I keep seeing ‘square root’ reels in one of my Facebook feeds. For example: √((82 + 82)/8).
So:
64 + 64 = 128
128 ÷ 8 = 16
√16 = ± 4
I was taught that a square root can be positive or negative. 4 x 4 = 16, and -4 x -4 = 16. Many people say, 'No. A square root is always positive unless the equation says otherwise, ‘Or 4 [in this case] is the principle root and the one asking is only looking for the principle root.’
Have I been doing it wrong all these decades? Are there only negative roots if there is another equation (e.g., √16 + 4 = 0) that would be false with a positive number?
I’ve asked similar questions before, and one answer was that a specific equation was ambiguous. I con’t see any reason why √16 can’t be ±4.
-4 is a square root of 16. But 4 is the square root of 16. y=\sqrt{x} is a function, which means that by definition it only has one output for any given input. By convention, that one output is taken to be the positive one. One could define a square-root relation, that was multivalued for any positive input, but usually, it’s convenient for things to be functions, and so we usually don’t do that.
That said, in a great many contexts where square roots show up, the positive and negative roots are both of interest. When that happens, we use a ± sign in front of the square root. For instance, the quadratic formula: x = \frac{-b±\sqrt{b^2-4ac}}{2a}
Yes, this. I’ve never met anyone in real life who was confused by this, but I’ve seen a lot of confusion and argument online, and I’ve speculated that at least some of it comes from people whose native language doesn’t use definite and indefinite articles the way we do.
16 has two square roots (that is, two numbers that, when you square them, you get 16). (It also has three cube roots if you allow complex numbers, and four fourth roots, and etc.) The radical sign √16 by definition denotes the principal square root, which is the one that’s positive. So we sometimes refer to this as “the square root of 16.”
It’s exacerbated by the notion of a ‘function’, which many math teachers themselves don’t really understand.
We like functions. They’re great. But that pesky “one and only one” mapping means some useful concepts, like square roots, sometimes gives us fits and occasional inelegance.
So, we get the general weirdness about how to handle square roots.
I’ll suggest that like most online math “puzzles” or “problems”, the underlying problem is the people writing them (and many people answering them) are believing that the lies to children (simplifications really) they were taught as tweens are the truth, the whole truth, and nothing but the truth.
So they vociferously defend ideas anyone with more math knowledge will simply laugh at.
Neither, really. Another way to look at the problem you quoted is to express the 8’s as powers of 2, and to recall that applying the square root function is the same as raising to the power of 1/2. So the question becomes, with crappy formatting using x^n to stand for x to the nth power, (2^7/2^3)^1/2 or 2^2 = 4
My take is @Johnny_L.A knew past the lies to children and saw that the “offical” answer was a lie to children. But admitted there was probably yet more he didn’t know. A wise posture to take.
A lot of pedagogy in every subject isn’t as simple as “either it’s truth or it’s lies to kids.”
Instead it’s layers of lies to kids starting in kindergarten and continuing all the way up until the “kids” are in grad school. A lot of onion layers of untruth get peeled off along the way, but many onions still have more layers than that. Hell, a lot of scientific research, again in many / any field(s) is just trying to peel one more layer off the onion of current bleeding edge understanding of [whatever topic].
FWIW, I’m not debating them. It’s just fun to do little math equations from time to time, and to see how my answer matches up with other people’s answers.
As I alluded to earlier, the notion of a ‘function’ is the crux of the issue. The square root in its basest form is not really a function.
One could make it a function by limiting to the principle square root. And probably this is implicit in the online question, but that’s different from the OP being outright wrong
Another option for dealing with things like the square root is to say that, yes, it is a function, with one singular output… but that that one singular output isn’t a number, but a single set of numbers.
The world is replete with times failure to keep track of all roots leads to bad answers. Context matters, and here is not negotiable.
The idea of a principal root should be as a second function applied to the square root function. Context should tell you if this is a valid construct.
Same logic for wrangling square roots of negative numbers. Context matters. Or coincident roots. The ± in the quadratic formula acts as a reminder that there are still two roots, even if they are coincident.
Programming languages don’t help. The vast majority of programming languages have desultory support for arithmetic let alone mathematics. I used to joke that the primary programming language of engineers is Matlab. The reality is that it isn’t a joke. And closely related, why Fortran is still relevant in high performance computing. A generation or two that regard the square root as single parameter function taking and returning a floating point value doesn’t help.
(All this from a previous life pedalling HPC to university researchers. Even the smartest have blind spots, and trivial errors like not watching the roots could cause much pain.)
Yup, context definitely matters. There are some contexts where the fact that any number has n different nth roots is vitally important. There are other contexts where only real numbers are relevant, or where only positive real numbers are relevant, and where therefore any number has only one or two nth roots. And neither of those contexts is wrong. The existence of multiple contexts makes it nigh impossible to make any sweeping, absolute statements.
Another example that you might appreciate: Physicists and other scientists never actually deal with numbers, as mathematicians generally understand them. The quantities that scientists deal with are all actually distributions of numbers. They can often be approximated as numbers with error bars on them, but in some cases even that approximation isn’t any good, and you need a full description of the error distribution. This fact is vitally important to doing good science, and yet, it’s never actually explicitly presented to scientists in our formal education.
Exactly. It depends what the number is used for, as numbers often represent things, or quantities. If a square room is 16 square meters, each side is 4m. It is possible to get the same area by going the opposite directions, but dimensions of a room is (usually) an absolute number so the negative root value is just an artifact of the calculation.
I’m positive that I never encountered the term “principle square root” in any of my math classes, an eon ago. If we wanted just the positive root we were to use the absolute symbol | n |. Is this something that developed from newer ways of teaching math in the interim?
Google Ngrams shows a gigantic rise from almost nothingness after 1980 (and an earlier short-lived blip around 1960) plus an almost equal drop by 2020.
