# What number is the sum of two roots

This is one of these questions that students ask and for which I have never found an answer that students would accept as convincing. Here is an instance:

Student: 2•3 is a number, namely 6. 2+3 is a number, namely 5. 1/2•1/3 is a number, namely 1/6. 1/2+1/3 is a number, namely 5/6. √2•√3 is a number, namely √6. So why isn't √2+√3 a number?

• When they ask you why it isn't a number, is your first answer that it is, in fact, a number? – G Tony Jacobs Jun 2 '17 at 23:01
• Certainly NOT. First, most people ( including math teachers) do not have a precise idea of what a number is. They mostly have mental images. Second, I quoted the question as closely as I could but the leading assertions would seem to indicate that what the question really is is "So why isn't √2 + √3 the √ of some number?" – schremmer Jun 2 '17 at 23:24
• If the students don't have a precise idea of what a number is, that should be addressed. "A number is a location on the number line" is as precise as they need for a good long while. – G Tony Jacobs Jun 2 '17 at 23:29
• Um... no, I'm not. What is wrong with that definition of "number"? It's comprehensible at a very early level, and it's sufficiently accurate to do a lot of mathematics. – G Tony Jacobs Jun 2 '17 at 23:40
• You asked whether I was joking about defining numbers as locations on a number line. Nobody else took it as a joke; try taking it seriously. Your students can understand that. Give them a little more credit. – G Tony Jacobs Jun 4 '17 at 4:52

I think G Tony Jacobs's answer is an excellent one, and I admit I don't quite understand what the OP's objection to it is all about. But I am going to take a stab at trying to explicate what I think might be underlying the objection that "$\sqrt{2}+\sqrt{3}$ is not a number".

Looking over the examples that precede the question, we can notice that they all have a kind of grammatical structure in common: namely, an expression that initially consists of a binary operation performed on two "single" object is evaluated or simplified to produce another "single" object. That is, they all have the structure of

[thing 1] (operation) [thing 2] = [thing 3]

The sum of $\sqrt{2}$ and $\sqrt{3}$ resists this grammatical structure, since there is no way to combine the two "things" into something that looks like a single term. This is probably related at some deep conceptual level to the well-documented phenomenon of students perceiving an equals sign as a command to "do something", separating an unperformed computation on the left from its result on the right. In this sense, the sum $\sqrt{2}+\sqrt{3}$ looks like a question, not an answer.

For another instance of this, consider the following problem, common in first-year Algebra:

Simplify each of the following expressions:

1. $5x^2 + 3x^2$
2. $5x^2 \cdot 3x$
3. $5x^2 + 3x$

Many students find it extremely frustrating and counterintuitive that while the first question can be answered as a single monomial ("Combine like terms to get $8x^2$") and the second question can be answered as a single monomial ("Multiply the coefficients and add the exponents to get $15x^3$"), the third problem cannot be simplified any further than it already is. We know that $5x^2 + 3x$ can be regarded as a single quantity -- and we use the word "polynomial" to describe such quantities -- but to (some) students, it looks like a question that has not yet been answered.

How to address this? In the algebraic case I described above, I usually begin by turning the question around on the student: Why should we be able to combine those two terms into a single term? For that matter, why is it possible to simplify the other expressions? This can lead into fruitful discussions about (for example) the distributive property, the associative property, the meaning of exponents, and so forth. But just because some expressions can be simplified into a single term does not mean that all expressions can be simplified into a single term.

Similarly, in the case of the OP, I might begin by asking: "Well, what do you think $\sqrt{2} + \sqrt{3}$ ought to be equal to?" In many cases what the student really wants to know is something like: "Why isn't it $\sqrt{5}$?" Once they have articulate what they think the sum ought to be, we can then explain why it isn't that: because when you square it, you don't get $5$. At this point I would probably follow G Tony Jacobs's line of reasoning.

(Possibly relevant: Why unlike terms cannot be simplified? and the second-to-last paragraph of my answer at https://matheducators.stackexchange.com/a/1058/29.)

• This is very good, and inspires a quite succinct answer - you can't combine them because they're not like terms. You can combine fractions, because they can always be made into like terms, by finding a common denominator. In the cases where you can add simple radicals to obtain another simple radical, e.g., $\sqrt2+\sqrt8=\sqrt{18}$, it's because they are like terms. – G Tony Jacobs Jun 5 '17 at 13:27
• @mweiss Re. "The sum of √2 and √3 resists this grammatical structure, ..." Exactly! As I just responded to Paul Garrett in Difference between whole numbers ... "The problem I have is that for developmental students this is typical of their first logical, critical questions for which they expect (hope?) there is a logical answer: they see a pattern which then breaks down and want to know why as, presumably, they want to learn from that. I am not the one who chooses their questions but I would rather not risk discouraging them as it takes extremely little at this point for them to clam up." – schremmer Jun 5 '17 at 13:52
• Building on @GTonyJacobs comment, I think that this highlights combining like terms (or units) as the fundamental idea about addition. My first question on StackExchange, in fact, was something of a wish that that was taken as axiomatic instead of the distributive property. – Daniel R. Collins Jun 5 '17 at 16:26

First of all, $\sqrt{2}+\sqrt{3}$ most certainly is a number. It is a real number, approximately equal to $3.14626$ Perhaps what you're asking is why the sum of two simple radicals isn't also a simple radical, when the sum of two integers is an integer, and the sum of two fractions is a fraction.

Roots of integers are examples of algebraic numbers - numbers that are the roots of polynomial equations with integer coefficients. The sum of two algebraic numbers is an algebraic number, but it doesn't have to be as simple as the addends are.

Something like this is already true with fractions: $\frac12$ and $\frac13$ are both unit fractions, but their sum is a more complicated fraction, with larger numbers in both the numerator and denominator than the fractions we started with.

Similarly, the numbers $\sqrt2$ and $\sqrt3$ are roots of the polynomials $x^2-2$ and $x^2-3$, respectively. Their sum, which is most simply expressed as $\sqrt2+\sqrt3$, is a root of the polynomial $x^4-10x^2+1$.

This particular algebraic number can also be written as $\sqrt{5+2\sqrt{6}}$, where you can see both $2+3$ and $2\times 3$ playing a role. (To see this, write $x=\sqrt2+\sqrt3$, square both sides, and combine integers.)

Just like you can't expect reciprocals of integers to be closed under addition, you can't expect roots of integers to be closed under addition. That's because we're involving more complicated operations than addition, namely, division and taking of roots. On the other hand, you can expect the sum of two rational numbers (fractions) to be rational, and you can expect the sum of two algebraic numbers (roots of integer polynomials) to be an algebraic number.

Does this help at all?

As requested in the comments, I'll put this in the voice I would use to address a high school or middle school student asking this question:

"First of all, $\sqrt2 + \sqrt3$ is definitely a number. Here, let's calculate its value... [calculator].... as you can see, it's not a very pretty one, but we can see a few decimal places: $3.14626\ldots$. Huh, it's kind of close to $\pi$, but a little bigger.

"Anyway, let's see if we can express this number in a nicer form:

\begin{align} x &= \sqrt2 + \sqrt3\\ x^2 &= (\sqrt2 + \sqrt3)^2\\ x^2 &= 2 + 2\sqrt2\sqrt3 + 3\\ x^2 &= 5 + 2\sqrt6\\ x &= \sqrt{5+2\sqrt6} \end{align}

(I'd talk through the steps of that algebra, making sure it's clear after each line.)

"Ok, so it's a square root, but it's a square root of something more complicated that what we started with. I guess that's fair. After all, when you add two fractions like $\frac12$ and $\frac13$, which are pretty simple, you end up with $\frac56$, which is more complicated - it's not just a $1$ on top, and both numerator and denominator are bigger than what we started with.

"Actually, it's pretty interesting, that the numbers $5$ and $6$, which are $2+3$ and $2\times 3$, both show up in the fraction $\frac56$ and in the radical $\sqrt{5+2\sqrt{6}}$

Second edit:

One more run at this, just to see how succinctly I can get the main point.

"To see what something is the square root of, square it:

$$(\sqrt2 + \sqrt3)^2 = 2 + 2\sqrt6 + 3 = 5 + 2\sqrt6$$

"As you can see, we don't get a whole number, because in FOIL*, we have middle terms giving us the $2\sqrt6$ part.

"It's different from a fraction, because if you look at the sum $\frac12 + \frac13$, there is a common demonimator $6$ you can multiply by that makes it the sum of two whole numbers: another whole number. No cross terms arise, because there's no FOIL going on."

(* FOIL = distributive rule applied to binomials; mnemonic for "First terms, Outside terms, Inside terms, Last terms")

Also instructive are the cases where it does work. For example, $\sqrt2 + \sqrt8 = \sqrt{18}$. You can "FOIL" it out and see why.

• The first thing I would tell a student, at any level, is that $\sqrt2 + \sqrt3$ is most definitely a number. – G Tony Jacobs Jun 2 '17 at 23:06
• What? I just did. It's everything below the line in my answer. Those are exactly the words I would use with a student. What do you want from me? – G Tony Jacobs Jun 2 '17 at 23:30
• It's not a speech; it's a conversation. I'm assuming you know how to talk to your students. I have a lot of success talking to students this way. I'm sorry if you don't. The pattern does continue, with added wrinkles at each level, and that's what a good teacher should reveal to their students. – G Tony Jacobs Jun 2 '17 at 23:39
• @G Tony Jacobs I think your answer is quite good, exactly what the OP asked for, with the edit (but even without was good). I'm sensing a big communication barrier that I don't see disappearing any time soon. I would have no qualms disengaging at this point. – pjs36 Jun 3 '17 at 1:25
• I agree, G Tony, you're dealing with a ranter not a questioner, so you're not going to satisfy him or her at any point. Your answer is fine (+1). – Daniel R. Collins Jun 3 '17 at 2:13

I like GTonyJacobs and mweiss's answers. It really boils down to the core fact that in addition, you can only combine like terms. However (to my frequent frustration), that is not actually taken as fundamental in a basic algebra course. Rather: The distributive property (of multiplication over addition) is. So I'd like to try an alternative answer that speaks directly in terms of that axiomatic basis.

The most likely misapprehension is that $\sqrt{2} + \sqrt{3} = \sqrt{5}$; in other words, that $\sqrt{2} + \sqrt{3}$ = $\sqrt{2 + 3}$. But that implies a belief that radicals distribute over addition, which is false. For a whole-number counterexample, consider: Is it true that $\sqrt{9 + 16} = \sqrt{9} + \sqrt{16}$? No, it's obviously not true, because by the order-of-operations (and grouping under a vinculum), the left-hand side simplifies to $5$, while the right-hand side simplifies to $7$.

In short, distribution (and hence combining) does work for multiplication-over-addition; but it does not work for radicals-over-addition. Different operations work differently.

One could hazard the guess that this question is precisely the one that led to Book X of Euclid's Elements, the famous crux mathematicorum (i.e. the "Cross of the Mathematicians").

While I do agree with the rejoinder that "$\sqrt{2}+\sqrt{3}$ is most definitely a number", and it is a very useful thing to tell the student this, it's still not entirely right in a pedagogic sense. First off, a student is not wrong to wonder about the fact that, apparently, $\sqrt{2}+\sqrt{3}$ can't be simplified. My guess would be that, subconsciously, most students just assume that $\sqrt{2}+\sqrt{3}$ can be simplified, until set straight that is, just as all comparable types of expressions encountered up to that point (like $\sqrt{2} \times \sqrt{3}$) can be simplified. The moment where a student asks why $\sqrt{2}+\sqrt{3}$ "is not a number", that's exactly the moment the student realizes he or she has been making this assumption all along. This is a valuable moment in the learning process, and one shouldn't throw it away by simply "correcting their mistake".

So here's what I would not do: I would not suggest to the student that he or she was wrong to say $\sqrt{2}+\sqrt{3}$ is not a number (after all, a precise definition of "number" usually hadn't been given anyway, although students would probably agree that the sum of two numbers is again a number), and I would also not go so far as to say or suggest that what he or she should have asked was something else.

The latter point is especially important to my taste. It is not easy to give a precise formulation to one's deepest thoughts straight-away, mathematical or otherwise; on the contrary, the most interesting observations can grow out of the stupidest-sounding questions. If a student is discouraged to give voice to their intuitions (which could easily happen after meeting with the remark that "what you really meant to ask was X", which is usually said for the benefit of the teacher anyway, who gets to show off their own unusual talent for reframing a pointless question into a meaningful one), simply out of a fear to be corrected, then the learning process is stifled.

Perhaps I should also add what my answer to your student would be, because it is such an excellent question (even if it isn't phrased in the best of all possible ways): "Why is $\sqrt{2}+\sqrt{3}$ not a number?" I think I would start by saying that there is no why to it, it is simply that way and no other. It's just a matter of biting the bullet. (Asking for a reason, they are running into the fact that they were expecting something that they didn't have a reason to expect -- you don't have to tell them this, but this is what they will take from it.)

I also think it is very useful to square the expression, just as G Tony Jacobs suggested, to show that it isn't a square root of something simple. But this approach is also kind of limited, because it doesn't exactly show there isn't some other simple form that it could take.

However, there are other difficulties here, and the main problem is that this is the type of question that opens up such a gigantic can of worms, that you simply can't do justice to it in the space of say ten minutes, or even a whole hour. Because one could equally well ask: why can't we write $\sqrt{2}$ any simpler? And there you've run into the irrationality of square roots of non-square integers. And if you want to prove that $\sqrt{2}+\sqrt{3}$ is not of the form $a+\sqrt{b}$, with $a$ and $b$ integers, that's a similar type of problem, but only much more messy.

I would go so far as to say that the only really satisfying way to answer this is to phrase these facts in the language of (basic) algebraic number theory. And you simply can't do that at their level. So you will have to treat the question as a moment where the student is discovering something that they won't be able to really get their head around unless they're going to do some serious mathematics. And it's not a bad thing if they decide not to do that. But it is good to leave them with a sense of wonder about the issue, and not take that away with some kind of facile reponse, because it is only right that they should feel this wonder.

• "I would start by saying that there is no why to it, it is simply that way and no other." -- This is my least favorite of all possible explanations. – Daniel R. Collins Jun 20 '17 at 15:19
• @DanielR.Collins: Why is that? – RP_ Jun 20 '17 at 15:33
• @René I already accepted mweiss' answer and I still think it is a good answer but if I had wanted to answer my question, your answer is exactly what I would have written because you see the student and consider what your response to the student is likely to do to the student. That is extremely rare as teachers are too fond of teaching to pay attention to the students. – schremmer Jun 20 '17 at 16:48
• @DanielR.Collins: I do not agree with your assumption that the "why" of a statement is answered by giving the proof of the statement. Why-questions ask for vaguer things, they are a sort of meta-questions. 'Why can't we integrate exp(x^2)?" The answer is not: "here's a proof", but it could be: "because differentiating functions tends to make them more complex (i.e. the number of symbols needed to write them down tends to increase), while integrating makes them simpler, and there are fewer simple functions than complex functions". – RP_ Jun 20 '17 at 18:45
• @DanielR.Collins: Also, there is no single optimal answer to a why-question, because different mathematicians may have developed different perspectives on the same piece of mathematics, and there's no way of telling which is better, except on the grounds of taste. To say that "there is no why to X" is really just a different way of asking: "why would you have supposed X is true in the first place?" I really don't see what is so wrong about that, as long as you're not dismissive of the question. – RP_ Jun 20 '17 at 18:48