How long does it take you to calculate the pythagorean theorem answer (to one decimal place) for any random set of two digits as your adjacent and opposite, all in your head. No paper, no calculator etc.

I was doing some in my head and wondered how long it took me. Did a stopwatch to check. It seems that for me it is between 1:15-2:30 (the square root takes the longest time so it depends on how complicated it is), idk if that is good or bad though. Wondering what it is for others now. So how long does it take you?

Also, any tips and tricks for doing mental math faster? My Physics E&M class Exams have a short time limit, so I wanna be able to do this type of basic stuff like square roots and shit faster just in my head just like addition or subtraction, but I seem to get stuck on decimals, fractions etc. Regular integer multiplication, addition etc. I can do really quick, almost intuitively, but anything involving non-integers fricks me over and ends up taking too much time. What do?

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>How long does it take you to calculate the pythagorean theorem answer (to one decimal place) for any random set of two digits as your adjacent and opposite, all in your head

about 5 seconds

1^2 + 2^2 = 3^2

the answer is 3^2

ta-da

>5 seconds to do 1+2

ngmi

Doing shit quick is what chimps do. Doing it right is what we do. Just get a fricking paper & pencil on the Exam, you don't have to do it in your head, it proves nothing if you get it wrong. So don't take the risk.

Well unfortunately then everybody wants us to be "chimps". Most tests and exams test for speed in addition to accuracy. I'm sure you remember the SAT. Not conceptually hard. If given infinite time, quite a large number of people would get perfect scores on the math sections. But being able to do it within the time limits matters. Our Physics exams are similar. I got a low score on one of mine last semester because I ran out of time, not necessarily because I didn't know it. And at our college, most people run out of time, that's just how the exams are made. In addition to testing for accuracy and if you can do it to begin with, they also test for how intuitively and quick you can do it. Because based on just accuracy with no time limitations, most of the class would get near perfect scores.

That's why I want to learn to do it quicker.

Yeah that's a very nice method, that's actually pretty close to how I do multiplication in my head, and it is indeed very quick like this. I feel like I can store a really large amount of Information, have good working memory. But when it comes to going beyond just simple integer multiplication, is it just practice until you have them memorized? no method? I feel like if one is able to address the technique, you don't really need to memorize by repetition as much, or is that the only way would you say?

>I feel like I can store a really large amount of Information, have good working memory. But when it comes to going beyond just simple integer multiplication, is it just practice until you have them memorized? no method? I feel like if one is able to address the technique, you don't really need to memorize by repetition as much, or is that the only way would you say?

Like for example, I mentioned square roots in OP post, what I do in my head right now is

>Trial and error by squaring numbers until you get a Whole number N, which if you go 1 above it, you go beyond the number under the Sqrt, let's call it Y

>Take the difference of N^2 and Y

>Find out what percent of N the difference is by doing N/diff

>Take the % and trial and error again until you find a decimal that, when added to itself and added to it's square root gives you the % number (this trial and error is shorter because the # is close to half of the % number)

>Or in other words, find x such that 2x +x^2 = %

>Add x to N, that's your answer

I think that's what it is but idk if I'm expressing it right.

I'm sure there is a better way to do this, this seems highly inefficient as I came up with it on my own this morning and takes a wild amount of time, more than a minute, sometime 2 minutes, for a simple pythagorean theorem.

Similar long winded methods for other complex decimals and fractions involved mental calculations. I can do them, but takes too much time is the problem and I blame my methodology tbh

any multiplication can be expressed as integer multiplication. I always work with number between 1 and 999 and the SI prefixes. I express most multiplication/division problems as ratios rather a*b or a/b. Use a slide rule for quick approximations of functions I can't calculate, I also remember some values that come in handy like pi to 5th decimal or ln(2) to 3rd decimal. Feynman explained something similar in his story about the time two dudes were asking him values of some hard to compute function for given x, but they asked him numbers that he knew by heart or could quickly compute using the ones he knew. I personally think that mental math should only be used to give you a hunch what the right answer could be and then focus on exact solution on paper. Like I said, I carry around a slide rule, but often there's no need to use it if there are no trig functions or roots or a^x or logs. I'm a molecular biologist, so most of the calculations I do are related to biochemistry labs (efficiency, molar mass, dilution of solutions), but what I do comes in handy whenever there is arithmetic involved.

> multiplication: know picrel by heart and apply the a(b+c)=ab+ac

> division: since you know picrel you know closest multiple of divisor — 67/7, closest is 70, which tells you instantly that 67/7 is 7 + 3/7, 3/7 is 30/7 *.1, 28 is the closest and remainder is 2 — 7.4 + 20/7 *.01, closest to 20 is 21 which is too high, but 21-7 = 14 => 7.42 + 6/7 => 7.428 and so on. This is what you do with long division, but you can do most of the process mentally and just add consecutive decimal digits.

> roots on numbers without integer roots, non-integer powers of numbers and logs can be roughly calculated by linear differential approximation, which requires you to know ln2 by heart

If you need to be faster than what the above allows then there is a problem with exams you take

> closest is 70, which tells you instantly that 67/7 is 7 +

obviously false, tells you instantly that 67/7 must start with number 1 less than 7 => 6, but I hope you get the idea

actually the division is all wrong, I made a lot of errors, but if you work on not making mistakes and don't suffer a stroke like I did you can reliably calculate a/b to 10 decimal places in 10 sec

I see, that is a very helpful response, thanks for writing it out anon.

>or ln(2) to 3rd decimal.

>non-integer powers of numbers and logs can be roughly calculated by linear differential approximation, which requires you to know ln2 by heart

I can kind of intuit why ln2 is important due to it's connection to the squared value, but I never used it tbh for mental math. How exactly must one use it for roots, logs and powers etc? I can already kind of tell that it will be really quick and helpful but can't fit the puzzle in by myself on what to do with it for solving.

>30/7 *.1

The division stuff is very useful, this above part especially. I didn't do it before but doing it like this does make it way quicker as compared to doing it like how I did it, which was literally imagine the image of a table like on the left in picrel in my head (idk what they're called and this is the only good result that showed up on google for 'division table') and literally write it all out in my head like middle school but just mentally lol, which is time consuming but gives accurate decimal answers. For the most part, 2 decimal places should be good for me, this makes it a bit more intuitive and quicker, will def save a lot of time, thanks.

Yeah I see the error. Instead of going closest number, should have been closest # <67. 70 could still be helpful though if one doesn't remember 63 off the top of their head. 7*10=70. 70-7=63=7*9, so 9 the integer. 4 left over. 4/7=(40*.1)/7. Closest to 40 while still < 40= 7*5=35. So 5 is first decimal. 9.571428 And so on (literally did this in like 5 secs and it is accurate as well). That correct right? I think I get the method, simple calc errors happen sometimes. This is def faster though, I appreciate it.

You seem good with fractions though, anything for fraction addition/subtraction? That always fricks me over as well especially when it's large and incongruent denominators.

You should look for the closes number, and if it is obviously too large you take one less, it's easier to remember that 63 is 7*10 - 1 than 6*9, and doesn't add much computation time, you got it right.

> The division stuff is very useful, this above part especially.

this is literally long division that we learn in grade school, weird that one of most useful "tricks" is taught and used since grade school yet everyone seems to not recognize its potential.

> I can kind of intuit why ln2 is important

picrel, and it's useful for changing log base 2 into log base e.

picrel is really useful, any value that grows exponentially as a function of some x can be approximated just by multiplying the current value by the change in x and adding the current value, provided that change in x is sufficiently small

> 63 is 7*10 - 1 than 6*9

once again an error, obviously 63 is 7*10 - 7*1, I think writing on the keyboard makes me more disconnected with the contents.

> picrel for general approach to linear approximation

in 2nd line of III there is another error, should be, a^x0 * ln2 * Deltax + a^x0

>tests

uhhh what are you even talking about pal? Exams don't exist in real life. No one if gonna point a gun at you in the Amazon jungle and ask you to compute a triple integral in your head.

if you are good at a(b+c)=(a*b)+(a*c) then you can do most arithmetic by writing consecutive answears,

> 13*27=260+70+21=351

it's harder purely mentally because you have to store more values, but is very easy with pen and paper. If you practice it for a bit a(b+c)=ab+ac with a slide rule for logs, squares and trig is much faster than using a calculator, and makes you feel more certain of the answer. No more of "didn't I accidentally input wrong digit in previous calculation?"

>13*27=260+70+21=351

Why wouldn't you break this one into 27×10+27×3, im not sure how you got 3 things adding each other with your formula you have

12*20 + 10*70 + 3*7

you can chain the formula how many times you like. 27*10 is instantly apparent to be 270, but 27*3 you must break into 20*3 + 7*3 for it to be instantly apparent to be 60+21=81, you can view 27*3 as 30*3 - 3*3 = 90 - 9 = 81, you do the same thing but attack from the back. It is not a hard rule or anything, but notice that when you do long multiplication on paper you multiply consecutive single digit numbers by each other. Multiplication by 1, 10, 100, 1000 ... doesn't change the digits, so is computed instantly, multiplication by anything else changes the digits, and rarely can be done instantly

> you can calculate 123*2 = 246 really fast, 789*2=1578 is still fast but slower since multiplication of each digit afects the digit to the left. I personally find it faster to split 13*27 into 10*27 + 3*20 + 3*7 rather than 10*27 + 3*27 and spend a while trying to calculate 3*27

Why would I ever do that?

>the square root takes the longest time so it depends on how complicated it is

Nobody ever did that. You looked it up in a table.

I literally listed right here

how I was doing it

Why would I use a chart anyway instead of a calculator?

The Amazon Jungle isn't real.

I took 7 and 8 because I think using low digits is cheating.

It took me about 10 to 15 seconds. As these = 106, the answer fairly trivially must be >10 without any further calculation; but 11 would be too high by the same token, as with a naive zero calculation magnitude shift itd be 110.

Thus besides the initial squaring/addition, we only have to consider which natural number squared evaluates to something within the space of numbers whose most significant digit is 6. This is autistictese for "is somewhere in the 60s".

Very good joke.

Such people don’t actually exist, i.e. work with numbers on a regular basis (like at a a school) but don’t know how to calculate roots of numbers <100.

For an approximation of [math]sqrt{n}[/math]that's easy to compute mentally, let [math]s[/math] be the nearest perfect square to [math]n[/math]. Then [math]sqrt{n}approxsqrt{s}+frac{n-s}{2sqrt{s}}[/math]. The error is less than [math]0.1[/math] for [math]nge3[/math], and less than [math]0.05[/math] for [math]nge7[/math]. This works well up to the point you've memorized the squares, which is probably at least 1000. More generally, you can take some best guess [math]a[/math] and refine it as [math]a=frac{a+n/a}{2}[/math]. As long as your initial guess is somewhat reasonable this converges extremely quickly and is easy to compute mentally. E.g., you want to take [math]sqrt{5300}[/math], your initial guess might be [math]70[/math], so computing one step is just:

[math]5300/70=75frac{5}{7}[/math]

[math]frac{75frac{5}{7}+70}{2}=72frac{6}{7}=72.857...[/math] which is close to the true value of [math]72.801...[/math].