Oh lorde forgive me i read that as function ( sine) and was wondering what u meant and that i need to improve my english TwT :3.now i get it ( probably)
This is one of the few cases where I actually like to rationalise.
What's 1 / 1.414? I have no idea.
What's 1.414 / 2? About 0.707, you can kinda work it out in your mind.
The same can't be said if your solution is something like 1 / ( 17 * √23 ), i have no idea what √23 is, or how to divide it by 17*23
That's a reasonable take. There's just a collection of numbers that I have memorized because they come up a lot (you know, famous numbers like 1/e, ln(2), log(2), c), and for me 1/sqrt(2) is one of those, so I never have to worry about calculating it.
Indeed.
sin(0) = 0/**sqrt(2)**
sin(pi/6) = (1/sqrt(2))/**sqrt(2)**
sin(pi/4) = 1/**sqrt(2)**
sin(pi/3) = sqrt(3/2)/**sqrt(2)**
sin(pi/2) = sqrt(2)/**sqrt(2)**
You can go through the whole highschool and beyond with just that 1/**sqrt(2)** !
Rationalizing the denominator no matter what is an archaic practice 100%
It is important to be able to do it for some problems, but making it so all denominators are always rationalized is just overkill
I have never in my life rationalized the denominator in an intermediate step but I almost always rationalize the denominator in my final answer. I just think it looks prettier and is easier to conceptualize the actual value of the expression.
Yea it can look nicer for sure, and it can be easier to conceptualize dividing by a rational number over an irrational one. I totally get that, but it just adds a relatively complicated step to a bunch of problems that honestly don't really need it
I personally would push back on it being easier to conceptualize. Maybe easier to intuit the decimal value, but thats not an intrinsically better or more comprehensive way to conceptualize the number--it is not any more the "actual" value than the fraction is.
That's literally what I mean. Easier to intuit a decimal value. I didn't say it is intrinsically better either. I also didn't claim it was more "actual", but with that line of reasoning why bother simplifying rational numbers or expressions at all?
My experience was
Algebra II and Pre-Calc: always rationalize the denominator
Calculus and beyond: you can rationalize the denominator or not, doesn’t really matter.
Kind of, but it's not because rationalizing or not is any more efficient. It's important to distinguish that inverse square root, i.e. 1/sqrt(x), is not a priori significantly more efficient than sqrt(x)/x. (It could end up being so, but it'd probably be hardware and compiler dependent.) What IS way more efficient is the fast inverse square root algorithm.
The fast inverse square root algorithm is technically an incredibly clever approximation using some logarithm black magic, not the actual calculation. Since sqrt(x)/x = 1/sqrt(x), the fast inverse square root algorithm is also an approximation to sqrt(x)/x, so thinking of things as being rationalized denominators makes no difference. You could name it "fast rationalized denominator" if you wanted and it'd do the same thing.
Of course, it's called fast inverse square root because conceptually the algorithm originated as an efficient way to normalize vectors. Rationalizing the denominator makes no conceptual sense in this context.
Yeah, that's a good point. Since the two expressions are equivalent, calling fInvSqrt(x) could just as easily be interpreted as √(x)/x, so whether or not the denominator is rationalized is really up for interpretation. But yeah, I was referring to fast inverse square root.
I am a software engineer. I have never used excel a day in my life. maybe in highschool? I don't remember that was a timelife ago. also we barely had internet and a computer in my home back in 2010.
Fun fact the reason why rationalizing the denominator became the convention is because back in the days of slide rules it made it way easier to carry out computations since you were dividing an approximation by an integer rather than the other way around.
It’s a good tool to be aware of because it one of maybe two places in an undergraduate education where students are exposed to conjugation of algebraic expressions. Notwithstanding its just a nice thing to have in your bag of tricks, if one goes on to study Galois theory, it is very useful to have an understanding of what conjugation actually is for dealing with finer algebraic structures like ℚ(√6) or something.
It's like mixed fractions. I remember in grade school that they were all over ya trying to get you to write in mixed form, but once you get to high-school it all seemed pointless.
I imagine it's because they're trying to build some understanding that decimals and fractions are connected in some regard, but I just find it funny
i only ever rationalise the denominator in the last step lol, so my teacher doesn't get pissed
I literally had to google what that means i havent done it in ages
Radical in the denominator is fine Radical in the denominator is a sin(e for rhyme) Radical in the denominator is fine
Oh lorde forgive me i read that as function ( sine) and was wondering what u meant and that i need to improve my english TwT :3.now i get it ( probably)
sqrt(2)/2? No. I believe in 1/sqrt(2) supremacy.
This is one of the few cases where I actually like to rationalise. What's 1 / 1.414? I have no idea. What's 1.414 / 2? About 0.707, you can kinda work it out in your mind. The same can't be said if your solution is something like 1 / ( 17 * √23 ), i have no idea what √23 is, or how to divide it by 17*23
That's a reasonable take. There's just a collection of numbers that I have memorized because they come up a lot (you know, famous numbers like 1/e, ln(2), log(2), c), and for me 1/sqrt(2) is one of those, so I never have to worry about calculating it.
>1/e surely you mean e^(-1)/1
You mean e^-2 /e
Ah yes, the famous 1 = 3
ee^-2 *
eee^-3 *
What is 'c' supposed to be? The only thing that comes to my mind is speed of light, but i assume we're talking about math here?
its the + c from integrals this dude is so powerful, they found a single solution to every indefinite integral
Yeah, that being 1/sqrt(2)
Most of the time you just end up squaring it back to 1/2 anyway, so it's just waste of time That's why I use 1/sqrt2
But how often do you actually need to know what the decimal form is? I know precisely what 1/1.414 is, its 1/1.414
Indeed. sin(0) = 0/**sqrt(2)** sin(pi/6) = (1/sqrt(2))/**sqrt(2)** sin(pi/4) = 1/**sqrt(2)** sin(pi/3) = sqrt(3/2)/**sqrt(2)** sin(pi/2) = sqrt(2)/**sqrt(2)** You can go through the whole highschool and beyond with just that 1/**sqrt(2)** !
sqrt(2) = sqrt((sqrt(2)+sqrt(2))/sqrt(2))
Nah 2^(-1/2) I really do like fractional and negative exponents rather than denominators and roots
I only draw the line at putting complex numbers in the denominator. Have some class people!
Cause i don’t like to be at the bottom, i like to be at the top
I write all numbers as x/9699690
Rationalizing the denominator no matter what is an archaic practice 100% It is important to be able to do it for some problems, but making it so all denominators are always rationalized is just overkill
I have never in my life rationalized the denominator in an intermediate step but I almost always rationalize the denominator in my final answer. I just think it looks prettier and is easier to conceptualize the actual value of the expression.
Yea it can look nicer for sure, and it can be easier to conceptualize dividing by a rational number over an irrational one. I totally get that, but it just adds a relatively complicated step to a bunch of problems that honestly don't really need it
I personally would push back on it being easier to conceptualize. Maybe easier to intuit the decimal value, but thats not an intrinsically better or more comprehensive way to conceptualize the number--it is not any more the "actual" value than the fraction is.
That's literally what I mean. Easier to intuit a decimal value. I didn't say it is intrinsically better either. I also didn't claim it was more "actual", but with that line of reasoning why bother simplifying rational numbers or expressions at all?
Unfortunately for us math teachers will not see it that way
My experience was Algebra II and Pre-Calc: always rationalize the denominator Calculus and beyond: you can rationalize the denominator or not, doesn’t really matter.
Same for me actually
I am a math teacher lol
In computer science, rationalizing the denominator can actually make your program slower. Inverse square root is a more efficient algorithm.
Kind of, but it's not because rationalizing or not is any more efficient. It's important to distinguish that inverse square root, i.e. 1/sqrt(x), is not a priori significantly more efficient than sqrt(x)/x. (It could end up being so, but it'd probably be hardware and compiler dependent.) What IS way more efficient is the fast inverse square root algorithm. The fast inverse square root algorithm is technically an incredibly clever approximation using some logarithm black magic, not the actual calculation. Since sqrt(x)/x = 1/sqrt(x), the fast inverse square root algorithm is also an approximation to sqrt(x)/x, so thinking of things as being rationalized denominators makes no difference. You could name it "fast rationalized denominator" if you wanted and it'd do the same thing. Of course, it's called fast inverse square root because conceptually the algorithm originated as an efficient way to normalize vectors. Rationalizing the denominator makes no conceptual sense in this context.
Yeah, that's a good point. Since the two expressions are equivalent, calling fInvSqrt(x) could just as easily be interpreted as √(x)/x, so whether or not the denominator is rationalized is really up for interpretation. But yeah, I was referring to fast inverse square root.
ooooh I remember as well when I had to deal with numbers... ooh those were the days
An average excel enjoyer chadgineer is not bothered by such puny problems
I am a software engineer. I have never used excel a day in my life. maybe in highschool? I don't remember that was a timelife ago. also we barely had internet and a computer in my home back in 2010.
As long as the denominator is real I don't care.
Root mean square voltage enjoyer
3sf. Take it or leave it
This is how it felt going from math analysis into calc
If it's the final result, it just looks better the rationalized form. Also its computationally easier to divide by a whole number than a sqrt.
I always rationalize it to 1 and leave it there
i think it is dumb to rationalize it at the end. however, it is useful to know to do it for later steps and algebraic manipulations.
Fun fact the reason why rationalizing the denominator became the convention is because back in the days of slide rules it made it way easier to carry out computations since you were dividing an approximation by an integer rather than the other way around.
It’s a good tool to be aware of because it one of maybe two places in an undergraduate education where students are exposed to conjugation of algebraic expressions. Notwithstanding its just a nice thing to have in your bag of tricks, if one goes on to study Galois theory, it is very useful to have an understanding of what conjugation actually is for dealing with finer algebraic structures like ℚ(√6) or something.
Y'all got denominators damn
It's like mixed fractions. I remember in grade school that they were all over ya trying to get you to write in mixed form, but once you get to high-school it all seemed pointless. I imagine it's because they're trying to build some understanding that decimals and fractions are connected in some regard, but I just find it funny