mathematics, we are often less concerned with the easily determined fact $\sqrt{2}$ is somewhere between $1.4$ and $1.5$ than we are with other questions about it.

In particular, some branches of mathematics stop thinking about the real numbers altogether and stop thinking about arranging things on the number line, and just want to talk about addition and multiplication. They start off in the rational numbers, $\mathbb Q$, equipped with their ordinary addition and multiplication and move further. Quickly, one can create questions which have no solution, like:

What $x$ satisfies $x^2=2$?

which can't be solved in the rationals. However, a very natural question is, "Well, supposing there were a solution to that, what properties might it have?" So, we define a new number, $\sqrt{2}$ and extend the rationals by it to the field $\mathbb{Q}[\sqrt{2}]$. What's this mean?

Well, now we're considering any number which can be written as a polynomial, with rational coefficients, of $\sqrt{2}$ - or equivalently, the numbers that can be written as a sum or product of rational numbers and $\sqrt{2}$. So, we're now interested in things like $\sqrt{2}+1$ and $\frac{1}2-3\sqrt{2}+\sqrt{2}^3$ and how addition and multiplication might work with them. Provably, every such number is of the form $$a+b\sqrt{2}$$ for rational $a$ and $b$ and we define addition and multiplication as $$(a_1+b_1\sqrt{2})(a_2+b_2\sqrt{2})=(a_1a_2+2b_1b_2)+(a_1b_2+b_1a_2)\sqrt{2}$$ $$(a_1+b_1\sqrt{2})+(a_2+b_2\sqrt{2})=(a_1+a_2)+(b_1+b_2)\sqrt{2}$$ which might not look like much at first, but suddenly, we have a new field in which we can perform addition and multiplication (and we even find results like division if we look harder) - and we find curious things like defining $\overline{a+b\sqrt{2}}=a-b\sqrt{2}$ preserves all the structure of multiplication and addition, which tells us that $\sqrt{2}$ and $-\sqrt{2}$ are somehow interchangeable.

This branch of mathematics is too large to summarize in any adequate way, but essentially, my point is that the algebraic properties of a number - that is, how it responds to addition and multiplication - are very worthwhile in their own right, and hence, though the notation involves "inventing" new numbers that we can't write in any satisfying closed form like we can the rationals, the definition "$\sqrt{2}$ is a number which, when squared, gives $2$" actually has a lot of interest to it.