Why do you rationalize the denominator? (here's my reason)
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- Опубликовано: 16 окт 2024
- Have you ever wonder why we bother to rationalize the denominators?
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> Doesn't remember 1/4
> Expects students to remember sqrt(2)
Arya Boudaie He did remember what 1/4 is, but he needed to do the long division anyway to be able to illustrate his upcoming point.
Wow! Great explanation on this. It's one of those things anyone in high school takes for granted, but when put into words like this, it really helps to visualize why we have to take that extra step!
ruclips.net/video/a5XymlfO1oM/видео.html
MushroomManToad thanks!!!
First reason: The value of 1/sqrt(n) is harder to estimate than the value of sqrt(n)/n.
Second reason: There's several ways to represent the same number (for example sqrt(n)/n = 1/sqrt(n)) and to make it easier to compare the results, mathematicians agreed on using the one with a rational denominator. It's one of many conventions in mathematics.
Yea, there's also the "abstract algebra" reason on why we rationalize the denominator.
Let me guess... field extensions? ;>
yup, in field extensions: it's easier to use (n, sqrt(n)/n) as a basis than (n, 1/sqrt(n))
yup
Why.....dividing by irrational numbers gets out of hand.
It works fine if we use geometry (two intersecting cords in a circle determine two similar triangles; choose one segment as the unit and you have obtained the four arithmetic operations with exact precision for irrational lenghts too)
@@quest4knowledge-xh3eu try explaining that to 7th grade me.
Sure, do you hang on Discord?
Nice job making even the complete basics interesting and informative
1) It looks nicer.
2) It's easier to calculate.
FACTS
Thank, very informative. This is the kind of content I like on RUclips.
Why are the shitty prank channels more famous than you?
sigh.. i do not know...
I know right. People these days.
Probably because the general audience doesn't like watching math videos in their free time. But we are not the general audience.
Maga DzhabraFTW because their content is entertaining to some while this is entertaining to others
No idea
This is one of my favourite channels on RUclips :)
Thanks!!!!!
Another good reason to rationalize denominators is you're less likely to end up with two expressions that are equal but are written differently. For example, 1/sqrt2 and sqrt2/2 are equal in value but don't look the same. That's a very simple example but there are more complicated expressions out there where it wouldn't be so obvious that the two expressions have the same value. So you are less likely to miss the fact that two expressions are equal if you rationalize the denominator for both of them.
You are my favorite youtube content creator c:
Thank you~!!!!
Mjm, interesting.
I usually tell my students that the reason why we rationalize denominators and simplify radicals is to be able to obtain a more succint result when doing "combinated calculations" (not sure if that's how you refer to that in English, just doing literal translation here).
Say, if you had to do sqrt(8)+sqrt(2), you wouldn't believe you can "compress" it anymore at first glance. But after simplifying you get 2sqrt(2)+sqrt(2), and you know how to continue.
But still this is a very interesting way to approach it! I'd love to make my students watch it, but I don't quite rely on their English level. Plus, we do long division differently here (not in concept but in the way we write it).
Thanks, good video!
I watched the blooper at the start several times lol, so unexpected to see you curse for some reason xD
Lol, it's the "behind the scenes"
I could watch it over and over non stop.
Do a 1-hour remix with a song :D
he also swore in recent videos :O
Bon Bon I will wait for the video from you!
this shows that it's useful, not necessary, to rationalize the denominator. I kind of like 1/sqrt(2) and it's perfectly acceptable if you leave your answer that way on any college math test.
I dunno. "Root 2 over 2" just _sounds_ nicer. It's got a better rhythm than "1 over root 2."
You probably know about these already, but memorization is not the only option for calling up sqrt(2). For any real number N, set x_0 equal to any reasonable approximation for sqrt(N), such as N/2 or even 1. Generate the iterative sequence x_(n+1)=((x_n)^2+N)/2x_n, and the sequence will rapidly approach sqrt(N) as n gets large. You're essentially bringing x and N/x closer together by averaging them, until they're equal in the limiting case. There's also a square-rooting algorithm akin to long division but it's a bit more involved
Nice to hear one of many rationales behind commonly-used practices. Many teachers/tutors, including myself, just spout it about without ever wondering why we do in the first place. Thank you, blackpenredpen.
Textbooks and exams often manufacture rationalising the denominator questions so that you end up with a nice neat integer at the end, which tricks you into think that's the whole point.
Integer or something much simpler than the original expression anyway.
In your videos it comes across that you really care about your students. Thanks for your quality videos! :)
Thanks!!!!!
What I would like to know, is how to calculate square roots by hand. For example, if you had to calculate the approximation of sqrt(2). How would you do it with pen and paper?
Ah, that's a great idea for a video. I will do some search on it and present to you guys later
risto sainio There's a bunch of ways to do it with Calculus.
Newton's method and the intermediate value theorem are some of the most basic and early ones they teach you.
There is another very complicated but cool and accurate way to do it that fits on a paper like any other kind of long division or whatever. The reason it gets complicated is because with each digit of precision, you also have to multiply bigger and bigger numbers in your head or by long multiplication.
I know the basic way is to "guess".
Say sqrt(2) is about 1.5, then we multiply 1.5*1.5 (w/o a calculator of course), we will find it too big,
so we say sqrt(2) is about 1.4, then we multiply 1.4*1.4
and then we start to put more and more numbers....
1.41*1.41....
etc....
all of the above, without a calculator
There are several ways the most basic one is:
1.Make an initial guess
2.Square your guess
3.if its too big guess lower if its too small guess bigger
4.Repeat 2-3
Needless to say this process is very slow and painful.
There is a way faster method but it requires knowledge of derivatives to be understood but it can be applied mindlessly. Here it is:
1. Make an initial guess x(0)
2.x(n+1) is given by x(n)- ((x(n))^2 - 2)/2x(n)
(Just draw a graph and it will make sense)
The bigger your n the better x(n) approximates root2
The consensus among those who have looked into the origin of rationalizing the denominator agree that it is muddy at best. The process starts showing up in the literature some 300 years ago, but it is not very clear. The popular opinion is that someone was creating a look-up table of some sort and in order to use the table the value could not have a radical in the denominator. So, the method of rationalizing the denominator was born. It is certainly possible, Mr. blackpenredpen, that you have hit upon the type of table that was being created. As you point out, it is more difficult to divide some number by an irrational number than an integer, so if you were building a reference table for decimal values of ratios, you would certainly want to create those values using the simplest method. Regardless of the origin, it seems that we have perpetuated the practice ever since, and we have developed more modern reasons for doing so such as standardizing fractions, simplifying the arithmetic in dealing with fractions that have radicals, and a few others that show up in the search results of why we rationalize denominators. However, as near as I can tell, there is no mathematical mandate to do so. Yet, I have met math teachers that get rabid about the process and tell their students that they must always rationalize the denominator. (I have a colleague at the community college where I teach that is that way. By no means am I implying that you do that - I am a fan and very much appreciate the time and effort you put into your excellent tutorials.)
You have presented a specific case where rationalizing the denominator is desired. But does that extrapolate to all cases? I say no. Let's test it. Let's take your example of 1 over the square root of 2. You might say to your trig students, "I have a right triangle, and its sine is 1 over the square root of 2. What is the length of the hypotenuse?" As your trig student, I know that the sine of an angle in a right triangle is the length of the side opposite the angle over the length of the hypotenuse, so I tell you the length of the hypotenuse is the square root of 2. You would say that is correct. However, if I had learned algebra from our mythical, rabid teacher mentioned above, I might pounce on radical in the denominator and rationalize it first, and then I might say, "The length of the hypotenuse is 2," since the rationalization gives me the square root of 2 over 2. So now, I have magically changed the dimensions of the triangle from the original. Somehow, that does not seem like it leads to a correct answer. As another example, perhaps I am sitting in your calculus class, and you are teaching us about using U-substitution in solving an integral problem. Let's say the problem has a radical in the integrand and that is the part that needs to be substituted. I am going to need a substitution that produces a radical in the denominator, so that it will cancel out the radical in the integrand. This is certainly no place for that rabid algebra teacher's imperative to rationalize denominators.
What your presentation demonstrates and what I am attempting to show is that math should not be taught in a vacuum! To be of value, math must have practical applications, and of course math does that by being the language of science, particularly physics. It is one thing to solve problems in an exercise set out of a textbook; most of us can certainly use the practice and especially in algebra. However, in the real world, the context of the problem is at least as important as the math representing it. Some mathematicians may disagree with that statement, but here is my context: If the job is to go out into the back yard and dig a hole with a shovel, mathematicians may be more interested in the shovel than the hole, and those of us who like to dig may be more interested in the hole. In reality, they are both sides of the same coin! The moral here is: Do not apply arbitrary "rules" to problems before analyzing the context of the problem to see if the "rule" should apply.
That beginning is hilarious
LOL!!! Yea, now you know the "behind the scenes"!
BlackCursesRedFace? :J
I love how you talk as fast as my brain thinks about this stuff. It is rare to see a math channel like that for me at least.
lol! Thanks!!!
When I started the video, I thought the playback speed was at 1.25X. But it was normal speed, surprise, surprise...
Secondary reason: The rounding error will be less, compared to the correct calculation, when the original rounding occurs in the numerator, especially if the original denominator is a subtraction, and most especially if the result is small, like 3 - 2 sqrt(2).
Tertiary reason, again when the denominator has a subtraction in it: Almost invariably, one chooses a multiplier that produces a positive result. If I had 5 - 3 sqrt(6), I wouldn't know it was negative until I did the approximation.
But why do you not compute sqrt(0.5) instead of 1/sqrt(2)?
Great video as usual, keep up the great work!!
thanks!
I have to be a third derivative now, sorry in advance..... why not multiply both numerator and denominator by 25 for the first one... i would rather tetrate in my head than do long division on paper ....great content, as always.
Wow. This one was interesting :)
Can you also do a video on why we also do it with i in the denominator?
i is the square root of -1, so it is the same.
@@bobh6728 i is not an irrational number though
@@shayanmoosavi9139 complex numbers can be written in the standard form a+bi, so if we have i in the denominator, it will not be similar to the standard form.
Complex non-number actually
Que bacana,nunca pensei que poderia fazer isso,quer dizer ,nunca ouvi falar sobre essa maneira de resolver uma racionalização.Parabéns.
Óia um br
Great explanation. Thanks
Well that’s true, but in most case (at least that I encounter) I want to do the least amount of computation so having 1 as the numerator makes it so that I only need to multiply the sqrt2 on the bottom and I’ll be done
I wish you had been my maths teacher sixty years ago!
I allways divide the other way arround the 1 at the left and the 4 in the right inside of the "box"
Love these vids brah.
thanks!!!!
is it used/useful in modern mathematics research?
nicktohzyu no. I just wanted to answer some HS algebra problem
Hello from Singapore!
Good video because I am learning SURDS for MATH CLASS and it explains it well
Amaths? Ahhah, secondary/poly/jc?
I am happy to hear this!! thank you!!!
That was so funny!! The intro :D
hahahaha
ruclips.net/video/a5XymlfO1oM/видео.html
Limits rationalize method
never seen division done like that before.
Yea. I know there are a few ways to set up long division.
With proliferation of Calculator, do you think it is still relevant irrelevant to rationalize denominator?
I've long maintained that the obsession to "rationalize the denominator" is a mathematical dinosaur in this age of abundant hand-held devices - a holdover from generations gone by (including my own!).
And that it's really useful only when you simply *have* to calculate the decimal expansion of such an expression.
ffggddss All mathematical conventions are dinosaurs so to speak, but they're useful in that they allow for rigor and consistency
+ Angel Mendez-Rivera
Some mathematical conventions are arbitrary, to some degree. Many are actually well-reasoned.
Rationalizing denominators is very high on the arbitrariness scale. It's essentially useless nowadays.
ffggddss I agree.
Rationalizing the denominator is useless from a truth perspective because an irrational number by definition cannot be represented by a ratio. If you rationalize the denominator, you can rewrite it as irrational number*some rational fraction, which solves the problem.
Soooo what about dividing by _rational_ numbers that have a periodic pattern (quite a long one), repeating to infinity? :q
E.g. by `1/69 = 0.(0144927536231884057971)(0144927536231884057971)…` which has a period of 22 digits.
We can't really "rationalize" a denominator which is already rational, can we? (or should I say "isn't it?" :D )
(Inb4: yes, I know we can just do the algebra fist to reduce the fractions. But that was not the point.)
Bon Bon Your question isn't much of a question, given that you answered it yourself. To divide by periodic decimals, you simply turn them into fractions, simplify the division, and then start long division. You can't do that with irrational, so you're forced to do it the brute way and do long division anyway, which causes algorithmic problems as the video demonstrates.
"But that was not the point."
The point was that it's not the irrationality in the denominator that really causes the problem, but an "ugly" decimal. The same "ugly" decimals appear for certain fractions too. So the only case in which long division is really useful is when we divide by some short and finite string of digits :q And _that's_ the real problem. Irrational numbers in the denominator is just one of the things that may cause it - the one we cannot easily avoid. But we can rationalize the denominator, as much as we can remove rationals with "ugly" (infinite) decimal expansion from the denominator, by doing some extra algebra. Which only hides the problem with long division and pretends it's the irrationals that causes it, not that it's something inherently wrong with the long division itself.
Bon Bon No, it is not the fact that the decimal is ugly that makes it difficult, but rather the fact that the decimal will in most cases be unknown in its completion. If a decimal is periodic, you can complete the division by noticing the periodic repetition in the denominator, not with irrationals. So the problem really does stem from irrationals. Not to say that long division isn't problematic, but with rational numbers, most of these problems simply DO NOT exist.
Bon Bon Also, long división solves more problems than it creates, especially when you deal with polynomial division and rational functions.
Hello! Can you solve the riemann summatory of sqrt(x) in the indefined form? I mind, just living the result in terms of a and b.
That intro had my dying.
Hahahahaha! I am glad that you like it!!!
Hey BPRP can you do a video explaining why the harmonic series diverges? I'd like to see an explanation from the master himself.
Akshay Kumar ok!
The answer is simple. We know that 1/1 + 1/2 + ••• + 1/x > Integral([0, x], dy/y). For x -> infinity, the integral diverges, since it has the form of Ln(infinity). Then the sum must also diverge as a x -> infinity.
the intro
Even my Maths teacher don't know this, randomly she is teaching, thanks for your awesome explanation.
True
How do u switch pen so quickly?
More HS math please....?🙂
I have a few more coming
How can we aproximate a irrational/rational to the power of a irrational number, like pi^e ????
Good question
Suppose we have one over pi. How to rationalize denominator in this case?
We can't rationalice Pi because Pi is a TRASCENDENT number, wich can't be derivative by any operation. I hope it'll help.
how many takes does an average video take you?
It depends!
Now I am getting better at it, so maybe like 1 to 4?
And also depends on how much tea/coffee I drink
Blackpenredpen, does exists a complex number which squared would be equal to 1?
Kachiro Gómez 1+0i
Kachiro Gómez -1+0i
Kachiro Gómez No, there are no non-real complex numbers that, when squared, evaluate to 1. That is, 1 has exactly two square roots: -1 and 1. This true for all numbers and all given n-roots: every number has n n-th roots in the complex group
However, there does exist the split complex number j with the property that j^2 = 1, but j is neither -1 nor 1.
Why we put 9 and what is the meaning of carry over
Incidentally, in another video about rationalizing denominators (specifically, a binomial involving both a cube root and a square root; ruclips.net/video/nEWDSCP64rY/видео.html ), there’s one commenter pretty near the top, Kevin Tong, who I think gives a better explanation of the practical needs for rationalizing denominators.
Dat chicken at 00:04
I love your videos! well explained, clear, and with a tip of humour that is never bad! (plus i think you're very cute, but that's not relevant...)
What about 1/e? Or better yet 1/sqrt(e)?
Morbius907 Taylor series, or more fun, use the limit definition of e
The reason why we rationalise the denominator is because the denominator is the unit of measure in which the fraction expresses its value, so the unit of measure in which the fraction works. And by definition a unit of measure, to be useful, must be commensurable. To use an incommensurable unit of measure is a contradiction in terms. It has no sense at all. Therefore, work with commensurable denominators will always be easier than work with incommensurable denominators.
I disagree. A better answer is that, we rationalize the denominator for the same reason that we simplify an answer in lowest terms... it's only by convention. We agree that that's the way we like it. And that's enough.
if I have an answer of:
5
⸻
√(2)
Instead of rationalizing the denominator by multiplying by:
√(2)
⸻
√(2)
to get
5√(2)
⸻
2
What if I express it as:
[ √(2) ]⁻¹
[ ⸻]
[ 5 ]
That fulfills your requirement that the denominator is an integer.
Another "justification" to rationalize the denominator goes like this:
Suppose I have an answer of:
1
⸻
√(2)
If the answer was a way to take a portion of a thing, say a pie (like an Apple Pie), it's problematic to divide the pie in √(2) pieces, and impossible to divide it into a whole number of √(2) size pieces. If I rationalize the denominator by multiplying by:
√(2)
⸻
√(2)
I get
√(2)
⸻
2
I am now able to divide the pie into 2 full pieces, but I am left to somehow take a √(2) quantity of those pieces... rationalizing the denominator was not really helpful.
The real reason, is that an answer can be expressed in an infinite number of ways... say by multiplying it by:
10 331 √5 √17
― or ⸺ or ⸺ or ⸺ or ...
10 331 √5 √17
So, a person reviewing that answer, say a teacher, looking at that answer from many students might have to do the reduction work on many/most of the students answers, just to see if each answer was correct. The solution is that there are certain conventions in place to minimize if not eliminate that.
‒ ‒ - - - - ― ― ⸺ ⸺ ⸻ ⸻
£ $ € ฿ ± Σ Ω Π Δ µ ← ↑ → ↓ ^ √ ³√ ∞ * ≈ ≠ ≤ ≥ ÷ •
₀ ₁ ₂ ₃ ₄ ₅ ₆ ₇ ₈ ₉ ₊ ₋ ₌ ₍ ₎ / ⁰ ¹ ² ³ ⁴ ⁵ ⁶ ⁷ ⁸ ⁹ ⁺ ⁻ ⁼ ⁽ ⁾ /
ᵃ ₐ ᴬ ᵇ ᴮ ᶜ ᵈ ᴰ ᵉ ₑ ᴱ ᶠ ᶢ ᵸ ᴴ ᶨ ᴶ ᶤ ᵢ ᴵ ᵏ ᴷ ᶪ ᴸ ᵐ ᴹ ⁿ ᴺ ᵒ ₒ ᴼ ᵖ ᴾ ᵣ ᴿ ᵗ ᵀ ᵘ ᵤ ᵁ ᵛ ᵥ ᵂ ᵡ ₓ ᵧ ᶻ
when they are people: they rationalize
when they are cool people: they realize
No words to thank you bro...😚
Something that seemed so pointless now makes perfect sense
You're right, 1/sqrt(607) is hard, but sqrt(607)/607 is cake
I just do it because I like it that way better 😂
My calculus teacher has a different explanation, "People who don't rationalize the denominator are monsters"
Oh
2^(-1/2)
2π is irrational, and is a fine denominator.
Why did he curse at the start?
And why we multiplied 7 by 1 one more time
I watch this video thinking I could teach this guy how to use a joycon as pointer instead of that bulge in his hands...
Qnd why did we put 8 after the 9
Good one. If You will be so kind and explain mathematician's one. Why? It appears from nower and it is not forbiden and solves everything. Literally life hack.
omfg i so like math but i am dying watching it on 0.5 speed XDDD
why?
cas it's like so slow and funny :_:
Why do you need to turn those fractions into decimals?
(Square root of 2) ÷ 2 = Square root of 0.5
You can turn the 1/1.414 into 1000/1414 to make it more obvious, that way it's still easy to calculate
wilkatis Yes, but sqrt(2) is irrational, so you still have an infinite number of decimals in the denominator.
Well, but that's kind of the point here - you don't have a calculator for it and you are approximating the number as far as you can remember it / to as many decimal places as you need if you remember more.
Rationalizing denominator just puts the sqrt2 on the top and it doesn't solve the infinite decimals problem, he still uses exactly the same approximation
wilkatis Of course it fails to solve the infinite decimal problem in completion, but it does partially solve it by minimizing the problem
√2 is √2
I still feel like it's utterly useless and is only a vestigial part of Mathematics. Great practice to get students to do more with radicals but in practice pointless
So, we shouldn't bother? cuz we got calculators
Isn't it?
EAT
You HATE using a calculator
I am newest comment😂
👍👍👍👍👍👍👏👏👏👏👏👏👌👌👌👌👌👌
Since we have a calculator and no one will divide without a calculator once we know how the division works, this argument is nonsense.
Thumbnail error
Why do you always hold your mic? Wouldn't it make more sense to use a headset or something so you could use your other hand?
Daniel Shapiro He has stated that he feels more comfortable when he carries the mic and his movements are more natural. He feels awkward with no mic, and also he would have the two markers (or chalks) on one hand since that's the "whole point" of the name of his channel
Oh, thanks!
Ricin Man Yes! That's correct!! You must have been following my videos for a while now and I thank you for that!!!
blackpenredpen Yeah, but it's the first time I put a comment cause it's also the first time I can comment about something I know and understand, lol
Rationalize 1/π :) rationalizing is useless just educative
Cannongabang It isn't useless, as the video clearly demonstrates. Algorithms yield better number approximations in more efficiency when the denominator is rational, simply due to the very nature of division.
Angel Mendez-Rivera half true: the true nature of division (costructive, rational numbers) has nothing to do with dividing real numbers a, b≠0 forming a new re number a/b.
And rationalizing can't always be done, so in general I'd say it is a wrong thing: as 1/π is pretty defined and one would need a different algorithm, more general for all the irrational numbers at the denominator ("rationalizing" would be this kind if thing, finding q rational in a costructive way such that
1/r - n
Cannongabang In the case of 1/π though, there already exist separate algorithms that allow us to calculate this particular number
i dont understand ur way of long division, i am used to another way
either way is okay. the point is to show it is "really hard" to divide an irrational number
yes, it is anglophon notation. quite weird indeed.
lol it's the way I learned long division, this way of long division is common in Asia, I guess
1/exp(sqrt(ln(2))).................. :o
hehehehe!
you speak so fast :)
Change Ur play back speed........
Pro tip: Don't. It just wastes time. There is no valid reason to do it.
Phoenix Fire The vídeo gave a valid reason.
I think this guy is crazy. He goes to university. See how hectic he is. "Irrational situation"
Stefan Reich yea but I love it! (I am a teacher btw)
Square root of 2 is irrational
Irrational denominators
Irrational numbers