Algebra teachers always want us to "rationalize the denominator", but why?

[@bprpmathbasics]

Why do we divide fractions this way? kzbin.info/www/bejne/i4rEi56brNaioNUsi=QPb8rPAJ73-XOIAs

[@deltalima6703]

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

[@bprpmathbasics]

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)

[@pikapower5723]

​@@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

[@deltalima6703]

@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.

[@kube572]

@@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.

[@aomacd7]

“I can do it, but I don’t want to do it” is the best lesson here

[@krozix23253]

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

[@williamhogrider4136]

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

[@BBQsquirrel]

As a math teacher, I always emphasise that a fraction is valid whether or not the denominator is rational. Having a rational denominator, however, allows for some further manipulations e.g. splitting a fraction to identify the real and imaginary parts of a complex number.

[@RobertXxx-uh6lr]

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.

[@IvyANguyen]

@@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.

[@kevinerose]

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.

[@TheDecagn]

@@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?

[@sumdumbmick]

complex vectors are not numbers, though. if you think they are, then you should also claim that 7 elephants is a number, or that 7x +3y is a number, but you do not. what this tells me is that you do not understand the words you're using, since they apply randomly. as such, you should reevaluate your conclusions about this topic.

[@jannegrey]

Important thing is to know when to do it in more complex math. Sometimes it's at the end to have a "better" answer. Sometimes you do it, because it helps you solve something in the middle of calculations - though just as often you don't do it, because then calculation might be easier. Multiplying things by 1 (so (sqrt(2)/sqrt(2)) in this case) is practically almost always allowed as far as I know.

[@akhilaryanfootball6181]

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

[@jannegrey]

@@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.

[@jannegrey]

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

[@bprpmathbasics]

@@jannegrey 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. 😃

[@jannegrey]

@@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 KZbin, not elsewhere.

[@johnwalker1058]

I always wondered why math teachers seemed so insistent that we "rationalize the denominator," but this made so much sense in explaining the "why" and not just the "because I'm the teacher and I said so" that is so pervasive in education. I wish I could like this more than once and thank you for making this video!

[@mjsteele42]

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.

[@tomaszadamowski]

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

[@doc0core]

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

[@Djorgal]

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.

[@HalfgildWynac]

@@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).

[@goes_by_santi3444]

I like this explanation better than what my algebra teach gave us way back in the day, which basically amounted to "irrational denominators are wrong". Missed opportunity to educate, or perhaps she actually didn't know herself. The real lesson here is that it's not enough to just know the rules, but it's also important to know why because then we can understand when and where the rules might not apply. Good stuff, and thank you.

[@Rot8erConeX]

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.

[@taquito5242]

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

[@boo-sd9ci]

Flag

[@samueljehanno]

@@taquito5242 Because that is logic

[@primeirrational]

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

[@stephenlesliebrown5959]

"When we talk about money it's easier, right?" Classic comment! Hard to imagine a pre-20th Century world without computers or calculators, but it did exist. Yet high precision was necessary for astronomy and other scientific stuff. Here's a similar idea: Give students the choice of working out one of two problems by hand. Either 456789/258637 or 456789-258637. They will probably prefer the subtraction. Then explain that the division problem can be solved by subtraction if there is a big book of base 10 logarithms in the library to use. (If asked why logarithms are still around in this century you can say they're handy for bringing an unknown exponent down to the level of the equal sign when solving an algebraic equation.) Best wishes to all 🙂

[@sender1496]

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).

[@analog_guy]

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.)

[@YonatanZunger]

Honestly, though, as a former theoretical physicist - we never did this, nor did we ever normalize "improper" fractions. For anything other than extracting decimal digits, it's far cleaner to leave them as-is. And if you need decimal digits, you can either do all of this, or learn to do fast approximate division in your head when all you need is a ballpark answer.

[@baptistebauer99]

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.

[@iyziejane]

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!"

[@silverhammer7779]

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.

[@silverhammer7779]

@@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.

[@eofirdavid]

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.

[@TheJakeSweede]

Thinking of it as sqrt(2)/2 also helped me with getting better at trigonometry and the unit circle, as the sine of many of the common angles 90, 60, 45, 30 (or radian equivalent pi/2, pi/3 etc) can be remembered as +sqrt(4)/2, +sqrt(3)/2, +sqrt(2)/2, +sqrt(1)/2, but then easily simplified

[@malvoliosf]

You missed my favorite thing about 1/√2 : a company was designing a new product and that number, 1/√2, kept showing up in the engineering calculations, so much so they decided to name the product the 707. It was so popular that the company, Boeing, now names the whole product line that way: the 707, 727, 737, 747, 777, 787.

[@ThiagoGlady]

why not 717?

[@malvoliosf]

@@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.

[@redfoxdeluxe697]

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.

[@mrcat5508]

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

[@malvoliosf]

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

[@JeffreyLByrd]

I rationalize denominators when applicable, but I must say, in this day in age where we carry powerful computers in our pockets, I don’t find “It’s hard to divide it by hand” to be a compelling argument for continuing the practice. When I was teaching and tutoring, my policy was always that the correct answer was correct regardless of form, but that starts to be a problem when you’re talking about integrals whose solutions can take wildly different forms based on how you handled the integration.

[@matthewmitchell3457]

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.

[@__christopher__]

In quantum information, I often used exactly that value (as well as other inverse square roots), and I never rationalized the denominator because what I really cared about was that it cancelled another square root of 2 when calculating the norm of a vector (in other words, I wanted the vector to be normalized). That fact would have been obscured by rationalizing the denominator. The bottom line is: There is rarely a universally best representation, only one that is best for a certain purpose.

[@marilynman]

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.

[@JHamron]

"square root of 2 is the most famous irrational number" > pi has entered the chat

[@fomori2]

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

[@pride7052]

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

[@areyamirinbruv]

​@@pride7052yes

[@nadjibam6384]

@@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...

[@conorpillay4315]

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

[@realdealsd]

You can use the difference of squares when your denominator is something in the form of a + √b. Multiply the numerator and denominator by a - √b because (a+√b)(a-√b) = a²-b.

[@MrDzsaszper]

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

[@keescanalfp5143]

​@@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 .

[@lanzji1345]

Fun fact: the teachers I learned that stuff from preferred 1/sqrt(2) over sqrt(2)/2 as sort of being more reduced ... impossible to reduce even more ... some justfication like this. I'm too old to remember exactly.

[@JayTemple]

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

[@CarmenLC]

√2/2 = 2/2√2 = 1/√2 = √2/2 …

[@gary-williams]

I always thought of the rules "rationalize the denominator" and "normalize improper fractions" as ways of "canonicalizing" a value. It's easier to grade assignments when students' answers are required to be in a particular form, for example, as the grader won't have to evaluate as many different expressions to check for equivalency.

[@doondoon859]

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. =) )

[@MikehMike01]

Nope

[@inyobill]

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.

[@KaiHenningsen]

@@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.

[@MyOneFiftiethOfADollar]

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

[@PhillipRhodes]

Ya know, I totally get the idea here, and can appreciate that in an algebra class or whatever it makes sense to require students to rationalize. But what annoys me is when some presenters treat it as though an answer like (1 / sqrt(2) ) is "wrong". As in, like, there's actually something mathematically incorrect about it. But there's not. It's just more awkward to work with... if you're doing the long division by hand. But approximately nobody (except teachers and poor, suffering, math students) does long division by hand. In the real world, anybody working with something that results in such an expression is eventually going to need the decimal approximation, and they're going to use a computer to work it out.

[@totally_not_a_bot]

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.

[@MikehMike01]

@@totally_not_a_botwhat a sad society

[@Harkmagic]

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

[@CouchPotator]

@ 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.

[@Sekla_]

i was searching about this for a long time!!! Thank you so much! Keep making videos your channel is so underrated :)

[@zzeroxxero]

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.

[@Wltrwllyngaeiou]

Write it as 2^-0.5 Don’t have to rationalize the denominator if there is no denominator

[@Harkmagic]

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

[@KaiHenningsen]

@@Harkmagic They're wrong, though.

[@HERKELMERKEL]

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

[@pi_xi]

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

[@carultch]

@@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".

[@jackkalver4644]

You can also calculate it as sqrt(1/2). It’s safe from decimal uncertainty. But when adding radicals, rationalization is helpful if not outright necessary.

[@mateush.]

he had a mental breakdown at 2:48 😂 amazing video!

[@Aristothink]

Thank you. I never knew why to rationalize the denominator. I know it's ugly to leave the denominator with a root and gets in the way if we want to add or subtract another fraction. But I have never thought about your explanation. Very good. Thank you for the marvelous video, as always very simple and informative. 👍👍👍👍👍

[@zerohz]

in pure mathematics it shouldn’t matter but in applied mathematics it is very convenient to rationalise

[@mtaur4113]

It's somewhat abstract, but interesting, that we can kick all the radicals back to the numerator, even when dividing by sqrt(2)+sqrt(3) and so on. This can possibly remove doubts about whether theoretical quantities are "nearly 0" or "exactly 0" when you wouldn't be so sure otherwise. The availability of computing power makes this less appealing in modern times, but what can you do.

[@gblargg]

Your humor looking at us through the camera always makes me laugh.

[@мультифора]

thanks a lot, i hate when you ask teachers "but why" and they just reply "thats the rule" or "thats how it is"

[@GNSD-h7k]

1:42 My Mind: Don't do it. Please! 😭

[@rosee430]

Thank you, sir. Not only was this very beneficial and made a lot of sense, but it was also very entertaining.

[@Shark-pj8in]

Always wondered why. Teachers never explained or told me.

[@the_mad_bunnyx9537]

In a world of calculators and solving equations by hand, having extra numbers in your expression is just an extra opportunity to make a mistake. Rationalizing numbers is irrational. You are much less likely to make a stupid mistake if you divide by (1/sqrt(2)) then if you divide by (sqrt(2)/2), for example. The same is true if you square it, or plug it in a calculator. None of these are hard either way, but if you do enough calculations or manipulations of equations it is only a matter of time.

[@Nikioko]

Pretty easy. What is 1/√2 + 1/√3? And what is √2/2 + √3/3?

[@january1may]

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.)

[@AllenKnutson]

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

[@Nikioko]

@@AllenKnutson And how much is that in decimals?

[@integer9590]

@@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..

[@Quantris]

@@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

[@crimfan]

Interesting putting this in terms of long division. Most of the time it's a lot clearer to have a rationalized denominator. There are some circumstances where it's easier to read or work with a square root in the denominator or other notational sins such as improper fractions, so this is one of those "sometimes honored in the breach rather than the observance" situations. For example, writing a function like f(x) = log(x-3)/sqrt(5-x) is a lot clearer than rationalizing it. That said, I encourage folks working with formulas to try variations to see what's clearer to write and read.

[@feedbackzaloop]

1/sqrt(2) is maybe the best example when rationalizing the denominator conflicts normalizing the fraction. Feels like we need an addition to PEMDAS in order to settle the conflict.

[@anthonyn.7379]

2:48 reminds me of when that one kid who's been told to stop talking five times already and the teacher had had enough 😂

[@birneytitus4785]

One historical reason for rationalizing the denominator may be that hand calculations are easier that way. For example, it's easier to divide 2 into sqrt(2) than to divide sqrt(2) into one.

[@lesnyk255]

All these years I've known about the insistence that we rationalize the denominator; and I've known that the longhand division was so much easier that way - but it never occurred to me that that was the very reason for the insistence! I solemnly promise never again to put my pen down until my denominators have been rationalized.

[@UnconventionalReasoning]

The other benefit with rationalizing the denominator is with adding fractions, which requires a common denominator. Having a "2" instead of "sqrt(2)" in the denominator makes it much easier to find the LCD.

[@Giannhs_Kwnstantellos]

(I 'making my guess, without having watched the video) because it's easier to calculate an aproach of the number (decimal form)

[@ASChambers]

I always say to my pupils, “When you can prove to me that you can successfully divide a pizza by root two, then you can stop rationalising the denominator…”

[@boltstrike2787]

You're forgetting that you cut off the decimal to 5 places in the first place for simplicity. Really you're looking at a divisor with an infinite number of decimal places and you would have to move the decimal infinity times to do that division. So it's not even that you wouldn't want to do it, you straight up CAN'T do it.

[@jimmymiller6068]

A more significant mathematical reason for why rationalization of a denominator is important is showing for example that 1/sqrt(2) is in the field Q(sqrt(2)). By definition sqrt(2) must be multiplicatively invertible if Q(sqrt(2)) is to be a field. Since 1/sqrt(2)=sqrt(2)/2=(1/2)*sqrt(2) we know that 1/sqrt(2) is in Q(sqrt(2)) by closure of multiplication.

[@jacoblitman4866]

A teacher could likely do it as a pair of math problems. "Question 1: Perform long division to solve 1/sqrt(2). Question 2: Perform long division to solve sqrt(2)/2."

[@Commenter3726]

When i see a multiplication or division of a square number, i always try to convert the numbers to a square root itself so √2/2 =√2/√4 and we know from the rule √a/√b=√a/b if a>0 and b>0 We can just put in √2/√4=√2/4 which is square root of 1/2

[@bryantaylor2523]

I thought i knew where you were leading us when you started doing the long division and moved the decimal, but then you just said because this division of large numbers is hard. That's definitely part of it, but i think maybe more important is the fact that the number is actually infinitely long so you'd need to add infinite decimal places to the numerator to even start which isn't possible. If you're using a rounded approximation for the irrational denominator then want a couple more decimal points of precision later, you basically have to start over (and with an even harder calculation this time). But if it's the numerator that's irrational instead and you want more precision in the answer, you can just extend your rounded numerator and continue the long division from where you previously stopped.

[@bayleev7494]

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).

[@taflo1981]

Is this something that's being taught differently in different countries? I went to school and university in Germany and am now teaching math at a university in Austria. I have never seen an instruction requiring you to "rationalize the denominator". Personally, I find 1/√2 to be more elegant than √2/2, but I would never require my students to use a specific style.

[@welcomb]

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

[@aperiodique4333]

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.

[@emryswilliams9190]

The US just forgot that calculators were invented.

[@Camarelli]

I used the rationalized form to memorize the sine & cosine values of the 3 key angles π/3, π/4 and π/6. So I knew the set of values was √1/2, √2/2 and √3/2 and then I picked up the value using logic, by drawing the trigonometric graph.

[@waynemv]

If one is going to accept an approximation in the end anyway, why not just start by using a rational approximation for the square root of two in the first place? We know 9801/4900 is very close to 2 (only off by 1/4900), and its square root is 99/70. If an even closer rational approximation is needed, it can be found using mediants. Decimals suck. Fractions are where it's at.

[@fatos-lite]

Thank you so much Sir ! I always wondered why and as it turns out, it does have a sense.

[@OleJoe]

Ok, here's the reason. When using a slide rule, you set 2 on the left side of the A scale and read its square root on the D scale. Once the square root is on the D scale draw the two on the C scale above the square root on the D and read the answer at the index.

[@georgesbv1]

also multiplying with conjugates. Usually done for square roots, but also for cubic roots

[@KaiHenningsen]

As far as I remember, "rationalize the denominator" never came up even once all through school and university. I think I see why that might be a practical tip if we need to get the result without using a calculator ... but how many people even know how to calculate sqrt(2) without a calculator? (I once did that in my head during a recession in high school. Once. Never before, never after. It's not very practical if it never comes up in practice, now, is it?)

[@spookyindeed]

Interestingly, in more theoretical classes, I often rationalize the numerator to find bounds on things, I don't think I have rationalized a denominator since like calc 1

[@ffggddss]

"... irrational, which means there's no pattern ..." Actually, there are decimals with patterns that are irrational. What makes an infinite decimal (or any fixed base, for that matter!) rational is having some finite position from which some finite string of digits repeats infinitely. Any other pattern, as well as absence of any pattern, makes it irrational. So your statement is correct, provided "pattern" is defined as above. And I'm with your thumbnail, on the "irrationality" of insisting on always rationalizing denominators. "1/√2" is a perfectly fine answer to some questions. Like, sin 45º = ? Or tan 30º = 1/√3 But sure, if you want to compute its decimal form, it's much easier to rationalize first. But for 1/√2 (or with other integers in place of "2"), there's a fairly simple way of generating "best" rational approximations, using continued fractions/Pell's equation solutions. [Any written decimal expansion of an irrational number is necessarily finite, so we're merely trading one approximation for another.] And although that goes deeper than you'd want to take a class of elementary algebra students, it can be thought-provoking to just lay out the easy iteration process and let those who are still curious about it, explore the reasons it works... b a b/a -- -- ---- 1 0 undef 1 1 1.00000 3 2 1.50000 7 5 1.40000 17 12 1.41667 41 29 1.41379 99 70 1.41429 . . . . . . . . . . Fred

[@ffggddss]

So for √2, you start with (b, a) = (1, 0). Then: Add b+a to get the next a. Add the previous a to the new one to get the new b. Repeat until you can't take any more. Or program a spreadsheet (Excel or Numbers) to do it. For other square roots, the rules change, but are very similar to these.

[@hith2re]

This guy is the goat of maths 🐐

[@MrTkwbear]

Thanks! I always hated doing this but now it finally makes sense.

[@dimBulb5]

Thanks! I've wondered about this many times. A teacher probably explained it years ago, but I forgot.

[@SiqueScarface]

As an computer scientist, you just scream "Numerical stability!"

[@douglasmagowan2709]

When you are in algebra class you must rationalize the denominator, and there are many good reasons for it. By the time you get to calculus, you can write 1/√2 and leave it like that. It is assumed that you know how to rationalize the denominator and so does your reader.

[@KaiHenningsen]

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

[@prathmeshgaonkar8026]

You teach better than any of the teachers i had

[@glasssmirror2314]

Thank you sir. Now I know why we should rationalize the denominator.

[@knurlgnar24]

As an engineering student I was never told to rationalize the denominator and sqrt(2) shows up everywhere in engineering. Things were also never written in decimal form if they were irrational. This seems like silly pedantry. Is the answer correct? Did you get it in the allotted time? Did you show how you got the answer? Good. It's a waste of time doing extra steps that can only result in more errors.

[@jimmonroe5193]

Rationalizing the denominator is a throwback to the days of tables and slide rules. If you're going to generate a decimal approximation of an irrational number, that's what a calculator is for.

[@husseinahme3484]

Thank you for explaining why we rationalize I’ve always wondered

[@elladunham9232]

There’s a similar logic as to why when doing fractions, the denominator should not be a complex number. Dividing by a complex number is extremely unintuitive but a complex number divided by a real number is easy. So we use a similar method of multiplying the top and bottom by the conjugate to get a real number in the denominator.

[@WerewolfLord]

I'll rationalise anything else except 1/sqrt2 when using it as sin or cos of π/4. Also you need to un-rationalise the denominator when getting the negative solution to x²-x-1=0 (at least if you want to see it as 1/φ).

[@mitchratka3661]

If rationalizing is trying to remove an infinitely repeating number with no pattern, then how does dividing it by 2 rationalize it? I'm in my 2nd year of college engineering, so I've obviously used it a lot, but I've always just considered it to be writing the fraction in an similar form to the other angles in radians (i.e. sqrt(1)/2, sqrt(2)/2, sqrt(3)/2, ...). Wouldn't dividing the number still include the infinitely long decimal? And how would you even go about proving that? I just started my discrete mathematics class and am doing basic proofs, so I thought I could still get something out of this.

[@areadenial2343]

Yes, but as the title says, we only care about rationalizing *the denominator*. The numerator can be as irrational as you'd like.

[@mitchratka3661]

@@areadenial2343 that makes sense, I guess I really didn't think too hard about the fact that it's the denominator. But that means the answer is still considered irrational right? So is there actually a reason why we care about rationalizing it, also why the numerator and denominator have different standards?

[@ConManAU]

It’s mostly a convention, but it also makes some operations simpler. For example if you have 1/(sqrt(3)+sqrt(2)) then there are a lot of situations where writing it as sqrt(3)-sqrt(2) is going to be more useful.

[@cbunix23]

@@mitchratka3661Rationalizing the denominator is just a holdover from the days of hand calculation. If you're doing calculations with a computer, it doesn't really matter numerically. And like ConManAU said, it does make some operations clearer if you care about that.

[@knurlgnar24]

I was in engineering school quite a while ago and never did anyone bat an eye about irrational numbers in the denominator. If you don't need to rationalize the denominator to simplify the math then don't. If you do then do. If you have an engineering professor that demands such idle pedantry then, well, I'm not sure what to say that won't sound insulting.

[@akshatpratapsingh5476]

thanks for the beautiful explanation

[@baptistebauer99]

In my classes I usually give the following reason: even though a fraction is valid regardless of how you write it, we're still trying to do something with it. Numbers don't exist in a vacuum in a high school math class. We might need to add or multiply these fractions later on, and particularly the "adding two fractions" can become very confusing. We need to have the same denominators, how are we doing that if the denominators have a bunch of square roots? Students, don't make your life more complicated than it needs to be: deal with integers as you have done your whole life, and do it by rationalizing the denominator. Otherwise it's a recipe for disaster.

[@hermannschaefer4777]

Well, hmmm.. I mean, it's not wrong to say, that it's easier to calculate sqrt(3) / 5 than 1 / 5*sqrt(3). On the other hand, it also depends on what you want to see/achieve/do next. Esp. a 1 divided by something can be nice in calculus/algebra, but it really depends on what you want. If, - IF - there is a dedicated rule to make it that way (because it's still part of the learning matter in lower classes) - OK. But as a general rule? Nope.

[@misterroboto1]

I thought the reason why teachers insist that you rationalize square roots at the denominator is to prepare students for when they'll have to deal with complex numbers in fractions.

[@Cas-Se78.97]

I always assumed the point was just to have one consistent standard, so that you don't have to spend the extra time converting to realize that 1/√2 = √2/2 = √(1/2) or that √(3/2) = 3/√6 = √6/2

[@Sg190th]

Even in Trig, that's the preferred coordinate when going 45 degrees or pi/4

[@deltalima6703]

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

[@Sg190th]

@@deltalima6703 Mhm. All multiples of 45.

[@stephenbeck7222]

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.

[@abc20100712]

That is exactly what I have been telling my students! However, I cannot really justify why we still need to do that in this calculator era. Any suggestions would be appreciated 😢

[@DeJay7]

If the reason only is for faster calculations by hand, then it's still very often preferred not to rationalise the denominator. If I have 1/sqrt(x) and I know (or suspect) that on the next step I will raise to a square, then if I rationalised it to sqrt(x)/x then I would get [sqrt(x)/x]^2 = x/x^2 = 1/x. If I didn't rationalise the denominator I would immediately get [1/sqrt(x)]^2 = 1/x. There's lots of examples like this. My opinion is that we should rationalise the denominator ONLY in our final answers, not in values in between, and even then maybe.

[@GamingWithUncleJon]

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

[@yeetdragon2413]

how do i I learn the art of holding and using two markers are once

[@AlRoderick]

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

[@deep24543542]

Rationale the denominator...1/sqrt(2) = sin(pi/4) take it or leave it.

[@NikkiTheViolist]

yeah that checks out

[@benbencom]

It's valuable to see things like this because as a kid you are taught to rationalize the denominator but you don't know why. Now, as an adult, you find out it is to simplify long division, which you never do. So the whole thing was a waste of time but it's still in textbooks because it was handy in the 1970s

[@Muhahahahaz]

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

[@GEMSofGOD_com]

I been in top Oxfords of the world, actually. Never heard this emphasized, ever. 1/sqrt(2) is perfectly 101% fine.

[@davidwitte8469]

Bro is flexing on us with the Expo markers.

[@zevfarkas5120]

Thanks for an entertaining explanation. This was a real time-saver back when calculators either didn't exist, or weren't allowed in exams (yeah, I go back that far...). Or you could have used a slide rule (if you don't need more than about 3 significant figures). We are definitely spoiled, with our calculators that divide by numbers like 1.414213 without complaining. ;)

[@gabrielcohen1538]

was furious at this not too long ago. thank you

[@TheGolux]

Does rationalizing the denominator still apply for other irrational numbers like pi or e that are less convenient to manipulate?

[@CouchPotator]

It depends on what your teacher wants. In the real world, anyone that deals with this kind of math would rather have 1/sqrt(2) as the answer.

[@zanti4132]

Denominators with transcendental numbers like pi and e can't be rationalized. If a transcendental number ends up in a the denominator, it has to stay there.

[@Kandralla]

If it's not clear from the rest of the comments, no one worries about this outside of primary/secondary school. The majority of people teaching it are likely not even aware of why it was even done in the first place. It's the mathematical equivalent of stone pineapples on pillars at the end of rich people's driveways, they have. You'll get the same answer either way.

[@zanti4132]

@@Kandralla I don't know.... there's something to be said for putting the number in a format that simplifies calculation. You can argue that 2/3 is the same as 74/111, so why simply 74/111 to 2/3. However, it is certainly true that 2/3 is easier to work with, and it just looks nicer.

[@DriverDad58]

First, love your videos! Is the long division easier if you have something like 1/(1-sqrt(3)) vs (1+sqrt(3))/-2? Or if it's complex? We still ask students to rationalize the denominator but good luck with the long division :) Seems it's just "the way it's done". Like mixed numbers vs improper fractions (although try to put an improper fraction on a blueprint or in a recipe and watch folks get very confused). In both of these cases, many classes stop making students do this when they reach a certain level of math. In AP Calc, they don't even have to simplify most answers after applying product, quotient, or chain rules.

[@totally_not_a_bot]

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.

[@DriverDad58]

@@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.

[@iorguemaxwell]

In other words, this business of continuing to rationalize is a legacy of the time when there were no calculators, or they were very expensive, and people needed to do calculations on paper. We need to update Mathematics teaching, changing the focus from "doing math" to "understanding" the math we are doing.

[@pavlopanasiuk7297]

I have an opposite question. I told my students how to rationalize fractions, and followed with mentioning I don't care about the form for your answer, for as long as it is simple and accurate. Rationale being no one in sciences calculates 1/sqrt(2) anymore, and we do that with calculator anyway. So am I bad or good?

[@carultch]

Even with modern computers, it makes your program more efficient to run, if you keep your denominators rational where possible. So there still is value in rationalizing the denominator. If you have to run thousands, if not millions of the same operation, the extra time to divide by sqrt(2) instead of multiply by it, can add up. It also makes your calculation less subject to the whims of rounding errors and floating point inaccuracies. All of this can be solved with more computational power, but it can be more cost-effectively solved with more efficient programming.

[@MrKalerender]

I'm going to rebut this and say this is not a good reason for rationalising denominators. Nothing you gave here gave any benefit to students except when you are trying to reduce fractions to decimal approximations by hand. So if your exam has some weird esoteric condition where you are both dealing with irrational numbers, AND you know approximate decimal values of those irrationals, AND you have to reduce all fractions to decimals, AND you can't use a calculator... then sure your reason is valid. But in my uni exams where we were not allowed calculators, fractional form was acceptable. The convention was to rationalise so I did - because that was the instructors 'norm' - but that didn't mean it was the 'right' thing to do. For the exams I assign students they have calculators, so all that forcing students to give decimal approximations of fractions would tell me is that they can work their calculator properly for this rather basic skill (not something I'm assessing in high school, primary sure). Sounds like you're looking for a problem to assign a solution to here.

[@matambale]

I still find it bizarre that 1/sqrt(2) is the same as one half of the sqrt(2)

[@RST]

I remember in trigonometry when I had to write √2/2 for sin(45°) and stuff I'd just write 2 to the -1/2 instead. I think it might have been because of this, or it might have also been me finding a funny way to represent inverse numbers. I don't remember.

[@danielamaya5241]

finally someone answering the real questions

[@antoniusnies-komponistpian2172]

Can any algebraic number be written in a form of a fraction where the numerator is not a fraction and the denominator is a natural number? 🤔

[@Vegeta-dn6lk]

Teacher : beacuse i say so! Random asian 20 years later:

[@kyokajiro1808]

the number with n "0"s in between the nth natural number all places after a decimal place makes an irrational number that has a pattern

[@Got-it747]

So much better❤

[@cbunix23]

You said irrationals don't have patterns, but that's not quite right. Irrationals can have non-repeating patterns, such as Liouville numbers.

[@Magmagan]

But why? I can't remember the last time I calculated √2/2 by hand.

[@MyOneFiftiethOfADollar]

From a standpoint of estimation, 1/sqrt(2) is not as easy to estimate, at glance, as sqrt(2)/2 (a number between 1 and 2). The instructions "rationalize the denominator" has exacted unknowable misery on teachers and students alike. "integerize the denominator" is a better description of what transpires in these "simplification" problems.

[@derekschmidt5705]

I feel like this is more an argument for _naturalizing_ the denominator than only rationalizing it.

[@deva-1711]

1 upon root 2 is actually sin 45 and cos 45 in respect to 45-90-45 triangle

[@elyaz9]

I've been watching this channel for a while now and I just realized he has enough markers to last him for the rest of his life 😅

[@Djorgal]

I'm not really convinced by this explanation that boils to down that it makes it easier to calculate an approximation. To me, a better reason 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.