https://www.reddit.com/r/askmath/comments/1p7rmvg/comment/nqzxbwd/
On a question about why does the √ function denote only the non-negative root, there is a user who stubbornly insists that the standard meaning of the √ symbol is not the function from [0, ∞> to [0, ∞>, but a multi-valued mapping.
R4: In fact, the standard meaning of the √ notation is to denote the principal root.
Is this badmathematics? More like badnotation...
There’s not really any such thing as ‘the standard meaning’, e.g. in algebra, √3 most likely denotes a formal square root, rather than a function applied to a value.
That said, √ is certainly most often for the principal square root, and using it to denote the multifunction is not that common.
Similarly, it’s inconsistent whether ∛ is the formal, principal, or real-valued cube root.
If we go with the convention that √ doesn’t denote the principal square root, but rather a formal expression that encodes both square roots at the same time, how do I call the number that before I used to call √3, which is, the irrational number with decimal expansion 1.732…? What symbol do I use?
‘The positive square root of three’ or ‘the principal square root of three’.
That’s not a symbol, that’s an entire full English sentence. The point of modern mathematical notation is to be able to do math with concise calculations and symbols rather than writing a poem every time you need to state an algebraic/numerical/geometric fact, like they did in the Middle Ages.
In an actual problem, how would I write down on paper or blackboard, in an equation or expression, the irrational number with decimal expansion 1.733… ? Can’t use √3 anymore, so what do I use?
You can use any symbol you want. if you're redefining things, feel free to denote it as π.
I’m not the one who wishes to redefine the √ symbol: they did. Or at least they were saying that there’s nothing wrong with it, and that it’s a valid convention that can be found in the literature.
I’m pointing out that actually it’s a horrible idea that is nowhere found in the literature and nobody uses it, for good reasons. An an example of why, I’m provocatively asking what I would call the positive real root of x2 - 3 in this convention.
If the only answer one can come up with is a sarcastic “in this convention the old √3 is now denoted π”, well, I think that just proves my point.
Most ‘actual problems’ consist mostly of words, not symbols, but if you do need a symbol, you write
I don’t know what your point is. If you’re in a field where √ unqualified is the multifunction, the obvious consequence is that √ unqualified is not anything else. The fact you’ve chosen that convention suggests you’ve decided it’s worth the trade‐off.
This sort of thing is incredibly common. Mathematical notation is totally context‐dependent, e.g.
I’m not going to respond to your other examples of situations where the same symbols can have different meanings depending on taste and context, because I agree with you. My point is not that mathematical notation in general has to be set in stone and can never be used differently.
My point is that in the specific case of the √ symbol, the proposed alternative convention is horrible, never used by anyone in any context (not even in complex analysis, where multi-valued functions arise naturally) and makes common expressions with square roots undefined and meaningless (you can’t even write something as basic as 1 + √5 anymore (twice the golden ratio), unless you also redefine what the + symbol means).
You also can’t do any algebra with it: is √2 + √3 four different values now? Is √2 + √2 zero? Is (√2)2 the Cartesian product of sets with two elements that becomes a set with four elements, each one a different pair? Silly me who thought that (√2)2 should equal 2!
I asked you how you would denote the “old” √3 with this new convention as a way to show that a good answer doesn’t exist. And in fact, your answer was:
What? First of all, what is f here? Secondly, what do you mean with “where √3 is the positive square root in this case”? The entire point of this discussion and the assumption at the basis of my question is that now √3 is NOT the positive square root anymore, as you proposed, and to find a different suitable notation for it!
There seem to be two points to your argument.
I'm not sure if you intend to have one of these statements imply the other. (Either such implication would, in my view, not hold.) However, it doesn't really matter, because I agree with 1, and 2 isn't true.
The answer to all these is: in those areas, this problem doesn't appear very often, and if they do, you think of something. Or you just ignore it, and hope everybody knows what you mean.
To be clear, I'm not claiming this is totally fine; I dislike it. But it's how it is, and I don't think it's the worst example.
I meant f to stand in for the large expression into which you wished to embed √3. What I mean is: when I said in the last paragraph ‘you think of something’, one of the things you might think of is to write √ for the principal square root for that expression only, and explain this by writing ‘where √3 is the positive square root in this case’). Of course, this might not work if you need to embed both into your expression, but there are other options too.
I believe what they're saying - and, /u/siupa, please correct me if I'm wrong - is that the default meaning of
√2is the principal root. In practically every context, people will read √2 as 1.414... rather than ±1.414..., unless clarification is given. It's not like, say, whether 0∈ℕ, where both conventions are on closer-to-equal footing in practice.I'd say there's also a bit of a conflation going on.
You can use
√2to mean 1.414... . Let's call this "Convention A".You can use
√2to mean some sort of vague idea of "a quantity that can be 1.414... and/or -1.414...", without bundling them into a set. Let's call this "Convention B".You can use
√2to mean the set {1.414..., -1.414...}. Let's call this "Convention C".Convention A is universally recognized as the 'default'. If you see
√2without any other context, pretty much every mathematician will assume that Convention A is being used. A lot of people who only vaguely remember high-school algebra argue otherwise, and claim the answer is Convention B.Convention B is occasionally used in places like complex analysis, where you want to deal with """multivalued functions""". But it's not the default, and it's also not really mathematically rigorous. You can't really algebraically manipulate this notation easily - at least, not with the full generality we usually do. You have to be very careful, because which manipulations are allowed is heavily context-dependent. Like, you can't replace "(x) + (x)" with "2(x)" anymore, because if x=√2, the first one can be 0 and the second cannot.
Convention C is where you end up if you try to formalize Convention B. It inherits all the problems of Convention B, but introduces some more too! You have to 'lift' arithmetic operations to sets, which we do indeed do sometimes in specific cases... but doing this in full generality causes a bunch of notational conflicts [for instance, /u/siupa's example of S² being the cartesian product of S with itself, and therefore (√2)² is a 4-element set].
In my experience, even in complex analysis, Convention B is not really taken to be the default - it's still Convention A. This is because Convention B means that square roots can't be algebraically manipulated, which we kinda like to be able to do.
Yes, all I was about to say in some way or another. Thanks!
R4: Contrary to what the linked user is claiming, the standard meaning of the √ notation is, in fact, to denote the principal root.
Absolutely not bad math. You can use the radix sign to refer to the multi-valued root function, you just need to state that that is what you mean.
Regardless of if it's right or wrong, I remember being taught in school that the symbol means both the positive and negative roots.
How did you write the quadratic formula?
Not gonna try to format it in Reddit text, but this image matches what I was taught: https://intomath.org/wp-content/uploads/2021/03/quadratic-formula-2048x1152.jpg
Then why put the ± if it's already implied by the symbol √?
That's a question to ask my high school math teachers, man, not me. Literally all I'm saying is that it's what I was taught, not that it's correct.
I think you’re misremembering
How would you even begin to know what I was taught in middle and high school? You don't even know who I am, let alone which teachers I had or what they said at the time.
Well of course there’s always the possibility that you’ve been taught something wrong and nobody in your class or any other teacher ever realized it, but I find it incredibly unlikely given that these elementary things have been established for over a century now. I’d bet 1:90 odds that you’re misremembering
By the way, I just saw the picture you posted about the quadratic formula they taught you, and it confirms what I said: that formula is inconsistent with the notion of the radical symbol meaning both the positive and negative roots. You see those +- symbols in front of it? It’s because the radical is always positive, so you have to put a +- double sign in front of it to signal that you’re taking either the positive or the negative root. If the radical symbol already meant both roots by itself, you wouldn’t need any +- symbols in front of it.
I update the bet odds to 1:110 ;)
I don't know, I had seen a lot of √4=±2 type stuff when it was getting taught. Maybe convention varies but that sort of thing could be why OOP was saying what they were.
I think you maybe saw a lot of x2 = 4 -> √ (x2) = √4 -> x = ±2 type stuff.
You raise an interesting point. I'm not sure how much weight it carries, but it's still an interesting point.
Many people probably have *accurate* memories of being told *incorrect* or nonstandard things by teachers.
Many people probably have *accurate* memories of being told by a teacher that 1 is a prime number, even though that's not standard.
Many people probably have *accurate* memories of a parent or teacher writing the name of those cartoon bears as the "Berenstein" Bears, even though that family actually spelled their surname "Berenstain".
In grad school I T.A.ed a class on introduction-to-formal-logic-and-set-theory, and one of the problems involved proving a theorem (I forget which one) about prime numbers using induction. Almost every student used 1 as the base case instead of 2, and I ended up spending a portion of the next discussion section explaining why 1 doesn’t count as a prime number, even if that’s what they had been told earlier.
My hypothesis is that people were taught "a prime number is a number divisible only by itself and 1" and then assumed because 1 fits that descrption, then it would be prime, which makes them think that's what they were taught.
Interestingly historically 1 was considered prime far longer than it has not, but it hasn't widely been considered since early 1900s. Some books might still have included it, so i would say if you went to school in the 1950s, meaning you are now around 90 years old, you have a small chance of actually having been taught this. In all other cases it's much more likely people are just misremembering.
You wildly overestimate the quality of teachers, especially at basics levels. One person misremembering quickly becomes hundreds who remember accurately being taught incorrectly.
People keep saying this, that it’s just a choice of convention, but I never understood what √4=±2 should mean if we take it seriously. Does it mean that now √3 isn’t a number anymore, but a pair of numbers? What do I write if I want to talk about the irrational number that has decimal approximation 1.732… ? I can’t call this √3 anymore under the new convention.
And this new convention gets worse! Does √2+√3 now have 4 values?
Can √2+√2 be 0? Oh no, now "√x+√x" isn't equal to "2√x"!
Yeah, it’s just a nonsensical mess. I have no idea why the notion that they’re just conventions that can be both found in the literature is so popular on Reddit. They’re not, one is simply the only one used and the other is useless at best and inconsistent and broken and worst
I guess it's treating √ as being an operator. Applying one to a number and getting a range of results is... Awkward but maybe less so.
This doesn’t answer my question at all though: if now √ is a mapping that takes as input a number and outputs a set of numbers, how do I call now what I previously called √3, the positive irrational number with decimal expansion 1.732…? I can’t call it √3 anymore, because now √3 = {1.732… , -1.732…}
If I’m understanding this correctly, the formal meaning of √x (the “principal square root”) is “the number y such that y≥0 and y2 = x.”
So, for example, √4 = 2, and ±√4 = ±2.
The important thing to keep in mind when solving equations, among other things, is that the inverse of “squared” is not merely √ but ±√, so for instance if we know that
x2 = 25
we can’t just apply the √ operation to 25; we have to apply the ±√ operation to 25.
Of course the whole idea of “principal square root” gets a little mushy when applied to complex numbers, because they’re unordered: there are two numbers y with the property that y2 = -1, but we can’t say that either one of them is “greater than or equal to zero,” so we just arbitrarily choose one to call “i” and call the other one “-i.”
Principal square root is extended to complex numbers by just specifying the root with the smallest argument (ie the angle when written in polar form).
Sure, but doesn’t that also rely on our arbitrary choice of which root to put on the right side versus left side of the graph?
No. How the physical graph is drawn has no effect on the math.
What determines which one has the smaller angle, then?
If I have a number x and a number y and I tell you that
x≠y
x2 = -1
y2 = -1
is there any test you could do to determine which one is +i and which one is -i?
You can calculate the argument of a complex number without making a measurement on a physical graph. The graph is just a helpful drawing, we can do math without it.
How?
It can be written as a piecewise trigonometric function and each piece can be defined by a power series with no reference to drawing anything. In practice if you want to know a value computers software uses clever tricks to calculate approximations very quickly.
I’m not sure if I understand this correctly, but isn’t this kind of a circular argument? It looks like the “piecewise” part of the function is being defined in terms of whether the imaginary part is positive or negative. How can this distinguish between “+i” and “-i” without already knowing which is which?
You define it this way
You don't start with the complex plane and try to figure out which imaginary unit is +i and which is -i. You start by saying "there exists a number i such that i2 = -1" and work from there. Now that you have i, you can do a lot of simple proofs that there must exist a -i, define things like the complex plane, and do all the other math we are familiar with.
Yeah, but what I’m saying is they’re interchangeable. If you rename -i as j and rename i as -j, you get a completely consistent and completely equivalent system. There are two square roots of -1 that are opposites of each other but there’s no fundamental underlying reason to think of one of them as a “positive” number and the other as a “negative” number.
There is no -i until you have +i. That isn't an arbitrary choice of which one to choose because there isn't a choice to be made yet. How do you identify "the other square root of -1" without the primary root? The definition of i makes it the primary root of -1.
You actually can apply the √ operation to both sides: you just need to remember that √ x2 = |x| and not x.
In fact I don’t even know what it would mean to apply the ±√ operation in a context of an equation, as that’s not something with a unique output.
Thank you; that’s a much more accurate way to put it.
This is why ( x2 )0.5 = |x|
Admittedly it's a poor historical choice, makes much more sense if it outputs the unordered pair of solutions, then it can be continuous in the complex domain :)
I think it is much more useful to have unambiguous notation for the positive one.
+sqrt()
|√x| does the job
Thanks, I hate it.
So, diagonal of a unit square has the length of |√2| (also, in speech would this still be pronounced "square root of 2" here, or should you spell out "absolute value/positive branch of ..."?)
And solutions to x2 - x - 1 = 0 are (1 + √5) / 2. Golden ratio φ is the positive solution, (1 + |√5|) / 2, while the other solution (1 - |√5|) / 2 is equal to -1/φ.
If √x means “the set whose elements are the square roots of the non-negative real number x”, doesn’t |√x| mean the cardinality of that set, which will always equal 2 except when x = 0?
Saying |√x| should denote the cardinality of the set that represents √x is like saying |-3| should denote the cardinality of the set that represents -3.
Wait no, that’s not at all the same scenario. -3 is a number, so if I see |-3|, I immediately recognize that |•| means the absolute value function, and so I understand that |-3| = 3.
However, if we work under the assumption that √x does NOT denote the principal square root of x, but rather both square roots of x at the same time, both the positive and negative, then √x is not a number anymore, but a set. For example, √3 would be the set {1.732… , -1.732…}.
And when taking |•| of a set, I interpret it as the cardinality of that set. So I would read |√3| = | {1.732… , -1.732…}| = 2. But what you actually wanted to denote was |√3| = |{1.732… , -1.732…}| = 1.733… However, the symbol |•| already has a different meaning for sets (cardinality) so there’s a notational conflict here.
you know it would be much easier if we used higher order complex numbers for the square root function.