Favorite mental math tricks/shortcuts

For the discussion of math. Duh.

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Blatm
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Re: Favorite mental math tricks/shortcuts

Postby Blatm » Fri Nov 14, 2008 1:59 am UTC

Which book? I read Surely you're joking, Mr. Feynman! (great book, by the way), and I don't remember seeing it in that.

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Re: Favorite mental math tricks/shortcuts

Postby brodieboy255 » Fri Nov 14, 2008 2:38 am UTC

successive numbers of the fibonacci sequence are approximate mile-to-kilometre conversions

5mi ~ 8km
8mi ~ 13Km

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Re: Favorite mental math tricks/shortcuts

Postby skeptical scientist » Fri Nov 14, 2008 2:55 am UTC

Interesting. That's because the mile/km ratio is close to the golden ratio.
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Re: Favorite mental math tricks/shortcuts

Postby joeframbach » Fri Nov 14, 2008 3:02 am UTC

Blatm wrote:Which book? I read Surely you're joking, Mr. Feynman! (great book, by the way), and I don't remember seeing it in that.

The part with the Math-Off between him and the abacus salesman in the restaurant. Cubic root of 1729.

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Re: Favorite mental math tricks/shortcuts

Postby qinwamascot » Fri Nov 14, 2008 3:21 am UTC

That's not too hard at all. But then again, I do 13th root competitions, so cube roots of 4 digit numbers are sorta boring (unless you want more than 3-4 digits).

Most of my favorites are little tricks I learned to estimate the arclength of curves. You can memorize the integrals for the easy ones, and the hard ones have relatively close approximations. My calc teacher was super impressed when I guessed the length of a catenary to 5 decimal places in under 30 seconds without using a calculator. But that one was a bit lucky :P

Actually if you do a lot of integrals they get pretty easy to do in your head, which is nice.

I've also found extracting square roots to be slightly faster than logarithmic calculations if I need more than a few digits.
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Re: Favorite mental math tricks/shortcuts

Postby Gojoe » Fri Nov 14, 2008 4:42 am UTC

qinwamascot wrote:That's not too hard at all. But then again, I do 13th root competitions, so cube roots of 4 digit numbers are sorta boring (unless you want more than 3-4 digits).

Most of my favorites are little tricks I learned to estimate the arclength of curves. You can memorize the integrals for the easy ones, and the hard ones have relatively close approximations. My calc teacher was super impressed when I guessed the length of a catenary to 5 decimal places in under 30 seconds without using a calculator. But that one was a bit lucky :P

Actually if you do a lot of integrals they get pretty easy to do in your head, which is nice.

I've also found extracting square roots to be slightly faster than logarithmic calculations if I need more than a few digits.
Are you trying to say that your trick is... just practice and memorize all your results? That is not really a trick...
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Re: Favorite mental math tricks/shortcuts

Postby qinwamascot » Fri Nov 14, 2008 5:45 am UTC

i guess you're right. hmm... then maybe the Alexis Lemaire 100 digit equivalence for 13th root of a 100 digit number (x13=y <=> x=y77 for the last 4 digits of x). I've gotten quite good at taking 77th powers now lol. Combine this with a little other stuff and 13th roots become relatively easy. I'm sure there's a faster method out there now, but this one is nice for beginners like me.
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Re: Favorite mental math tricks/shortcuts

Postby Alpha Omicron » Sat Nov 15, 2008 6:21 am UTC

A common one (which may already be above) for multiplication (works best on 2-digit x 2-digit problems):

Code: Select all

  34
x 71

= (1*4) + (3*1*10) + (7*4*10)+(3*7*100)
= 4 + 30 + 280 + 2100
= 2414


Also, Russian peasant multiplication:

Code: Select all

34 |   71 |
17 |  142 | 142
8  |  284 |
4  |  568 |
2  | 1136 | 
1  | 2272 | 2272

142 + 2272 = 2414
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Re: Favorite mental math tricks/shortcuts

Postby Simplex » Thu Jan 22, 2009 3:52 am UTC

I was playing with numbers in my head on a long train ride a few weeks ago, and came across something i felt like sharing with you. There's allot of shortcut methods already for doing multiplication / finding squares, but i figured out a nice niche one that i haven't seen posted here or anywhere else before. :)

x^2 = (x+3)(x-2)-(x-6)
(for all x>2, obviously :P)
10^2 = (13*8)-4 = 100

It becomes handier for doing squares of larger numbers, such as 97^2, which just becomes 100*95-91=9409. I've started using it for all numbers ending in 2's and 7's i want to find the square of. By the way, if anyone wants to actually prove why that works for all numbers >2, they get a cookie. :P

Another one of my favorites is x^2+2x+1 = (x+1)^2, which is a quick way of taking advantage of the difference between successive squares incrementing by 2 each time. ((1) +3=4, +5=9, +7=16, +9=25, etc etc.) Pretty sure i haven't seen that posted in this thread yet, sorry if it has been.

I'll admit my mind refuses to commit 16^2 to memory for some reason, i still just add 31 to 225 for it :oops:

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Re: Favorite mental math tricks/shortcuts

Postby joeframbach » Thu Jan 22, 2009 4:10 am UTC

Alpha Omicron wrote:Also, Russian peasant multiplication:

Code: Select all

34 |   71 |
17 |  142 | 142
8  |  284 |
4  |  568 |
2  | 1136 | 
1  | 2272 | 2272

142 + 2272 = 2414
This is a common implementation in computer processors, right? With the whole being-good-at-powers-of-two thing.

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Re: Favorite mental math tricks/shortcuts

Postby Alpha Omicron » Thu Jan 22, 2009 5:45 am UTC

joeframbach wrote:This is a common implementation in computer processors, right? With the whole being-good-at-powers-of-two thing.

I don't know.
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Re: Favorite mental math tricks/shortcuts

Postby chapel » Fri Jan 23, 2009 5:09 am UTC

(x^2 - y^2) = (x + y)(x - y)

Very useful for the 10th graders I teach.

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Re: Favorite mental math tricks/shortcuts

Postby joeframbach » Fri Jan 23, 2009 5:15 am UTC

chapel wrote:(x^2 - y^2) = (x + y)(x - y)

Very useful for the 10th graders I teach.

I usually use this in the opposite direction.
17*13 = 15^2-2^2 = 221

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Re: Favorite mental math tricks/shortcuts

Postby FrictionDefender » Fri Jan 23, 2009 7:19 pm UTC

Well, it's not entirely mental, but you can get any single-digit number x multiplied by nine by putting both hands out flat in front of you, lowering the xth finger from the left. The number of fingers left of the lowered finger is the first digit of the answer and the number right of the lowered finger is the second digit.
For example (please excuse my uninspired ASCII art):

Code: Select all

9*2:
| ||   ||||
||||/ \||||

One finger is left of the lowered finger, so the tens digit is one, and eight fingers are right of the lowered finger, so the ones digit is eight.

Of course, this is only useful if you're miserable at arithmetic, like me :( .

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Re: Favorite mental math tricks/shortcuts

Postby MarshyMarsh » Fri Jan 23, 2009 7:42 pm UTC

The above method has resulted in me never learning the 9* table. 1-20 I can do like !clicks fingers! that, but for 9 times table, allthough there in my memory, I always pull out my hands like a cheesy western.

On the subject of the number 9, a primary school method of determining if a number is a factor of 9 (and 3 subsequently) is to add up all the digits of the number, they should equal a recognizable multiple of 9:

81 = 8+1 = 9

22221 = 2+2+2+2+1 = 9

954936 = 9+5+4=9+3+6 = 36

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Re: Favorite mental math tricks/shortcuts

Postby joeframbach » Fri Jan 23, 2009 9:41 pm UTC

MarshyMarsh wrote:On the subject of the number 9, a primary school method of determining if a number is a factor of 9 (and 3 subsequently) is to add up all the digits of the number, they should equal a recognizable multiple of 9:

81 = 8+1 = 9

22221 = 2+2+2+2+1 = 9

954936 = 9+5+4=9+3+6 = 36

This is also recursive(is that the right word?).
ex. 8967958254 -> 63 -> 9

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Re: Favorite mental math tricks/shortcuts

Postby Xanthir » Fri Jan 23, 2009 11:12 pm UTC

FrictionDefender wrote:Well, it's not entirely mental, but you can get any single-digit number x multiplied by nine by putting both hands out flat in front of you, lowering the xth finger from the left. The number of fingers left of the lowered finger is the first digit of the answer and the number right of the lowered finger is the second digit.
For example (please excuse my uninspired ASCII art):

Code: Select all

9*2:
| ||   ||||
||||/ \||||

One finger is left of the lowered finger, so the tens digit is one, and eight fingers are right of the lowered finger, so the ones digit is eight.

Of course, this is only useful if you're miserable at arithmetic, like me :( .

I don't understand why this is difficult for anyone. My wife can't do X*9 easily either. If X is single-digit, the first digit of the answer is X-1, and the second is 10-X. I learned this when I was a wee one in elementary school, and though my times table is now firmly memorized, it proved very useful for several years.
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Re: Favorite mental math tricks/shortcuts

Postby Uskok » Sat Jan 24, 2009 1:21 am UTC

Simplex wrote:x^2 = (x+3)(x-2)-(x-6)
(for all x>2, obviously :P)
10^2 = (13*8)-4 = 100

It becomes handier for doing squares of larger numbers, such as 97^2, which just becomes 100*95-91=9409. I've started using it for all numbers ending in 2's and 7's i want to find the square of. By the way, if anyone wants to actually prove why that works for all numbers >2, they get a cookie. :P


Well, it works for any real number, larger than 2 or not. It's actually a factorization of the quadratic function x^2 + x - 6 :
x^2 + x - 6 = ( x + 3 ) ( x - 2 )
and its domain is the whole set of real numbers.

You could construct a function for numbers ending in other digits; for instance:
x^2 = ( x - 3 ) ( x + 2 ) + ( x + 6 )
would be helpful for numbers ending in 3 or 8.

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Re: Favorite mental math tricks/shortcuts

Postby poirelli » Wed Jan 28, 2009 4:12 am UTC

Blatm wrote:Which book? I read Surely you're joking, Mr. Feynman! (great book, by the way), and I don't remember seeing it in that.


I thought it was in What Do You Care What Other People Think?: Further Adventures of a Curious Character, the followup to Surely You're Joking. It's not as good though.

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Re: Favorite mental math tricks/shortcuts

Postby PM 2Ring » Thu Jan 29, 2009 7:03 pm UTC

joeframbach wrote:
chapel wrote:(x^2 - y^2) = (x + y)(x - y)

Very useful for the 10th graders I teach.

I usually use this in the opposite direction.
17*13 = 15^2-2^2 = 221


Me, too. If you memorize the squares up to 25*25, and use a few other formulas, like (50 + x)^2 = (2500 + 100x +x^2) and (100 + x)^2 = (10000 + 200x + x^2) so you can extend your square table, you can quickly multiply any pair of 2 digit numbers, as long as you don't mess up the subtraction. :) Oh, (x + .5)^2 = (x^2 + x + .25) also comes in handy in case your multipliers aren't congruent mod 2.

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Re: Favorite mental math tricks/shortcuts

Postby Dallen » Fri Jan 30, 2009 8:51 am UTC

Some of these may seem a bit cryptic or difficult to apply, but with practice it gets pretty easy to multiply some large numbers in your head.

Multiplying 2 digit numbers (method 1):

Example 1:

Take two two digit numbers. As an example, lets take 22 * 31.

First we can find the beginning digits of the answer by multiplying the tens digit of both numbers together.

2*3=6

now, we can find the end digits of our answer, by multiplying the ones digits together.

2*1=2

So now we have 6_2.

To get the middle digit, we multiply the tens digit of each number with the other number's ones digit and add the two products together.
2*1=2
2*3=6
2+6 = 8.

We now have the full answer, 682.

Example 2:

Take two two digit numbers. As an example, lets take 25 * 71.

First we can find the beginning digits of the answer by multiplying the tens digit of both numbers together.

2*7=14

now, we can find the end digits of our answer, by multiplying the ones digits together.

5*1=5

So now we have 14_5.

To get the middle digits, we multiply the tens digit of each number with the other number's ones digit and add the two products together.
2*1=2
5*7=35
2+35 = 37.

Notice how we have two digits for our middle digit. To fix this, we add the tens digit of our middle answer to the ones digit of the beginning section's ones number.

our answer is now 1775, which is the correct answer.



Multiplying 2 digit numbers (method 2):

Example: Lets take 81 * 95.

First we need to determine how much less each number is from 100.

81 is 19 less than 100 and 95 is 5 less than 100.

Now here we can either subtract 19 from 95, or 5 from 81. Either way we will get the same result of 76. This will be the beginning of our answer. To get the end of our answer, we multiply the differences with each other.

19*5=95.

Now we combine them, and we have 7695 which is the answer.

Note: This trick can be altered to work with numbers that have more digits as well, but I'm tired (Its 2:30 here) and I don't feel like going through it.



Squaring Large Numbers:

Example 1:
Lets square 105. We are 5 more than base 100, so we take that number and add it to the original number.

105+5=110.

This is the beginning of the number

We now take that number we added to the original number, and square it.

5*5=25.

This gives us the end of our number.

Combine them, and you get 11025. :D




Example 2:
Lets square the number 115. 115 is 15 more than 100 and so we take that and add it to our original number.

115+15=130.

This is the beginning of our number. To get the end of our number we square the number we just added to the original number.

15*15 = 225.

Since 225 is more than 2 digits (we are dealing with base 100, so 2 digits), we add the 2 in the hundreds digit to 130.


We get the answer 13225.

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Re: Favorite mental math tricks/shortcuts

Postby quintopia » Sat Jan 31, 2009 1:01 am UTC

To sum up Dallen's long post:

1: (10a+b)(10x+y)=100(a*x)+10(a*y+b*x)+(b*y)

2: (100-x)*(100-y)=100*(100-y-x)+(x*y)
(He mentioned larger numbers. Replace 100 with anything, say 1000, and the equality holds.)

3: (100+a)^2 = 100*(100+2a)+a^2



In other words, to multiply two numbers together quickly, rewrite them in a form that involves a power of ten, and then FOIL the binomial product.


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