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Mekatra Mekatra is offline
A*DIC*TED
Default Explain a thing! (Working on Math, Coda HELP!)   #1  
Okay so anyone can answer this if they get it, Coda just offered help lol.

So when you have a negative exponent Like 9^-2 It says I can write it as a fraction over 1 and then just flip it to make it positive.

WHY does this work? I get that it does, but because I don't know why I have a hard time remembering it.

Anyone? Did I explain that in a way that made sense?

Also if anyone else has things they would like explained but have always been afraid to ask or never had the chance, you are welcome to use this thread.
Answer my riddle to earn a prize!

You'll always feel it, but can never see it. It's strong but shatters easily. You can give it away but never hold it. What is it?

Pm me the answer, no cheating please!
Last edited by Mekatra; 09-30-2019 at 12:08 PM.
Old Posted 09-30-2019, 10:38 AM Reply With Quote  
Default   #2   Tohopekaliga Tohopekaliga is offline
Forward Thinker
Exponents are a notation, and not something specifically intrinsic to numbers.

So, when you have 4^3, you’re shorthand writing 1 * 4 * 4 * 4.

Negative exponents are divisions, so 4^-3 is 1 / 4 / 4 / 4, which is the same as 1 / (4 * 4 * 4), or 1/(4^3).


It also might help to consider that 4^4 * 4*^-2 = 4^2, since 4-2 = 2, also that (4 * 4 * 4 * 4) / (4 * 4) = (4 * 4)
Old Posted 09-30-2019, 11:19 AM Reply With Quote  
mdom mdom is offline
Jellosexual
Default   #3  
While a real explanation doesn't come, google told me that regular exponents are multiplication right, so 9^2 is 81 because 9 times 9 is 81! Negative exponents are the opposite! Because negative. So they're division! And what better to visualize division than fractions! So 9^-2 would be 1 divided by 9 times 9. Or something like that.
Old Posted 09-30-2019, 11:33 AM Reply With Quote  
Default   #4   Mekatra Mekatra is offline
A*DIC*TED
That is actually hugely helpful. Thank you much and I believe I can figure it out in a way that makes sense to my stubborn brain now. <3
Answer my riddle to earn a prize!

You'll always feel it, but can never see it. It's strong but shatters easily. You can give it away but never hold it. What is it?

Pm me the answer, no cheating please!
Old Posted 09-30-2019, 12:07 PM Reply With Quote  
Mekatra Mekatra is offline
A*DIC*TED
Default   #5  
Will someone remind me how to change the title on this thread? Cuz... I am blanking.
Answer my riddle to earn a prize!

You'll always feel it, but can never see it. It's strong but shatters easily. You can give it away but never hold it. What is it?

Pm me the answer, no cheating please!
Old Posted 09-30-2019, 12:09 PM Reply With Quote  
Default   #6   Coda Coda is offline
Developer
Sure, I can explain this pretty easily! :)

All right, so you should already know most of this, but I'm going to build this up in order to show the sequence of steps.

First off, the definition of exponentiation:
an = a × a × ... × a
That is, exponentiation is repeated multiplication. Next, a useful property:
ab × ac = ab+c
Proof:
(A) ab = a × a × ... × a (b times) (definition of exponentiation)
(B) ac = a × a × ... × a (c times) (definition of exponentiation)
(C) ab × ac = (a × a × ... × a (b times)) × (a × a × ... × a (c times)) (substitution)
(D) ab × ac = (a × a × ... × a (b+c times)) (associativity of multiplication)
(E) ab × ac = a(b+c) (definition of exponentiation)
But what if there's a negative number in the mix? We can't count a -2 times, after all, so we can't write it out as repeated multiplication. But let's apply (D) from the above proof...
(F) ab-c = (a × a × ... × a (b-c times)) (substitution)
(G) ab × a-c = (a × a × ... × a (b-c times)) (as proved above)
(H) (a × a × ... × a (b times)) × a-c = (a × a × ... × a (b-c times)) (definition of exponentiation)
Okay, so let's just think about this... What could we possibly substitute into a-c to make this equation true? We need to take away some a's. Let's solve for it:
(I) (a × a × ... × a (b-c times)) = (a × a × ... × a (b times)) × a-c (rearranging for formatting)
(J) (a × a × ... × a (b-c times)) = a-c (divide both sides)
____(a × a × ... × a (b times))
(K) 1/(a × a × ... × a (c times)) = a-c (cancellation)
(L) 1/(ac) = a-c (definition of exponentiation)
And there you go -- the property you're looking for!

Said in plain English: Taking a number to a negative exponent is a way to take away multiplications by the number instead of a way to do more multiplications by it. And we take away multiplications by using division!
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Old Posted 09-30-2019, 12:16 PM Reply With Quote  
Coda Coda is offline
Developer
Default   #7  
You rename a thread by editing the first post and changing the topic line.
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Mega Man: The Light of Will (Mega Man / Green Lantern crossover: In the lead-up to the events of Mega Man 2, Dr. Wily has discovered emotional light technology. How will his creations change how humankind thinks about artificial intelligence? Sadly abandoned. Sufficient Velocity x-post)
Old Posted 09-30-2019, 12:16 PM Reply With Quote  
Default   #8   Glitch Glitch is offline
Pixels
Pretty easily but look at all that math!




Old Posted 09-30-2019, 12:28 PM Reply With Quote  
Coda Coda is offline
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Default   #9  
It only looks like a lot of math because I was showing a step-by-step -- showing why it works instead of just saying it does.
Games by Coda (updated 10/1/2023 - New game: Adrift)
Art by Coda (updated 8/25/2022 - beatBitten and All-Nighter Simulator)

Mega Man: The Light of Will (Mega Man / Green Lantern crossover: In the lead-up to the events of Mega Man 2, Dr. Wily has discovered emotional light technology. How will his creations change how humankind thinks about artificial intelligence? Sadly abandoned. Sufficient Velocity x-post)
Old Posted 09-30-2019, 12:28 PM Reply With Quote  
Default   #10   Espy Espy is offline
Wanderer
Wait, heck, that makes sense. IDK if I just never questioned it, or had another way of thinking about it.
STONEWALL WAS A RIOT

Old Posted 10-01-2019, 06:48 AM Reply With Quote  
Coda Coda is offline
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Default   #11  
I, for one, had never questioned it. I just followed the properties handed to me and they all seemed to work, so I didn't bother thinking any harder about it. That proof there is actually something I came up with on the fly.

Although technically, that's not a proof of correctness -- step (F) does not follow from the rest. The definition of exponentiation that was asserted at the beginning implies that only natural numbers are valid, so if this were being treated as a rigorous proof this would be an error that disproves it, or at least it would establish that it's only true if b > c.

Instead, this is a proof that if you extend exponentiation so negative integers are allowed in addition to positive ones, then this is what a negative exponent must be in order to be consistent with the property we want it to have. You have to follow a similar procedure if you want to define an extension that allows fractional exponents. (Extending it from fractions to real numbers, on the other hand, can't be done with this procedure, and you have to introduce logarithms.)
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Mega Man: The Light of Will (Mega Man / Green Lantern crossover: In the lead-up to the events of Mega Man 2, Dr. Wily has discovered emotional light technology. How will his creations change how humankind thinks about artificial intelligence? Sadly abandoned. Sufficient Velocity x-post)
Old Posted 10-01-2019, 10:10 AM Reply With Quote  
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