Message: If you don't actually understand it don't act like you do.

"I don't understand why taking .17 of a 2 out of a bag of twos is a thing you're allowed to do," says Pela.

"Well, look at it this way.  A 16 is a bag of two 4s.  What happens if you take half a 4 out of the bag?"

"....you take a two out, and now you've got a bag that multiplies up to eight."

"A million is a bag of two thousands.  What happens if you take a third of a thousand out of the bag?"

"I don't know what a third of a thousand is. ...I mean I know what it is when it's 333. But it's not, here."

"A thousand is a bag of three tens.  What happens if you take a third of three tens out of a bag of twice three tens?"

"You have five tens left in the bag. So - a hundred thousand."

"Yep."  His smile goes away after a moment; it's impossible to have any sense of how well this is going when everybody is supposed to learn this at age five or six and they're adults.

"Well, if you can take half of a four out of a bag of fours, and a third of a thousand out of a bag of thousands, why not take 17 100ths of a 2 out of a bag of twos?"

"....I guess."

"I mean, there's the problem of figuring out that taking out 0.17 twos from a bag works out to multiplying the contents by roughly 8/9, but you can get that fairly precisely off nineteen twos being a bit less than twelve threes.  Possible self-study problem: rederive that yourself, convince yourself of it, prove it, without looking back at the whiteboard."

"Now, or after class?"

"How many people in this group think that's now so obvious that there's no point in proving it themselves?  Because if the answer is no, then yeah, maybe everybody pauses and tries to rederive the logic."

Most of them don't in fact find it so obvious there's no point in proving it!

They get to work on that. 

This would otherwise be a good time to Message Carissa to ask how he's doing teaching-wise, but apparently Asmodia, Ione, Meritxell, and Carissa also think that rederiving this claim is a good exercise for them to do.

Possibly Keltham is overcorrecting for how many fewer exercises ought to be required to grok logarithms if you first encounter them as an adult rather than a five-year-old.

When Carissa was a five year old she required one on one tutoring from her mother to have enough attention for anything at all complicated, constantly forgot things that ought to be in working memory and needed reminding of them, and had about ten minutes' attention span for actual thinking. Trying to teach her math in a group would have been a disaster. 

Anyway, what is 8 9ths as a bag of twos. 

Dath ilan didn't say it was easy to teach it to five-year-olds.  Civilization is staring at that problem and optimizing it roughly as hard as Civilization ever optimizes anything.

Figuring out how to have logarithms be fun to learn about starting one month earlier on maturation timelines is a perfectly respectable accomplishment for a +4sd researcher's entire life's work.  If any single individual made a discovery like that singlehandedly, it would get them well into the 'more money than one person can reasonably spend on themselves' category of rich.

Meritxell scribbles until satisfied that the log of (8/9) is going to be the difference between the log of 8 and the log of 9, and then until satisfied that that difference is the difference between 3 and 2*log(3), and then looks around for someone who looks stuck and helpfully helps! Paxti, are you stuck?

Paxti has worked out that 8/9 is 0.888.  She's worked out that nineteen 2s is 524,288 (by multiplying by 8 repeatedly to get to 18 2s, and then doubling the final result), which got her an answer that could've maybe been on the whiteboard she can't look at.

Paxti is currently working on computing twelve 3s via assembling a bag of six 9s.  After that she's going to divide out 19/12 the long way.  Maybe if she computes all the numbers Keltham said to compute, it'll be obvious once she's computed them how to put them together.

Message: hey, free hint, all of that's stupid. Further hints available for sale.

Message:  Keltham wandered over to look at what I was doing and nodded approvingly.  Fuck off.

A bit later on Paxti has managed to get 9^6 = 531603 (close enough), and 19/12 = 1.58something.  Now she just has to figure out how that all fits together with 8/9, or three 2s and two 3s.

Nineteen 2s equals twelve 3s.  You need more 2s than 3s to make up something, so that makes sense.  19/12 is the number of 2s in a 3.  There'll be 2*19/12 2s in a 9, so it's 2*19/12 = 19/6.  Subtract 3 2s for the 8, and... 19/6 - 3 = 19/6 - 18/6 = ...

Message Keltham:  I get that there's exactly 1/6th of a 2 in an 8/9, does that make any sense?

Keltham will come over and check how she arrived at that conclusion, but will soon approvingly inform Paxti that if, as is not actually the case, 2^19 exactly equalled 3^12, then yes, there'd be exactly -1/6 2s in 8/9.  Please observe that -1/6 is -0.1666... or about -0.17.

To try to see it a glance, consider that if there's 19/12 of a 2, in one 3, that's 1/12 more of a 2 than 1-and-a-half 2s:  19/12 = 18/12 + 1/12.

So in a 9, there should be 2/12 more 2s than in 8.  Though it's actually a bit more, because 3^12 is greater than 2^19, so there's a bit more than 19/12 2s in a 3.

It might be good to give everyone another similar problem, Carissa tells Keltham. To check if they really get it.

Three 5s is a tad less than seven 2s; tell me how to score a prediction of 2/5 on something that actually happens.  Sanity check, 2/5 is a noticeable-chunk less than 1/2, so your score should be a noticeable-chunk less than -1 2s.