I mean, if you give me something borderline nontrivial like, say 72 times 13, I will definitely do some similar stuff. “Well it’s more than 700 for sure, but it looks like less than a thousand. Three times seven is 21, so two hundred and ten, so it’s probably in the 900s. Two times 13 is 26, so if you add that to the 910 it’s probably 936, but I should check that in a calculator.”
I think what’s wild about it is that it really is surprisingly similar to how we actually think. It’s very different from how a computer (calculator) would calculate it.
So it’s not a strange method for humans but that’s what makes it so fascinating, no?
That’s what’s fascinating about how it does language in general.
The article is interesting in both the ways in which things are similar and the ways they’re different. The rough approximation thing isn’t that weird, but obviously any human would have self-awareness of how they did it and not accidentally lie about the method, especially when both methods yield the same result. It’s a weirdly effective, if accidental example of human-like reasoning versus human-like intelligence.
And, incidentally, of why AGI and/or ASI are probably much further away than the shills keep claiming.
Yes, agreed. And calculators are essentially tabulators, and operate almost just like a skilled person using an abacus.
We shouldn’t really be surprised because we designed these machines and programs based on our own human experiences and prior solutions to problems. It’s still neat though.
This is pretty normal, in my opinion. Every time people complain about common core arithmetic there are dozens of us who come out of the woodwork to argue that the concepts being taught are important for deeper understanding of math, beyond just rote memorization of pencil and paper algorithms.
This is what I do, except I would add 700 and 236 at the end.
Well except I would probably add 700 and 116 or something, because my working memory fucking sucks and my brain drops digits very easily when there’s more than 1
Know what’s crazy? We sling bags of mulch, dirt and rocks onto customer vehicles every day. No one, neither coworkers nor customers, will do simple multiplication. Only the most advanced workers do it. No lie.
Customer wants 30 bags of mulch. I look at the given space:
“Let’s do 6 stacks of 5.”
Everyone proceeds to sling shit around in random piles and count as we go. And then someone loses track and has to shift shit around to check the count.
That’s what I do in my head if I need an exact result. If I’m approximateing I’ll probably just do something like 70 * 15 which is much easier to compute (70 * 10 + 70 * 5 = 700 + 350 = 1050).
OK, I’ve been willing to just let the examples roll even though most people are just describing how they’d do the calculation, not a process of gradual approximation, which was supposed to be the point of the way the LLM does it…
…but this one got me.
Seriously, you think 70x5 is easier to compute than 70x3? Not only is that a harder one to get to for me in the notoriously unfriendly 7 times table, but it’s also further away from the correct answer and past the intuitive upper limit of 1000.
See, for me, it’s not that 7*5 is easier to compute than 7*3, it’s that 5*7 is easier to compute than 7*3.
I saw your other comment about 8’s, too, and I’ve always found those to be a pain, so I reverse them, if not outright convert them to arithmetic problems. 8x4 is some unknown value, but X*8 is always X*10-2X, although do have most of the multiplication tables memorized for lower values.
8*7 is an unknown number that only the wisest sages can compute, however.
I’ve always hated it and eight. I can only remember the ones that are familiar at a glance from the reverse table and to this day I sometimes just sum up and down from those “anchor” references. They’re so weird and slippery.
Going back to the “being friends” thing, I think you and I could be friends due to applying qualities to numbers; but I think it might be challenging because I find 7 and 8 to be two of the best. They’re quirky, but interesting.
Times 5 and times 10 tables are really easy for me. So yeah, in my mind it’s an easier comuptation.
That being said having a result of a little over a 1000 gives me an estimate for the magnitude of a number – it’s around a thousand. It might be more or less but it’s not far from there.
No it doesn’t, multiplication and division always take precedence over addition and subtraction. You’d need parentheses to clarify what is in the divisor since that can be ambiguous with line notation.
Pen and paper maths I’m pretty decent at, but ask me to calculate anything in my head and it’s anyone’s guess if I remembered to carry the 1 or not. Ever since learning about aphantasia I’m wondering if the lack of being able to visually store values has something to do with it.
Is that a weird method of doing math?
I mean, if you give me something borderline nontrivial like, say 72 times 13, I will definitely do some similar stuff. “Well it’s more than 700 for sure, but it looks like less than a thousand. Three times seven is 21, so two hundred and ten, so it’s probably in the 900s. Two times 13 is 26, so if you add that to the 910 it’s probably 936, but I should check that in a calculator.”
Do you guys not do that? Is that a me thing?
I think what’s wild about it is that it really is surprisingly similar to how we actually think. It’s very different from how a computer (calculator) would calculate it.
So it’s not a strange method for humans but that’s what makes it so fascinating, no?
That’s what’s fascinating about how it does language in general.
The article is interesting in both the ways in which things are similar and the ways they’re different. The rough approximation thing isn’t that weird, but obviously any human would have self-awareness of how they did it and not accidentally lie about the method, especially when both methods yield the same result. It’s a weirdly effective, if accidental example of human-like reasoning versus human-like intelligence.
And, incidentally, of why AGI and/or ASI are probably much further away than the shills keep claiming.
Yes, agreed. And calculators are essentially tabulators, and operate almost just like a skilled person using an abacus.
We shouldn’t really be surprised because we designed these machines and programs based on our own human experiences and prior solutions to problems. It’s still neat though.
I mean neural networks are modeled after biological neurons/brains after all. Kind of makes sense…
This is pretty normal, in my opinion. Every time people complain about common core arithmetic there are dozens of us who come out of the woodwork to argue that the concepts being taught are important for deeper understanding of math, beyond just rote memorization of pencil and paper algorithms.
Rote memorization should be minimized in school curriculum
deleted by creator
The problem with common core math isn’t that rounding is inherently bad, it’s that you don’t start with that as a framework.
How I’d do it is basically
72 * (10+3)
(72 * 10) + (72 * 3)
(720) + (3*(70+2))
(720) + (210+6)
(720) + (216)
936
Basically I break the numbers apart into easier chunks and then add them together.
This is what I do, except I would add 700 and 236 at the end.
Well except I would probably add 700 and 116 or something, because my working memory fucking sucks and my brain drops digits very easily when there’s more than 1
Nah I do similar stuff. I think very few people actually trace their own lines of thought, so they probably don’t realize this is how it often works.
Huh. I visualize a whiteboard in my head. Then I…do the math.
I’m also fairly certain I’m autistic, so… ¯\_(ツ)_/¯
I do much the same in my head.
Know what’s crazy? We sling bags of mulch, dirt and rocks onto customer vehicles every day. No one, neither coworkers nor customers, will do simple multiplication. Only the most advanced workers do it. No lie.
Customer wants 30 bags of mulch. I look at the given space:
“Let’s do 6 stacks of 5.”
Everyone proceeds to sling shit around in random piles and count as we go. And then someone loses track and has to shift shit around to check the count.
deleted by creator
But you wouldn’t multiply, say, 74*14 to get the answer.
Not, but I’d do 75*10 + 75*4, then subtract the extra.
The LLM method of doing it with multiple numbers without proper interpolation though makes it extra weird
I might. Then I can subtract 74 to get 74*14, and subtract 28 to get 72*13.
I don’t generally do that to ‘weird’ numbers, I usually get closer to multiples of 5, 9, 10, or 11.
But a computer stores information differently. Perhaps it moves closer to numbers with simpler binary addresses.
72 * 10 + 70 * 3 + 2 * 3
That’s what I do in my head if I need an exact result. If I’m approximateing I’ll probably just do something like 70 * 15 which is much easier to compute (70 * 10 + 70 * 5 = 700 + 350 = 1050).
OK, I’ve been willing to just let the examples roll even though most people are just describing how they’d do the calculation, not a process of gradual approximation, which was supposed to be the point of the way the LLM does it…
…but this one got me.
Seriously, you think 70x5 is easier to compute than 70x3? Not only is that a harder one to get to for me in the notoriously unfriendly 7 times table, but it’s also further away from the correct answer and past the intuitive upper limit of 1000.
See, for me, it’s not that 7*5 is easier to compute than 7*3, it’s that 5*7 is easier to compute than 7*3.
I saw your other comment about 8’s, too, and I’ve always found those to be a pain, so I reverse them, if not outright convert them to arithmetic problems. 8x4 is some unknown value, but X*8 is always X*10-2X, although do have most of the multiplication tables memorized for lower values.
8*7 is an unknown number that only the wisest sages can compute, however.
The 7 times table is unfriendly?
I love 7 timeses. If numbers were sentient, I think I could be friends with 7.
I’ve always hated it and eight. I can only remember the ones that are familiar at a glance from the reverse table and to this day I sometimes just sum up and down from those “anchor” references. They’re so weird and slippery.
Huh.
Going back to the “being friends” thing, I think you and I could be friends due to applying qualities to numbers; but I think it might be challenging because I find 7 and 8 to be two of the best. They’re quirky, but interesting.
Thank you for the insight.
Times 5 and times 10 tables are really easy for me. So yeah, in my mind it’s an easier comuptation.
That being said having a result of a little over a 1000 gives me an estimate for the magnitude of a number – it’s around a thousand. It might be more or less but it’s not far from there.
For me personally, anything times 5 can be reached by halving the number, then multiplying that number by 10.
Example: 66 x 5 = Y
(66/2) x (5x2) = Y
cancel out the division by creating equal multiplication in the other number
66/2 = 33
5x2 = 10
33 x 10 = Y
33 x 10 = 330
Y = 330
(72 * 10) + (2 * 3) = x
There, fixed, because otherwise order of operation gets fucky.
No it doesn’t, multiplication and division always take precedence over addition and subtraction. You’d need parentheses to clarify what is in the divisor since that can be ambiguous with line notation.
Well, I guess I do a bit of the same:) I do (70+2)(10+3) -> 700+210+20+6
I wouldn’t even attempt that in my head.
I can’t keep track of things and then recall them later for the final result.
Pen and paper maths I’m pretty decent at, but ask me to calculate anything in my head and it’s anyone’s guess if I remembered to carry the 1 or not. Ever since learning about aphantasia I’m wondering if the lack of being able to visually store values has something to do with it.
deleted by creator
I can visually store values and I still struggle. :(
I would do 720 + 3 * 70 + 3 * 2