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

Why do we divide fractions this way? ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-UYaZkewqojs.htmlsi=QPb8rPAJ73-XOIAs

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

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.

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.

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!

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.

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.

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.

"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 🙂

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

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.

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

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.

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.

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. 👍👍👍👍👍

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.

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.

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.

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

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

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.

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

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

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.

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.

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

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.

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

Always wondered why. Teachers never explained or told me.

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."

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.

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

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.

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

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.

thanks for the beautiful explanation

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

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.

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

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

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

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

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.

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.

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

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

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.

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.

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

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.

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 😢

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

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

This guy is the goat of maths 🐐

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.

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.

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.

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

So much better❤

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

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.

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. ;)

You teach better than any of the teachers i had

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

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?)

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.

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…”

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.

was furious at this not too long ago. thank you

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

1:42 My Mind: Don't do it. Please! 😭

finally someone answering the real questions

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.

Bro is flexing on us with the Expo markers.

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

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).

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.

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

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 😅

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 😂

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.

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

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.

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.

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.

i prefer √½ personally. The division is simple, and the square root is probably equally hard to do by hand no matter what

I prefer 1/√2 in general, simply because it uses slightly less ink. Saves time and the planet!

That was excellent

You are right, but there is more to the topic. Rationalizing the denominator is a concept from before calculators became common. People would look up square roots in a big table of roots. And then they would do long division. In such cases rationalizing the denominator makes this process much easier. However, no one is using big look-up tables to find the value of a square root anymore. They will just plug it in a calculator. So it is ridiculous to expect people to use calculators to calculate the square root but then do long division by hand for the rest. It is an antiquated custom that people still follow because our ancestors did it. And no one will stand up to reevaluate whether it is still useful in our current society.

"... 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

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

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.

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.

Good explanation, but 1/sqrt(2) is also a very! famous irrational number in engineering. Ok, in other subjects or using other squareroots I fully agree.

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.

Hello my name is Piter and I am from Brazil ...you're awesome teacher.This is the first video that i watch

Good question

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?

when the denominator is as complicated as sqrt(2) then simplifying makes sense, but when we get answers like 10/4, then i don't agree with insisting on simplification. 5/2 is not easier to work with

Well done.