math: why do we divide the way we do?

Code:

    2.8
   ___
5| 14
   -10
   ---
    4 0

i'm a bit confused behind the logic of this "long hand" way to division.

ok, so
5 goes into 14 2 times, so you write a 2. you then subtract the amount of times 5 can go into 14 from 14, and you get 4. Up to this point, i understand it.

why, now, do you bring down a 0, and see how many times 5 can go into 40? why does this give you your fractional part, logically?

Technically it's how many times it goes into 4.0, but it's a lot easier to look at it as 40 instead. The decimal point does actually come down, but we never show it because it's a lot easier to learn that way.

A whole number is not just the integer part that you see.

4/100 = 25

But it's 4.00... /100.00...

The zeroes are not made up, they are really there since there is 0 amount of tenths, hundreths, etc. That's why you bring the zero down because it is really there and it allows you to continue the division.

The algorithm uses this relationship:

For all values of k, x/y = k/y + (x-k)/y

For example:

Code:

To divide 84732/7, let:

84732/7 = 10000 * (8/7)     + 4732/7  (We lop off all but the last
        = 10000 * (1 + 1/7) + 4732/7   four digits.)
        = 10000 + 10000/7   + 4732/7
        = 10000 + 14732/7             (At this point, we write down '1'.)

14732/7 =  1000 * (14/7)    + 732/7   (We lop off all but the last
        =  1000 * (2 + 0/7) + 732/7    three digits.)
        =  2000 + 0/7       + 732/7
        =  2000 + 732/7               (At this point, we write down '2'.)

  732/7 =   100 * (7/7)     + 32/7    (We lop off all but the last
        =   100 * (1 + 0/7) + 32/7     two digits.)
        =   100 + 0/7       + 32/7
        =   100 + 32/7                (At this point, we write down '1'.)

   32/7 =    10 * (3/7)     + 2/7     (We lop off all but the last
        =    10 * (0 + 3/7) + 2/7      one digit.)
        =     0 + 30/7      + 2/7
        =     0 + 32/7                (At this point, we write down '0'.)

   32/7 =     1 * (32/7)    + 0/7     (We lop off all the digits.)
        =     1 * (4 + 4/7) + 0/7
        =     4 + 4/7       + 0/7
        =     4 + 4/7                 (At this point, we write down '1'.)

    4/7 =   0.1 * (40/7)    + 0/7     (We 'lop off' past the decimal
        =   0.1 * (5 + 5/7) + 0/7      place.)
        =   0.5 + 0.5/7     + 0/7
        =   0.5 + 0.5/7               (We write a decimal point and a
                                       '5'.)

  0.5/7 =  0.01 * (50/7)    + 0/7
        =  0.01 * (7 + 1/7) + 0/7
        =  0.07 + 0.01/7    + 0/7
        =  0.07 + 0.01/7              (We write down a '7'.)


And so on and so forth.

Substituting expressions back up, we get the answer:

Code:

10000 + 2000 + 100 + 00 + 4 + 0.5 + 0.7 + 0.01/7
= 12104.57 + 0.01/7

Ok.

Our numbers are base 10. So:

12500 is

1 * (10^4)+
2 * (10^3)+
5* (10^2)+
0 *(10^1)+
0 *(10^0)

for decimals

.01 is
0 * (10^-1)+
1 * (10^-2)

I'm not sure where the other stuff from above came from, but this is why we drop zeros. If the value was not zero, you would drop that value.

did anyone knows that there is a similar way of finding the square root of any number?
http://www.nist.gov/dads/HTML/squareRoot.html

>>Technically it's how many times it goes into 4.0, but it's a lot easier to look at it as 40 instead.

ok. since there are 5 8's in 40, then there should 5 0.8's in 4.

Originally Posted by white

did anyone knows that there is a similar way of finding the square root of any number?
http://www.nist.gov/dads/HTML/squareRoot.html

I think I'll stick to local linear approximation.