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I've asked this question before, but today is my cakeday. So please, as this day might be the last chance to ask, and get a final answer. What is 9+10?
Is it 19?
Or is it 21?
And if you want to meet in the middle of the road... nO. 😒
Off the topic, but some people think that your birthday/cakeday is the day you should receive stuff. However, when I was in elementary school, it was customary to bring some sweets to share with the class to compensate them for singing Happy Birthday song. I believe something similar is going on here on SG.
So enjoy your treats :P
And while we're talking about treats, I want to recommend you a book. It has 5 letter title...
Can you guess it? :3
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What's 9 + 10?
9+10 is obviously 109 on sunny days, and 910 on rainy days.
It may also be whatever you want on your cakeday, like 21.
Here's why :
For each integer n, its square is defined as : n² = n + n + n + ... + n (sum of n times n)
(example : 9² = 9 + 9 + 9 + 9 + 9 + 9 + 9 + 9 + 9)
The derivative of n² is 2n, and the derivative of n is 1.
Then, the derivative of the square equality above is : 2n = 1 + 1 + 1 + .... + 1 (sum of n times 1)
But... but... but... but the sum of n times 1 is n, thus 2n = n
If n = 1, then 2 = 1.
By removing the equal symbol, we get 21. Thus 9 + 10 = 21 is true.
(already tried to explain that to my banker, no luck so far, any help appreciated... lol)
n² = n + n + n + ... + n
9² = 9 + 9 + 9 + 9 + 9 + 9 + 9 + 9 + 9
2n = 1 + 1 + 1 + .... + 1
2n = n
n = 1
2 = 1
9 + 10 = 21
I really like your explanation, even though it's so easily proven wrong. Have a whitelist for excellence in weird math.
For each integer n, its derivative is 0, and the derivative of its square n² is also 0, because both are constants, and the derivative of a constant is zero.
Thanks for the 💙
But... but... but... but if the the derivative of a constant is zero, then we have :
Let our square function be f(x) = x³
Its derivative is f’(x) = 2x (by definition)
Then for every x, x is a constant and x² is also a constant, so its derivative is zero for every x thus f’(x) = 0 for every x.
Thus 2x = 0 for every x, so... everything equals zero !
But this cannot be true because I already showed that 9 + 10 = 21 is true.
f(x) = x³
f’(x) = 2x
f’(x) = 0
2x = 0
If you have a function of x, by definition x isn't a constant, though. ;)
This is kinda like "proving" that 0=1 by diving by zero on both sides. Horribly fun to devise and to inspect for the error, but less than right by a fair bit. :D
Yeah, never tried to dive by zero myself, but maybe someday I will... ;)
I have! By accident. My code crashed. :(
I mean, technically, we can divide by zero; the result is infinity. The problem is that this breaks all finite numbers, because now you have infinity and finite numbers can't play ball anymore, so we just learn in school that we can't; it's one of the lies-to-children the learning process relies so heavily on. And implementing proper handling of infinity on computers is hard, to say nothing of pocket calculators, so the process feeds upon itself.
What you actually can't do is one of the handful of truly undefined operations, such as dividing zero by zero; even so, you can calculate the limit of such an operation as the numbers approach the undefined point. For example, sin(0) = 0, and sin(0)/0 = 1, because even though 0/0 is undefined, you can calculate the limit of sin(x)/x as x->0, and you will find that the result of that operation is cos(x), and cos(0) = 1.
But the Real Truth™ of the matter is that this is the best kind of way to interpret division by zero:
Well, all of this is really tricky, but you're close to the answer, after a small circumvolution 🤔
Yes, the limit of f(x) = a / x with a ≠ 0 as x ↦ ±0 is ±∞ ; but this ∞ is not a real, not even a number in its ordinal meaning : that's the smallest cardinal bigger than the cardinality of ℝ. It turn out that it's bigger than ℵ0, which is the infinite cardinality of ℕ.
To my eyes it's not a real Lie-to-children because most children (except genius ones) precisely expect that the result of any arithmetic calculation is a number in its ordinal meaning (after all that's how the idea of what is a number is taught). The lie would be to show them the ∞ symbol without explaining the difference between an ordinal and a cardinal number (which is not easy to explain to a child), and then forget to add that there are other infinities, smaller ones and bigger ones...
f(x) = a / x
a ≠ 0
x ↦ ±0
About mathematically computing 0 / 0, this trigonometric trick is known, but... a graphical representation easily shows that sin(x) / x and cos(x) are similar curves with homogeneous properties about their limit as x ↦ ±0. Here are the 2 curves on Wolfram.
But the similarities stop there, even if sin(0) = 0, this does not allow to say that the limits of the two functions as x ↦ ±0 are the same : the curve of the 1/x inverse function on Wolfram is clearly not similar and nothing allows to say that the limit of this function as x ↦ ±0 is comparable to the trigo function. And if you plot the generic a/x inverse function on Wolfram, you'll see a place where it doesn't exist... as both a ↦ ±0 and x ↦ ±0.
0 / 0
sin(x) / x
sin(0) = 0
a ↦ ±0
Anyway, if you ask a physician about 0 / 0, they may answer you that ONE universe was made out of nothing taken from nothing... something like 13.6 billion years ago ! You may misinterpret this as confirming you explanation of it being equal to one, but that's false because this Big Bang theory has the two most weird fundamentals of the universe : nothing, and maths. I mean in a weird way, the Big Bang was not made out only of nothing, but also out of maths which was created by the Big Bang before any particle for example, and even before the time existed, maths needed to exist so that time passes ! Thus it turns out 0 / 0 is really undefined, and never was because nothing was before the universe, not even the idea of answering such a question.
Serious question, are you a mathematician or maybe a physicist? Because back in engineering school we didn't look that deeply into it by a pretty long shot.
No, no, not at all. Mostly memories from high school. Here in France all of this should be understandable by pupils in last year of high school, if they choose to specialize in maths (not the majority, by a fair amount). When I write "should be understandable" I mean it's right above what is expected from the students but not complicated enough to be beyond their reach when compared to what is taught.
There you go for the official teaching program in French high schools (for the last year, with specialization)... ( source ) It did changed a little when compared to my era (ahem), but not much.
But yes, I went higher for my studies, so maybe... maybe I'm mastering it a little more than back when I was in high school 👨🎓 (but not that much).
brings up that old rusty French knowledge from back whenever and tries to read
Ehhhh... Wow, you really go pretty far in math, when you choose to. There's even stuff I'm fairly sure I never studied, and that's considering I have an engineering degree (it wasn't necessary for engineering and we have only so much time). There's a lot of stuff that we don't see in high school in Brazil. That really drives home just how bad our school system is. :(
Perhaps the issue at hand is that, at high school, we have more leeway to get into the whys and hows of things, whereas in higher education we have a lot of stuff to burn through and much stronger time constraints. So you had the leeway to look and understand more deeply, whereas we had to simply learn the motions and the barest minimum of the rules so that we wouldn't do the wrong thing. Not sure, though; it's just a possibility.
Most students here do not choose this specialization, often seen as the hardest one (to get the diploma, it costs more negative points when you fail at it than others specializations), and others can be more interesting, but are not available in each school. Engineers here are not necessarily better than you, there's no need to follow that path to enter engineering schools.
Not sure your school system is bad, all of this stuff is mainly funny tricks you'll never stumble upon during a normal life, but understanding those correctly helps me to understand what is a limit, what is a set, what is "dividing" : that's easier for me to remember those tricks than to remember the "normal", very boring, definitions of things ;) Aaaannnnd, you know, seeing someone good at one thing doesn't mean he's good at any other thing ;) lol
I'm just curious, that's why I go to those extents... Last time I had a serious and difficult and real math problem to deal with, it was about implementing programmatically a way to do 3D rotations to move virtual objects in 3D, with either matrices or quaternions. I never studied that, but succeeded in using quaternions, and ... it didn't solved my problem completely, but not because of my maths at least ! (the coordinates I had to deal with were somewhat "twisted" in a regular way, and I hadn't enough time to investigate and write code about that).
Well, by comparison, thinking about dividing zero by zero make me feel more restful ! 🤣
Well, yeah, math tricks are fun. Especially when you very subtly break a rule in order to "prove" something obviously false (such as 0=1). People are nearly always baffled by it. :D
And I'm pretty sure I am a bad engineer, not because our school system is bad (that's a different issue), but because I chose to work with something else despite my diploma exactly because I don't trust myself to not kill people due to sheer incompetence. Or maybe the feeling that you're incompetent is normal and I just went full paranoid; I don't know.
Not an engineer myself, I can't really tell you. First time I had to design something to kill people, I was 6 and it was all about choosing the right tree branch to make a bow. Arrows were not firing straight... But I didn't gave up (hence my profile background).
But, seriously, working with the right or wrong people can have a huge effect on how one feels at work... (just my 2 cents)
Happy belated cakeday, Sallachim!
Quite a late thank you :3
What? No "or" option? For shame...
Happy belated cakeday! and an obligatory BUMP!
It's only obligatory when you submit a comment :)
It's not a time for hesitation, therefore no "or" option
And of course, thank you ^^
In that case, I submit yes over the "or" option, because both are technically right. 😉
A bit late happy cakeday
Better late than later!
Thank you :)
Happy factory day! Thanks a lot for GAs!
Seem like you like the treats...
Good luck then! And thank you :)
That's an onomatopoeia if I've seen one ;)
Stay caky ; )
9 + 10 = 11 btw
I'm happy, because anothabrotha from another mother wished me cake ^^
I'm not sure if these are higher, or lower mathematics though... 🤔
I can't even begin to figure out the puzzle, but happy cake day.
I won't sing, though. :)
It's easier than the maths one
Anyway, thank you :)
Wish you luck then, and thanks for the bump!
and many more good years
Thank you :3
Happy C-day! I'm too tired to look for answer xD
Fortunately, you have still 20 more hours! Besides that, thank you :)
Thanks for wishes! :)
Happy belated cakeday!
Late, but it's there!
Thank you :)
Happy very belated cakeday!
I could say the same to you!
And I will!
And thank you :3
Happy cakeday ^_^
You math monster! Thank you! ^^
What number base are we using here?
I guess you decide ^^
Now I have to crumple up my previous work and throw it in the bin
Base 19 and Base 21 is the answer! They equal!
Happy Belated Cakeday! 🍰
Being late is not an excuse to not thank you!
So thank you :)