The meaning of the square root (√) symbol

I made this video because this misconception is so widespread. What were you taught about the square root symbol (√)? Let me know below.

Music:
LAKEY INSPIRED - Better Days

Outro:
"Lateralus" as performed by Sakis Strigas
Originally by Tool

My current equipment for making videos (affiliate links):

[HitchcockTheBirds]

I was taught a slightly different way:
the √(x²) is |x|, so if you want to solve x²=9 you do the following:
x² = 9
√(x²) = √9
|x| = 3
x = ±3

[huntermlp]

I was taught the square root by the definition and then that the square root of x^2 is the absolute value of x.

[katieblair]

I have master's in applied physics and have always treated the root as positive or negative. Having this explained to me by an actual mathematician was awesome! Definitely changed my perspective. 👍

[aryamankejriwal]

Another way to verify that the square root refers to only positive roots (and not both) is that the quadratic formula specifically uses plus-minus before a square root.

[jamieg]

While I 100% agree with what you're saying, the way you said it looked a little crazy. 😂

[tacostastegood]

4:43 this is what made this a good video. You took what people struggled with and explained why it wad the wrong notation. Many people say "it's jusy positive. Learn that." But because people don't understand why their method is wrong they can't let that go and will make the mistake again in the future.
Thanks for clearing this up. I don't know whether I didn't learn this or whether I simply forgot. 23 y/o engineering student looking up the definition of square roots lol

[estebanmartinez]

Just discovered your channel and it's awesome but something is bothering me... Do you flip the video while editing or have you learnt to write backwards?? xDDD

[perappelgren]

This is so good! The definition of the square root cannot be repeated too many times and there can never be too many warnings of the insane consequences of not applying the definition with rigor.

[saschabaer]

Well, imo there are two conflicting definitions of the symbol. There is √·: ℝ⁺ → ℝ⁺, the function you described, defined only on the positive reals. And then there is the multifunction √·: ℂ → 2^ℂ, defined on all of ℂ but returning a set of (typically) two values; that one’s the preimage of ·². It may also, thirdly, refer to a branch of the latter, in which case it is again a regular function on the complex numbers, but not continuous. It is annoying that all of these are used, but imo they all have their place.

[haihaut]

Hey so first big thank you for your Channel! I am a mechanical engineer and I'd love to learn more about math (more detailed than in my studies) . It's a real shame a lot of other students in my major see math as an obstacle rather than something fascinating (of course not all of them don't value math thank God). I would love to hear more about in depth topics in math (Topology, Abstract Algebra, Complex Analysis, Deduction and reasoning etc). Or a video where you recommend books for different math topics. Keep up the good work and thanks a lot for your entertaining and informative videos :D.

[AndrewKroll]

35 years ago, your explanation is exactly how I learned that.
The other strange way was probably made up by either people who want π=3 as a state law, or flat earthers. 😂

[prestondebetaz]

Great channel, been needing some great math humor. Best of luck in the future!!

[JakeFace]

got into an argument with my teacher about this. He decided that sqrt(x^2) was +/-x. I tried to argue that it was |x|

[konstantinkostov]

The tool ending is what every math channel needs

[altuber_athlete]

Hi! Can you confirm all of the following is correct?:

The square root "y" of a positive real number "x", is any number whose square is "x". And since a negative times a negative equals a positive, the positive real number "x" has two square roots. For example, 4 has two square roots: +2 ans -2, because (+2)^2 = 4 and (-2)^2 = 4.

However, when the symbol "√" is used, it only stands for the positive square root. So, "√4" doesn't actually mean the square rootS of 4, only the positive square root of 4.

[plaustrarius]

The better notation...? How does the 1/2 power feel about this?

A better question, what are you're thoughts on complex logarithms? That's an infinite number of values you would be ignoring, not just one.

[Edgawliet]

bro, did you have a glass wall between you and the camera?

[tricky]

What about sqrt of monads of numbers, can sqrt of the singleton number x be a singleton collection of two numbers, and by extension sqrt of a collection of numbers be a collection of two collections of numbers?

Sorry I don't know the proper terminology, I'm a haskell programmer so I have a creole of category theory at hand

[frozantini]

i love how you have Tool's Lateralus track in the outro

good vid

[JamalAhmadMalik]

At the start of the semester, I got into somewhat of an argument with my complex analysis instructor--she being inclined on believing that \root of 1= +\-1.
You saved my day ;)