

How to Divide any number By 9, 90, 900 and so on in just 5 seconds  - mquaes
http://mathema-tricks.blogspot.com/2012/01/dividing-by-9-90-900-and-so-on-into.html

======
enkrs
Unrelated note, you can do simple multiplication by 9 with your fingers: place
your palms on a table and imagine your fingers are numbered 1-10 from left to
right.

4 * 9? Lift up your fourth finger - you have 3 to the left and 6 to the right
- answer is 36.

8 * 9? Lift up your eight finger - 7 to the left and 2 to the right - answer
is 72.

I hope it's not totally common knowledge thing.

~~~
neovive
Never knew this one. Thanks for sharing!

------
rytis
This is obviously cool and all that, but I see one problem with these kind of
shortcuts - children (who this site targets) have no idea why this works and
how. They memorise it and go about their lives using it. They know it works,
but have no idea why.

I'm ranting about it because I have 2 kids who learn all sorts of similar
stuff at school, but teachers fail to explain why these things work. (and in
most cases I suspect they don't actually grasp the idea themselves)

~~~
Jach
I'd correct you by saying that they'll memorize it for as long as it's useful
for passing a test because the teacher wants them to solve something a
particular way. They'll soon forget it. I learned long division in 3rd grade,
relearned it in 6th grade, learned the algebraic polynomial version in 10th
grade (then was shown/told synthetic division which is cooler)... and at this
point I know I'd struggle doing any of those. I'd rather use a calculator and
get back to doing whatever it was that required the answer. In these crazy
times where even most elementary kids have phones with calculator capabilities
there's no excuse to be parading these mental tricks as "math".

I do agree though that a proof of this would be much more interesting than the
technique itself. If I have kids I intend to teach them math myself. I don't
think I was introduced to the idea of formal proof until 8th grade geometry,
and everyone hated _those_ proofs since it was a bookkeeping exercise of
writing down every algebraic step and what theorem/axiom allowed you to
perform it.

------
finnw
The same technique for dividing by 9 in base 10, also works for dividing by
255 in base 256, so you can quickly calculate N % 255 by adding the bytes of N
together.

Useful for algorithms like this: <http://stackoverflow.com/q/295579/12048>

------
lkozma
Unrelated, you can square quickly a (reasonably small) number ending in 5:
multiply the number without the 5 to the number one larger and append 25 to
it. Example:

    
    
      85*85 = (8*9).25 = 7225

~~~
onemoreact
Did you mean it's only fast when it's a small number, or does it fail at some
point?

EX: works for 10005^2 and 12345^2.

~~~
lkozma
It works for any number, I meant "reasonably small", because for those it can
be done mentally.

~~~
onemoreact
Cool, thanks I was sitting there trying to think why it would fail for large
numbers. But, (x * 10 + 5)^2 = x * x * 10 * 10 + 2 * x * 50 + 25 = x * (x + 1)
* 100 + 2 .

~~~
onemoreact
edit '+ 2 .' should be '+ 25.'

so: (x * 10 + 5)^2 = x * x * 10 * 10 + 2 * x * 50 + 25 = x * (x + 1) * 100 +
25.

------
jgrahamc
Also fun is squaring two digit numbers in your head:
[http://blog.jgc.org/2010/03/squaring-two-digit-numbers-in-
yo...](http://blog.jgc.org/2010/03/squaring-two-digit-numbers-in-your-
head.html)

------
jsvaughan
An alternative way (but more intuitive) to find approx a / 9 in your head:
result = (a x 0.1) + (a x 0.01) + ...

so e.g. take a random number, 22103 answer is about 2210 + 221 +22 + 2 = 2455

~~~
lotharbot
To be more complete: .1a + .01a + ... + (digit sum / 9) will give it to you.
That's actually what this method is doing, it's just poorly explained.

22103/9 = 2210 + 221 + 22 + 2 + (2+2+1+0+3)/9 = 2455 + 8/9

So your ones digit ends up being 2+2+1+0+(any whole part left over from the
digit sum), your tens digit is 2+2+1, your hundreds digit is 2+2, and your
thousands digit is just 2.

------
jgrahamc
Why this works: [http://blog.jgc.org/2012/03/how-to-divide-by-9-really-
really...](http://blog.jgc.org/2012/03/how-to-divide-by-9-really-really-
fast.html)

------
why-el
Unrelated, but relates to 9: if you take any two digit number,say 59, and you
deduct the sum of the two digits 5 and 9, it yields 45, a multiple of 9.
Interestingly, this is true for any two digit number. I used this trick in a
game where I labeled all multiples of 9 with a sign, asked my sister to
perform the operation above, and claim that I can guess the sign, because all
multiples of 9 share the same sign. :)

~~~
rimantas

      > Interestingly, this is true for any two digit number
    

Not only for two digits numbers. Also there is a trick based on that: ask
someone to think of a number (two-digits, or greater). Ask to subtract the sum
of digits from that number. Ask to multiply result by any number. Ask to
cross-out any digit, except 0 and then say remaining digits to you in any
order. Then you say what digit was crossed out. The secret is simply: you add
the digits told to you till you get one digit number, and then subtract it
from nine.

E.g.:

    
    
      Original number => 42
      Subtract the sum => 42 - 6 = 36
      Multiply by any number => 36 * 666 = 23 976
      Cross out 3 => 2976
      Name the digits in any order => 2679
    

So you got 2679. Calculate digital root: 2 + 6 + 7 + 9 = 24, 2 + 4 = 6.

9 - 6 = 3 <= there is your answer.

See also: <http://en.wikipedia.org/wiki/Digital_root>

<http://en.wikipedia.org/wiki/Casting_out_nines>

------
lignuist
Cool Trick. Reminds me a bit of vedic math:

<http://en.wikipedia.org/wiki/Vedic_math>

~~~
swatkat
Yes! And, this reminds of amazing "Mathemagic" series of videos by Khurshed
Batliwala. Here's one: (a+b)²=a²+2ab+b² - But Why?
<http://www.youtube.com/watch?v=49_TJymgXgM>

------
obtino
Interesting fact: If the sum of all digits in the dividend is a multiple of 9,
then it can be assumed that the dividend is a multiple of 9.

------
pg_bot
Why don't you just divide by three two times? This would be a lot easier to
remember and most likely will be faster than the other method.

~~~
Jach
Or get an approximate answer by dividing by the appropriate power of 10, which
is pretty much the fastest method. I think the "5 seconds" editorialization is
BS.

------
no-espam
Professor Trachtenberg (early 1900) comes to mind for doing speed math.

~~~
no-espam
Wow, downvoted for posting info related to a professor who was held in nazi
concentration camp and kept his mind lucid with math?

<http://en.wikipedia.org/wiki/Trachtenberg_system>

~~~
downx3
+1 for the Trachtenberg system. I only wish we were taught these methods at
school, in addition to the 'normal' way...

------
Sembiance
Uhm, 5 seconds? I can type X/9 into Google or my smart phone in 1 to 2
seconds.

~~~
thewisedude
I think I can reason why you are down voted. May be it was the delivery style
of the message than the message itself. But I think it was unfair to you as
you have a valid point. If you strictly want to asses the fastest way to
arrive at this result in today's day and age. Your solution is probably one of
the fastest ways to do that. I think the author's mathematical tricks would
have been very useful in a world where personal computing is not where it is
today... say prior to the advent of calculators.

