I have a question, maybe its important, maybe not. Why is tanh(π^e)=1?

It’s not exactly equal to 1. tanh has a horizontal asymptote at y=1 so when the input is “big enough”, the result will be “like 1”. Try tanh(100)

​@@bprpmathbasics so i dropped out of school after completing middle school but i tryed home schooling as far as i could tell most algebra using letters instead of numbers didnt make sense because we have infinite numbers so there would always be a number to use so there was no reason to represent with letters so why use letters instead of numbers i think it just makes it more confusing

@pikapower5723 you would say "x is an element of real numbers" then you know that x is some real number(s) and x just becomes a placeholder for that. If you said "1 is an element of real numbers" and used 1 as your variable it would quickly get mixed up with all the other numbers and lead to errors. You could use (1) but that is already taken as a short form of multiplication, {1} is used for nested brackets, so same problem. You would be left with $1 or something, which computer programming does actually use. TLDR: there is no good reason really, written letters are used, then greek letters when those are used up. Historically its because paper was very valuable and anything to shorten equations saved money.

@@pikapower5723 Lets say the letter is x. If we solve an equation using x to represent something else, we've effectively now solved the equation for all possible values of x. So you can essentially solve the equation as you would with a regular number but by maintaining it as a variable, we can do useful things such as graphing it to tell us how the answer of the equation changes as we change the initial number. There's obviously other reasons in more complicated math, but that's really sort of what it boils down to - we can do interesting things with variables that we can't really do with numbers, regardless of whether or not solving one specific case is harder/easier.

Wait no reply till now? 😮 Let me do it - Yes

Why does is feel similar to " If you're good at something, never do it for free " 😂

That is just optimization trick to get result as a number thus dictated by a CPU execution algorithm and numerical error analysis. From the idea point of view it is nonsense to enforce anyone for converting from one compact form to another equivalent.

@@RobertXxx-uh6lr I used to get docked a point for leaving sqrt(8) as that instead of changing it to 2sqrt(2). The latter is longer! sqrt(8) is usually just 2 characters on paper whereas 2sqrt(2) has 1 more character.

It's all about being neat and professional. Suppose you are doing a calculation and you "estimate" that the answer should be around 2.8. Well, it is more difficult to gage if your solution is correct base on sqrt(8) rather than 2(sqrt[2])! When you have it nice an neatly defined, then you can have more confidence in your solution. It is a basic example, but you would want to be able to look at your solution with a glance to see if it is in the ballpark figure of what you are expecting to see.

​@@kevinerose That is very subjective. If you can't realize 2√2 and √8 are the same thing then you shouldn't even be doing what you're doing with these numbers.

@@xRafael507 think a bit more ahead of what he's talking about, you don't immediately realise the simplication of a bigger example such as √180, much easier to work with 6√5 no?

Bro how is your comment two weeks ago it's only been 9 min💀💀

@@akhilaryanfootball6181 bprp often re-releases his videos or gives links to unreleased ones. Which means you can comment earlier than the video has "officially" been up.

@@tdj461 various. But it can even be a link on another YT video - since I'm certain I didn't use twitter or reddit.

@@jannegrey593 No. I made some unlisted videos last month and put them in a public playlist. You can still find some if you go through my algebra basics playlists. 😃

@@bprpmathbasics From memory I followed the train of your pinned links, but I might have also gone with playlist. Point is, I found it on KZfaq, not elsewhere.

I've taught high school math for more than 20 years. Honestly, at the beginning of my career, my answer to "why do we rationalize the deniminator?" was "because it's one of the rules of math." That was my answer back then because, at the time, I never had anyone provide me a legitimate explanation when I was a student. Somewhere along the way, I learned the reason provided in this video and have ALWAYS explained it any time I've taught this concept since. So, I would submit that it's possible that it wasn't "I'm the teacher and I told you so" and instead it was a lack in the teacher's own understanding.

It's always "because learning it now with easy stuff simplifies things later on".

Because most teachers don't understand the stuff they are teaching.

I'm not really convinced by this explanation that boils to down that it makes it easier to calculate an approximation. To me, a much better explanation is the unicity of the answer. For example, it's far from immediately obvious that 1/(1+√2) and √(3-2√2) are equal. That's why we want a unified and unique way to simplify them, and if we follow the rules of simplification, they both are -1+√2.

@@Djorgal Yeah, that's what I have always thought. No way is inherently better if you get the answers right but having one "standard" form lets you see what your fraction is and maybe immediately spot the numbers that are in fact the same (and check whether your number matches the one in the textbook).

And I supposed when asked with the follow-up question "why not just have a book with all fractions", you could argue that it would require 2 degrees of freedom as opposed to a base-10 log, requiring 1 degree of freedom (even including reverse operations).

A book! Groucho said, "Outside of a dog, a book is man's best friend. Inside of a dog, it is too dark to read". And if you are an old-timey engineer, you just use a slide rule! (It has the logarithms already built in, but it doesn't work very well inside of a dog either.) Three or four digits of precision are more than you need anyway, said the engineer. 🙂 (Some physicist has already slaved to do the hard part such as working out a constant of nature to something like twelve decimal places.)

I have an undergraduate degree in physics and experience in teaching math to middle and high school students. In my math classes, I would ask my students to rationalize the denominator and give as an argument "if you need to add them to something else later on, it's a lot easier that way". However in quantum mechanics I realized we would specifically avoid normalizing the 1/√2 fractions - and it makes sense: these fractions usually pop up as coefficients for some quantum entanglement process, and to find the total probability we need to add their squares. It turns out that (1/√2)² + (1/√2)² can be done immediately, this trivially becomes 1/2 + 1/2 which is equal to 1: checking for total probabilities is extremely easy that way. In general we can just add whatever numbers are under the roots in the denominator and it's far easier to check if everything adds up to 1.

As a physics grad student, I worked on a project that involved a quantum system of dimension 2^9 = 512, so the Hamiltonian matrix was 512 x 512. The professor I worked with had a found a result that was easy to check numerically on matrices of this size, but 512 x 512 is too large to calculate by hand. Using symbolic algebra I was able to crunch it, to show that the professor's numerical result of 0.62132... was exactly equal to (3/2)(sqrt{2} - 1). The humorous professor looked at my expression and said "this is worse than before, I can't even tell what number this is!"

As an engineer, I can tell you with dead certainty that, for anything we do in the Real World, it doesn't make any difference. And there are plenty of engineering calculations that have radicals in the denominator.

@@iyziejaneExactly. AFAIC, you haven't solved a problem until you have reduced it to a hard number that has Real World significance. As they say in constructive mathematics, once you have proven the existence of a number, you should be able to show how to find the number. A radical expression is how to find the number; it isn't the number itself.

This normalization is important in mathematics, since it tells you that 1/sqrt(2) is in the rational vector space spanned by sqrt(2). Or even better, you can show that the numbers a+b*sqrt(2) where a,b are rational is a field, which is very useful in number theory. That being said, normalizing such numbers just to compute their decimal digits seems like a waste of time. I hope that this is mostly a quick exercise before moving on to more important stuff.

why not 717?

@@ThiagoGlady There was (and is) a 717 but it has not been produced in great numbers. I think there are about 200 of them flying. The 757 and 767 exist and are fairly popular. There is no 797, as far as I now, but I am sure Boeing is working on it.

Always loved the Math Idea behind the naming, but unfortunately it's not true. It's 700 because that's the Boeing reference number for Jet aircraft. And then Marketing decided Seven-Oh-Seven just sounded better.

@@redfoxdeluxe697yeah that’s what I heard before. Never heard this explanation and honestly one root two doesn’t look much like 707

@@mrcat5508 0.70710 does not look like 707 to you?

Actually the reason I learned was because of computing power; it's easier for a computer to use a rational denominator just like when doing it by hand. Of course, it's definitely possible for a computer to work with an irrational denominator, and with the speed of modern computers it won't make an iota of difference unless you're writing a program that does tens of thousands of divisions with square roots. 2 years into a computer science degree, I still haven't had to rationalize a denominator yet. Maybe it mattered with those old computers in the 1940s and so rationalizing the denominator became a relevant skill, so they taught it in high school and then they just never bothered to remove it from the curriculum.

My go-to math moment of "knowing the rule" vs. "knowing why the rule exists" is PEMDAS. Once you know *why* higher-order operations are always done first, it allows you to internally rationalize where things like permutation, computation, or factorials would go in the order of operations.

im curious, why is pemdas the​ way it is? @Rot8erConeX

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@@taquito5242 Because that is logic

@@Rot8erConeXexactly, I never used PEMDAS since what to use first just felt very natural.

I would say that I prefer to cancel numerators and denominators first and then go with the rationalization for the final answer. The same goes for rounding, always go for fractions until reaching the final answer, values are more precise. Additionally because even computers are bad are doing division, rationalizing also saves computing time and has less numerical noise.

((2)^1/2) < pi .... or .... square root of 2 eats pie! Therefore, square root of 2 is now more famous.

​@@fomori2wait... So if i eat the mona lisa will i be more famous than it

​@@pride7052yes

@@pride7052 if you eat it, the next generations would never get the chance to see it. Whereas for you, they can pay a visit to the prison...

He said it was the most famous square root number because it was irrational, not the most famous irrational number

I get what you're saying. If 2/4 isn't in simplest terms, then neither is sqrt(2)/2.

√2/2 = 2/2√2 = 1/√2 = √2/2 …

Math teacher here! "Canonicalizing" a value is precisely the reason I would give my students, except I would say that this is convenient for the STUDENTS instead of the teachers. (I don't want to make students feel like teachers ask them to do that because teachers are too lazy to check. =) )

Nope

The student should be able to challenge any answer masked as "wrong". Teaching young Mathlings that there is only one "correct" answer does not promote understanding. If a teacher can't tell that 1/sqrt(2) = sqrt(2)/2, they probably shouldn't be teaching anything beyond arithmetic.

@@inyobill I'm accustomed to the idea that when a student does an enormous derivation/proof/whatever, and it goes wrong because there was a sign error near the beginning, a math teacher will check all the steps and still give most points, just pointing to the one sign error why it wasn't all points.

understood, but that is tantamount to saying "making things easier on grader(lazy teacher)" is more important than student comprehension.

There are people out there who would say that is wrong because the radical means something different from a 1/2 power.

@@Harkmagic They're wrong, though.

yes indeed we can use powers for every irrational number.. no need to user "sqrt" symbol .. soi who invented that useless symbol ?

@HERKELMERKEL The square root is only a positive number while the power to 0.5 has two solutions.

@@pi_xi How does 2^0.5 have two solutions, while sqrt(2) only has one solution? They both only have one solution, according to every calculator I've ever used. If you want 2 solutions to the equivalent concept, you'd have to write it indirectly as "given x^2 = 2, solve for x".

I also explain to my students that it is easier to comprehend conceptually as a standard. If you split 1 thing between radical two parts, it is harder to comprehend compared to a radical 2 amount split into two parts.

as long as you do not multiply the numerator and denominator by 0, of course ;)

​@@MrDzsaszper, yeah, we both should not forget to add the strict condition b ≠ a² . in the case of b = a² , the denominator would simply be equal to a + √b = a + √(a²) = a + |a| either = 2a , if a > 0 , or = 0 , if a < 0 . so yet a quite dangerous case for the denominator .

This is why my calc professor doesn't care. You calculator is handling it anyway, so whatever.

However, if the computer uses floating point (which is what they usually do), then the computer's result will be more accurate (i.e. have more correct digits) if you rationalize.

@@totally_not_a_botwhat a sad society

​@@MikehMike01unless you're plan to teach kids the square root algorithm the problem will always be solved by a calculator.

@ No, for calculators, computers use decimal data types. And when they do use floats, 64 bits of precision is way more than enough to accurately position subatomic particles on the scale of the universe. In other words, it doesn't matter.

If the calculator does the operation 1 or 2 times, sure it is fast. Try to make the same operation for 10^1000 times and you will start to see the difference, the computation time will stack because the potential operations needed for division are greater. It also helps to reduce numerical noise as the computer has a limited number of digits it can store and in each iteration it can potentially lose information.

Yeah, wanted to come in with the same idea - once you start adding together expressions with radicals in them, especially if they're _different_ radicals, it's a lot more convenient if you've rationalized the denominators first. (In actual math textbooks, I've seen some similar problems that worked on the same idea but were more complicated - but "1/sqrt(2)+1/sqrt(3)" is probably the easiest option where this explanation comes up and I literally thought of this exact addition as well.)

It's (√2 + √3) / √6, of course. What was this supposed to prove.

@@AllenKnutson And how much is that in decimals?

@@Nikioko I don't see the point tbh. Just grab a calculator...? Idk how it is for you guys, but my teacher NEVER asks us for a decimal answer, soo..

@@Nikiokoif you want it in decimals then don't bother wasting time simplifying the fraction....just punch 1/√2 + 1/√3 in to your calculator from the beginnning

For dividing (1+sqrt(3))/-2, I'd factor out the negative so you have -((sqrt(3)-1)/2) 1.732-1 is easy, 0.732/2 is easy, then negate it for -0.366 However, I'd normally just plug it into a calculator and call it a day. Rationalizing is great to know, sometimes handy, usually a waste of time.

@@totally_not_a_bot Nice way to do it! Maybe there's an easy way to do it with complex numbers. I'll have to think about that a bit.

It’s not the only reason, just one example It also helps to have a standard form for your answer, otherwise you can’t compare different results. Depending on the context, other standard forms might be used instead. But in pure mathematics, where we are not applying our calculations to a specific science, the rationalized form is the standard we choose

That's for values of "many" approaching zero.

mhm! computational complexity increases much faster with sqrt(2) in the denominator (i think it's something like O(n²) compared to O(n) where n is the number of digits).

Exactly. Same here. Something must be wrong with US math

I had all my schooling and then 5 years of physics at an university in France. This rationalizing the denominator business doesn't ring the tiniest bell.

The US just forgot that calculators were invented.

yeah that checks out

Interesting. Could go 45° the other direction too. (Counterclockwise instead of clockwise)

@@deltalima6703 Mhm. All multiples of 45.

I don’t think there is a universal preference in trig for 1/sqrt(2) or sqrt(2)/2. You can find textbooks using either.

It's not a "rule" for intermediary calculation. You wouldn't want to do a decimal approximation until the algebra was done anyway.

Believe it or not he actually did a video about how he does it on his main channel a few years ago.