For whatever reason, I'd like to become a human calculator or at least improve my math skills so that they kill.
Lately, I have simply been focusing on multiplication (inverse division, except I haven't done much division itself) memorizing tables and doing practise.
Back in primary school, I could do up to higher 2-digit by 2-digit or even 3-digit by 3-digit multiplication (while others were working with 0 - 10 * 0 - 10), but now I've just slowed down.
Any one a human calculator or have any pro tips (they'd be much appreciated)?
I am not one at all but I did grow up with a guy that was. He always said his secret was two things. First, he'd quiz himself constantly. Any time he saw numbers he'd do some sort of math problem with them in his head as a game. For example, if he looked at a clock and the time was 6:47 he'd quickly multiply 6x47. Things like that. Second, he played a lot of card games so he was always doing quick adding and subtracting.
I love numbers. I do basic arithmetic when I'm bored. I look for number patterns in license plates (4835? 4 doubled is 8 minus 3 is 5, which is 1 more than 4, which is 1 more than 3; if you square 3 and subtract that 1, you get 8; things like that. It gets intense when you factor in a numerical value to any letters, like X = 22). If someone asks me a math problem, I visualize working the problem out on paper in my head. My teachers always took points off for not showing my work because "there's no way you could've figured that out in your head". Except I did figure it out in my head. I think the easiest way for you to do larger problems in your head is through visualization, honestly - for example, if I want to know what 53 times 6 is, then I'll separate 50 from 3 and turn it into 5; 5 times 6 is 30, add on a 0 for it being multiplied by 10, then multiply 3 and 6 for 18 so we get 318. Set aside numbers or line them up in your visual memory as you're calculating so you don't forget your placeholders!
Another example, and I'm making this up at the top of my head:
312
*41
____
Well, you can take the 1 and multiply 312 by that, so we know to add 312 to the final product of whatever 40 times 312 is. Multiply 4 by 2, we have an 8 in the ones column, 1 by 4, 4 in the tens, 4 by 3, 1 in the thousands, 2 in the hundreds. Add 1248 to 312 for 1560 and you have your answer. It's all about splitting it up into smaller problems.
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I love numbers. I do basic arithmetic when I'm bored. I look for number patterns in license plates (4835? 4 doubled is 8 minus 3 is 5, which is 1 more than 4, which is 1 more than 3; if you square 3 and subtract that 1, you get 8; things like that. It gets intense when you factor in a numerical value to any letters, like X = 22). If someone asks me a math problem, I visualize working the problem out on paper in my head. My teachers always took points off for not showing my work because "there's no way you could've figured that out in your head". Except I did figure it out in my head. I think the easiest way for you to do larger problems in your head is through visualization, honestly - for example, if I want to know what 53 times 6 is, then I'll separate 50 from 3 and turn it into 5; 5 times 6 is 30, add on a 0 for it being multiplied by 10, then multiply 3 and 6 for 18 so we get 318. Set aside numbers or line them up in your visual memory as you're calculating so you don't forget your placeholders!
Another example, and I'm making this up at the top of my head:
312
*41
____
Well, you can take the 1 and multiply 312 by that, so we know to add 312 to the final product of whatever 40 times 312 is. Multiply 4 by 2, we have an 8 in the ones column, 1 by 4, 4 in the tens, 4 by 3, 1 in the thousands, 2 in the hundreds. Add 1248 to 312 for 1560 and you have your answer. It's all about splitting it up into smaller problems.
I use the same method and the same thing used to happen to me back in school.
You fouled up your second example. You forgot your own advice!
12480 + 312 = 12792
Another example, and I'm making this up at the top of my head:
312
*41
____
Well, you can take the 1 and multiply 312 by that, so we know to add 312 to the final product of whatever 40 times 312 is. Multiply 4 by 2, we have an 8 in the ones column, 1 by 4, 4 in the tens, 4 by 3, 1 in the thousands, 2 in the hundreds. Add 1248 to 312 for 1560 and you have your answer. It's all about splitting it up into smaller problems.
Sorry, but how else would you do your multiplication then?
My mother sent me some 'Indian multiplication' thing (not at all Indian and not really a thing; in fact, more of a reference and that was it; she didn't bother finding what she had read on a variant of WhatsApp). That prompted me to look into it and I found this: http://www.youtube.com/watch?v=bnP1Ji1a-2w
Crazy!
It's more of trick, and it's essentially the same thing as (normal) multiplication. As Iso said, it's like breaking up the algorithm for multiplying into multiple further steps.
(Even with enough practise, which is essential to be wicked good at math, I don't think I'd be any slower with standard multiplication, admittedly.)
Okay, in addition to becoming good at arithmetic, how do you become awesome at nigh instantaneous recall of the products of two-digit multiples? Again, I have just been doing the rote learning charts, practising for familiarity, and doing speed tests (with random number generator websites or my calculator).
While I was preparing for case interviews at consulting firms, quick mental math was heavily emphasized in all the material I read. I'm not naturally gifted at mental arithmetic, so the only solution I had was simply to practice constantly. The good news is that because it's such a common part of the case interview process, there are lots of online tools to help practice. A quick Google for "consulting math practice" or some variant should give you tons of results. Good luck!
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While I was preparing for case interviews at consulting firms, quick mental math was heavily emphasized in all the material I read. I'm not naturally gifted at mental arithmetic, so the only solution I had was simply to practice constantly. The good news is that because it's such a common part of the case interview process, there are lots of online tools to help practice. A quick Google for "consulting math practice" or some variant should give you tons of results. Good luck!
Thanks, Brandon. Most of it sounds like conventional wisdom; practice makes perfect and knuckling down. I'll definitely look into 'consulting math practice' and the like.
I'm not going to pretend to be a human calculator, but a lack of adding machine as a kid meant I had to learn to find some shortcuts. One of the biggest things I can recommend is 'simplify' and 'factor'. 312 * 41 is harder than, say, 104 * 41 * 3. The further you can simplify without adding undue steps the easier your life will be. 52 * 41 * 2 * 3 should be even easier, I think.
The other bit is 'chunking' it like Iso said. (300 * 41) + (12 * 41) is easier than 312 * 41.
But the biggest thing here is 'simplify'. Make it as simple as possible so you're not juggling 5 digit numbers in your head.
More of a concern is the advice that you kindly provided above is how most people normally do multiplication.
Surely, people don't sum 312 forty-one times or 41 there hundred and twelve times??
Most people don't do mental math at all. I don't understand what you're getting at - I was trying to explain the process I go through when I'm performing mental math (that of the breaking it into bits and keeping a mental tally of all of the numbers I'm juggling). Did I not explain myself well?
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Is there any reason you want to become better at math? The ability to do math in your head is not a particularly valuable skill in a world of calculators. You would be far better off learning about algorithmic complexity, as that is something computers have a harder time doing and is more applicable to real life.
More on topic, I usually try to find the unique prime factorization of all the numbers I am multiplying/dividing. That makes it a lot easier for me (unless I'm dealing with large prime numbers, of course).
Is there any reason you want to become better at math? The ability to do math in your head is not a particularly valuable skill in a world of calculators.
I very much have to disagree on this one. While you may not notice it, you're typically using math on a regular basis throughout the day for everyday things. Even something as simple as glancing at your car's estimated remaining mileage and saying "I have enough for X amount of trips before I have to fill up." Or looking at your bank account and quickly calculating if you have enough for your remaining bills.
Since I work in retail and one of the things I have to do regularly is sell a card that gets 10% off a purchase, I regularly have to do a quick 10% off in my head. Not hard, but then I have to factor in the taxes on the remainder so I can give people estimates on their bill. Learning fast ways to do mental math helps.
Sure, I could use a calculator. But if I can get my answer faster than it'd take me to pull out the calculator and do the math, I'd rather do it in my head.
Mental math also lets you have a rough idea what answer the calculator should give you. If you see 312 x 41, you should be able to estimate that the answer will be around 12000ish. That way when you use the calculator, if you make a mistake and it puts out the answer 1200, well you don't just take it at face value.
Being able to estimate a correct answer, then confirm it with a calculator is a good skill to have, rather than simply taking whatever answer the calculator gives you.
I think if you want to be REALLY good, you need to practice thinking of numbers spatially (like Daniel Tammet) or train on an abacus until you don't need an abacus. An abacus is probably more likely for 'normal' folk as Daniel is a synesthete, but I think it's possible if you trained really hard. That would make numbers so beautiful, in a direct and sensory way. Would be pretty cool
I'm a bit late, but I tend to do some pretty big math for "normal people" in my head which tends to impress them.
I find "simplifying the multiplication" makes things harder.
For example, yesterday, I was doing 7^7 (which I got wrong, I got 831343 or something, instead of hold on... 823543 (someone check that for me, it litterally took me like 10 seconds, but I'm unsure (I remember being wrong and what I said, not the real answer)) and I ended up with 343*2401 (which should give you the number above. I feel like the way people tell you to do it would be (((((7*7)*7)*7)*7)*7)*7). I tend to try to make the two sides as close as possible, to get the more manageable numbers. 6^6 for example, I'd do 216^2. (real fast, ummm, 46656?)
But that's just me :/
Edit: for 7^7, my steps were this:
7^7 = 343 *49*49 (this is the part where I don't have to calculate, cause it's easy to do in your head. Kind like whats 1+1, you never calculate it, you just remember it)
then:
=343*2401 (then i split it as:)
=343*(2500-100+1)
=857500-34300+343 (yesterday, what I did wrong is I got mixed up (like always) and 8575 became 8750 when I did the subtraction)
=823543
I'm a bit late, but I tend to do some pretty big math for "normal people" in my head which tends to impress them.
I find "simplifying the multiplication" makes things harder.
For example, yesterday, I was doing 7^7 (which I got wrong, I got 831343 or something, instead of hold on... 823543 (someone check that for me, it litterally took me like 10 seconds, but I'm unsure (I remember being wrong and what I said, not the real answer)) and I ended up with 343*2401 (which should give you the number above. I feel like the way people tell you to do it would be (((((7*7)*7)*7)*7)*7)*7). I tend to try to make the two sides as close as possible, to get the more manageable numbers. 6^6 for example, I'd do 216^2. (real fast, ummm, 46656?)
But that's just me :/
The issue is probably retaining all of those numbers faithfully. I know what is holding me back from doing insane (properly insane) mental arithmetic is that; I'm okay with clever little algorithm or tricks.
Mental math also lets you have a rough idea what answer the calculator should give you. If you see 312 x 41, you should be able to estimate that the answer will be around 12000ish. That way when you use the calculator, if you make a mistake and it puts out the answer 1200, well you don't just take it at face value.
Being able to estimate a correct answer, then confirm it with a calculator is a good skill to have, rather than simply taking whatever answer the calculator gives you.
Very, very good post. Thank you very much for such a helpful and thought-out post.
I think if you want to be REALLY good, you need to practice thinking of numbers spatially (like Daniel Tammet) or train on an abacus until you don't need an abacus. An abacus is probably more likely for 'normal' folk as Daniel is a synesthete, but I think it's possible if you trained really hard. That would make numbers so beautiful, in a direct and sensory way. Would be pretty cool
I don't believe you can become a synesthete. If you could, the likelihood of becoming or learning to become is likely to be inversely correlated with age. Though I know Jed Fonner (from the MIT Media Lab) are trying to develop toys to simulate an experience of synesthesia.
The abacus part is interesting. I might look into this.
But the biggest thing here is 'simplify'. Make it as simple as possible so you're not juggling 5 digit numbers in your head.
I even simplify when adding and subtracting. Take 300 - 257 = 43, I turn the problem into (300 - 260) + 3 = 40 + 3 = 43. I have been doing math problems this way in my head for so long now that simplifying is automatic for me. I find it's a much faster way for me to do problems in my head.
Okay, I guess the most key question here is, "but can you guys do these questions, unassisted, without pencil or paper (and certainly without any calculating apparatus), within a maximum of three seconds?". That, my friends, is the speed we're aiming for.
This thread has already mentioned most of the basic useful tricks, which form the bread and butter of my mental calculations. The most useful trick I use which hasn't been mentioned yet is exploiting the difference of two squares: a^2-b^2=(a+b)(a-b). So for instance, 47*77=(62-15)(77-15)=62^2-15^2=3844-225=3619. For calculating the squares themselves you can save time with the multinomial expansion of the expanded form, so 62^2=(6^2)10^2+(2*2*6)10^1+2^2=3600+240+4=3844 shortcutting 62^2=(60+2)^2=(6*10+2)^2=(6^2)10^2+(2*2*6)10^1+2^2. 62 is a pretty simple example, so for instance I'd just reduce 1377^2 to (13^2)10^4+(2*13*77)10^2+77^2 by chunking the two digit powers together.
This combines with the above techniques because you have to either add or subract one if the numbers you're multiplying have different parity: 87*34=87(33+1)=(60^2-27^2)+88=3600-729+87=2958. As you may have noticed things are much easier if you can get a or b to have fewer non-zero digits, so you might want to make 855+331 into 851*331+4*331=591^2-260^2+4*331 rather than 593^2-262^2. The main increase in convenience is that you only have to memorize squares. As demonstrated with the 1377 example you can use this recursively, with the sizes you choose determined by what you think you can square quickly. Having a lot of the 2 digits squares memorized is a common byproduct of using this alot, but if you memorized some 3 digits squares you'd get a speedup and if want to mostly reduce things to one digit squares that also works.
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1) Whenever you encounter an opportunity to do basic math, use it. (35% off a shirt that's $40? Two cheeseburgers and a medium fries, plus tax and tip, split two ways? etc.)
2) Simplify then compensate. For example,
57 x 32 is like 50 x 30, except you need to add 7 x 32 and 2 x 57.
94 x 27 is like 94 x 25, then add 94 to that twice.
312 x 41 is like 300 x 41, plus 10 x 41, plus 2 x 41.
Lately, I have simply been focusing on multiplication (inverse division, except I haven't done much division itself) memorizing tables and doing practise.
Back in primary school, I could do up to higher 2-digit by 2-digit or even 3-digit by 3-digit multiplication (while others were working with 0 - 10 * 0 - 10), but now I've just slowed down.
Any one a human calculator or have any pro tips (they'd be much appreciated)?
Another example, and I'm making this up at the top of my head:
312
*41
____
Well, you can take the 1 and multiply 312 by that, so we know to add 312 to the final product of whatever 40 times 312 is. Multiply 4 by 2, we have an 8 in the ones column, 1 by 4, 4 in the tens, 4 by 3, 1 in the thousands, 2 in the hundreds. Add 1248 to 312 for 1560 and you have your answer. It's all about splitting it up into smaller problems.
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I use the same method and the same thing used to happen to me back in school.
You fouled up your second example. You forgot your own advice!
12480 + 312 = 12792
Sorry, but how else would you do your multiplication then?
My mother sent me some 'Indian multiplication' thing (not at all Indian and not really a thing; in fact, more of a reference and that was it; she didn't bother finding what she had read on a variant of WhatsApp). That prompted me to look into it and I found this:
http://www.youtube.com/watch?v=bnP1Ji1a-2w
Crazy!
It's more of trick, and it's essentially the same thing as (normal) multiplication. As Iso said, it's like breaking up the algorithm for multiplying into multiple further steps.
(Even with enough practise, which is essential to be wicked good at math, I don't think I'd be any slower with standard multiplication, admittedly.)
Okay, in addition to becoming good at arithmetic, how do you become awesome at nigh instantaneous recall of the products of two-digit multiples? Again, I have just been doing the rote learning charts, practising for familiarity, and doing speed tests (with random number generator websites or my calculator).
Cheers.
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Surely, people don't sum 312 forty-one times or 41 there hundred and twelve times??
Thanks, Brandon. Most of it sounds like conventional wisdom; practice makes perfect and knuckling down. I'll definitely look into 'consulting math practice' and the like.
Many thanks!
The other bit is 'chunking' it like Iso said. (300 * 41) + (12 * 41) is easier than 312 * 41.
But the biggest thing here is 'simplify'. Make it as simple as possible so you're not juggling 5 digit numbers in your head.
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Most people don't do mental math at all. I don't understand what you're getting at - I was trying to explain the process I go through when I'm performing mental math (that of the breaking it into bits and keeping a mental tally of all of the numbers I'm juggling). Did I not explain myself well?
{мы, тьма}
2012: Best (False?) Role Claim - Worst Town Performance (Group) - Best Mafia Performance (Group) - Best SK Performance - Best Overall Player
2013: Best Non-SK Neutral Performance
2014: Best Town Performance (Individual) - Best Town Performance (Group) - Most Interesting Role - Best Game - Best Overall Player
2015: Worst Mafia Performance (Group) - Best Read
2016: Best Town Performance (Group) - Best Town Player - Best Overall Player
More on topic, I usually try to find the unique prime factorization of all the numbers I am multiplying/dividing. That makes it a lot easier for me (unless I'm dealing with large prime numbers, of course).
I very much have to disagree on this one. While you may not notice it, you're typically using math on a regular basis throughout the day for everyday things. Even something as simple as glancing at your car's estimated remaining mileage and saying "I have enough for X amount of trips before I have to fill up." Or looking at your bank account and quickly calculating if you have enough for your remaining bills.
Since I work in retail and one of the things I have to do regularly is sell a card that gets 10% off a purchase, I regularly have to do a quick 10% off in my head. Not hard, but then I have to factor in the taxes on the remainder so I can give people estimates on their bill. Learning fast ways to do mental math helps.
Sure, I could use a calculator. But if I can get my answer faster than it'd take me to pull out the calculator and do the math, I'd rather do it in my head.
My helpdesk should you need me.
That being said, I don't know of anything such formulas off the top of my head.
Being able to estimate a correct answer, then confirm it with a calculator is a good skill to have, rather than simply taking whatever answer the calculator gives you.
I find "simplifying the multiplication" makes things harder.
For example, yesterday, I was doing 7^7 (which I got wrong, I got 831343 or something, instead of hold on... 823543 (someone check that for me, it litterally took me like 10 seconds, but I'm unsure (I remember being wrong and what I said, not the real answer)) and I ended up with 343*2401 (which should give you the number above. I feel like the way people tell you to do it would be (((((7*7)*7)*7)*7)*7)*7). I tend to try to make the two sides as close as possible, to get the more manageable numbers. 6^6 for example, I'd do 216^2. (real fast, ummm, 46656?)
But that's just me :/
Edit: for 7^7, my steps were this:
7^7 = 343 *49*49 (this is the part where I don't have to calculate, cause it's easy to do in your head. Kind like whats 1+1, you never calculate it, you just remember it)
then:
=343*2401 (then i split it as:)
=343*(2500-100+1)
=857500-34300+343 (yesterday, what I did wrong is I got mixed up (like always) and 8575 became 8750 when I did the subtraction)
=823543
The issue is probably retaining all of those numbers faithfully. I know what is holding me back from doing insane (properly insane) mental arithmetic is that; I'm okay with clever little algorithm or tricks.
Very, very good post. Thank you very much for such a helpful and thought-out post.
I don't believe you can become a synesthete. If you could, the likelihood of becoming or learning to become is likely to be inversely correlated with age. Though I know Jed Fonner (from the MIT Media Lab) are trying to develop toys to simulate an experience of synesthesia.
The abacus part is interesting. I might look into this.
Okay, I guess the most key question here is, "but can you guys do these questions, unassisted, without pencil or paper (and certainly without any calculating apparatus), within a maximum of three seconds?". That, my friends, is the speed we're aiming for.
This combines with the above techniques because you have to either add or subract one if the numbers you're multiplying have different parity: 87*34=87(33+1)=(60^2-27^2)+88=3600-729+87=2958. As you may have noticed things are much easier if you can get a or b to have fewer non-zero digits, so you might want to make 855+331 into 851*331+4*331=591^2-260^2+4*331 rather than 593^2-262^2. The main increase in convenience is that you only have to memorize squares. As demonstrated with the 1377 example you can use this recursively, with the sizes you choose determined by what you think you can square quickly. Having a lot of the 2 digits squares memorized is a common byproduct of using this alot, but if you memorized some 3 digits squares you'd get a speedup and if want to mostly reduce things to one digit squares that also works.
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Z x 9 = (Z x 10) - Z.
1) Whenever you encounter an opportunity to do basic math, use it. (35% off a shirt that's $40? Two cheeseburgers and a medium fries, plus tax and tip, split two ways? etc.)
2) Simplify then compensate. For example,
57 x 32 is like 50 x 30, except you need to add 7 x 32 and 2 x 57.
94 x 27 is like 94 x 25, then add 94 to that twice.
312 x 41 is like 300 x 41, plus 10 x 41, plus 2 x 41.
47 x 77 is like (50 x 77) - (3 x 77).
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25: 25.1%
26: 25.3%