mathmisery.bsky.social
Mathematician in Industry, Fractional Executive, Math Education
#MTBoS #elemmathchat #iTeachMath #mathsky #mathchat #mathed #educoach #datascience #machinelearning #EduSky
126 posts
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291 following
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what no conclave function jokes? nothing about "higher order popynomials"? probability "mass" functions can take on a whole new meaning!
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bring on the cardinal(ity) puns and revive Pascal's wager
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I like it!
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I assume you meant 70 + 11 for the right hand side.
The commutative property is that "a [operator] b = b [operator] a". In the example you gave, you have "a [operator] b = c [operator] d" which is more a different representation than the values "a" and "b" commuting.
hope that helps!
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edinburgh castle?
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yeah this is nice and fun! i think it comes down to which fact you have memorized to short circuit calculations. the (36 + 1)*(25) is convenient per your earlier post.
here's another one leaning on the trick for squares of multiples of 5
(25 + 10 + 2) * (25) = 625 + 250 + 50 = 925
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not a place but "chai tea latte" kills me
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thank you! will check it out. apologies for not replying sooner, I'm on sporadically!
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these are cool. how do you do these? [i don't mean code, but what's the general idea?]
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I use this one:
$1,000,000 per year for the next 1,000 years is $1,000,000,000. So imagine what personal wealth of $400,000,000,000 means --- 400,000 years of $1M per year :D
[and for the aCtuAllY crowd, yes, ignore interest / investment income and taxes but also ignore looting from war, etc.]
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highly recommend :)
www.amazon.com/Mathematical...
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this is such a fun book to go through and i'd be happy to take a recreational course in it :)
#iTeachMath #MathSky
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congrats!
a rando's two cents:
- manage stakeholder expectations: DS isn't magic, data existing != insight existing, data quality > fancy buzz word algorithm
- advocate for a budget and your own engineers
- easy way to say "no" --> "which business kpi are we looking to improve with this project?"
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you could ask why / how the rth equation begins with r^2, and this could mean counting the total number of numbers being added across the LHS and RHS. then see that the count is 3,5,7,9,11,... and then have the conversation about the sum of 1 + 3 + 5 + 7 + ... + 2r+1. just spitballing!
#iteachmath
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for high school maybe we see that the first term on the left hand side of rth equation is r^2?
then to see if they can write the LHS as
r^2 + (r^2 + 1) + ... + (r^2 + r)
and the RHS as
(r^2 + r + 1) + ... + (r^2 + r + r)
LHS = r^2*(r + 1) + (1 + ... + r)
RHS = (r^2 + r)*r + (1 + ... + r)
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you can also see how the average size of the largest piece drops as the number of cuts increases
you can also ask questions about the distribution of resources
there are ways to change the how we cut [this graph assumed pulling from a Gamma(1,1)]
all told, it's a fun exploration
#math #mtbos
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On the other hand, if A guesses 20%, B guesses 25%, and the largest piece is revealed to be 23%, A wins. And both players lose if they are both over.
You can test your intuition if the string were cut into only two pieces [so 1 cut]. On average, how large is the largest piece?
#math 🧵
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There's a push-your-luck game you can make out of this. You can get the largest piece if you have the highest guess less than the largest piece. Think "Price Is Right".
For example, A guesses the largest piece is 20% of the string, B guesses 21%. The largest is revealed to be 23%. B wins.
#math 🧵
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Here's a distribution of the largest cut as a percentage of the string. Interestingly, it's not too crazy to see the largest piece be 25% of the string even if we were targeting an average length of 5% (1/20). In fact, seeing the largest size to be near 5% is more unusual than 25%.
#math 🧵
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You have to get into the Dirichlet distribution to make this work [there are other methods]. In any case, without going to much into the weeds, here's an image of 40 pieces of string, each cut into 20 pieces at random [and each string is sorted from smallest cut to largest].
#math 🧵
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It wouldn't be unusual to say that a fairly cut piece of string would have all cuts of equal length. But random cuts with the desire to have each cut be the same length, on average, means that we have to get our heads wrapped around what that spread looks like.
#math 🧵
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for whatever crazy reason, i just got a notif of your message. you're welcome!
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things get wonky with [0,1] since in some sense "1" doesn't belong; so go with [0,1) and when you have to decide what to do with rounding a number x, you get [0,0.5) vs [0.5, 1). which means that x = 0.5 rounds to 1.
anyway, curious to know how you all resolve this at the elemed level
#mtbos 10/10
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whoops, "closed" the thread too early.
in reality, you can map any interval [a,b] to [0,1]. so in the case of rounding 20 to the nearest 8, the range you're working with is [16, 24] which can be remapped to [0,1] via (x - 16)/(24 - 16). this makes x = 20 map to 0.5.
#mtbos 9/n
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you can twist the rounding question a bit to something like "round 20 to the nearest 8". Is that going to be 16 or 24?
and you can have a rich discussion about which way the inequalities should be set up for rounding.
#mtbos 8/8
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*walk across a number line
[dropped the word "line" in the previous post]
#mtbos 7/n
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without having to get into cadlag functions, you could make the heuristic argument that as you walk across a number from left to right you are approaching the "half way point" between two integers from the left. and so the exact halfway point should mark the beginning of "rounding up"
#mtbos 6/n
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extending these and focusing just on what lets you round to 10, we would have these rules
round to 10 if
[a] 9.5 <= x < 10.5
[b] 9.5 < x <= 10.5
[c] 9.5 < x <= 10.5
[d] 9.5 <= x < 10.5
now there's a bit of a quagmire because the tussle is around which way 0.5 rounds
#mtbos 5/n
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IOW compare
[a] round to 10 if 10 <= x < 10.5 and round to 11 if 10.5 <= x < 11
vs
[b] round to 10 if 10 < x <= 10.5 and to 11 if 10.5 < x <= 11
vs
[c] round to 10 if 10 <= x <= 10.5 and to 11 if 10.5 < x < 11
vs
[d] round to 10 if 10 < x < 10.5 and to 11 if 10.5 <= x <= 11
#mtbos 4/n
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the latter needs clarity around the definition / convention for "nearest". on the one hand, 10.5 is equidistant to 10 and 11 (distance of a 0.5). but you have to decide if you are going to work with half open, left closed intervals, or half open, right closed intervals.
#mtbos 3/n
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understandably it is confusing because sometimes the conflation is in the word "round". the question "round 10.5 to the ones place" is different from "round 10.5 to the nearest one". the former needs clarity if "round" means "smallest integer above" or "greatest integer below".
#mtbos 2/n