Now, since to write down the long-division algoritm is a bit difficult (and besides, we do it differently in Norway from the US), I will show you the RATIONALE behind the long division technique instead; long division is simply a condensed version of what I'll present.
1. What is meant by "division" in this case?
Ordinarily, "division" is meant to be that process that rewrites a number, given as a FRACTION, into the equivalent DECIMAL REPRESENTATION of that number (which, as it happens, is a particular TYPE of fractional representation).
2. Let's look at the number given by fraction 3/4
We want to write 3/4 in its decimal representation, that is to find digits a_{i} between 0 and 9, so that we have:
\frac{3}{4}=0.a_{1}a_{2}a_{3}...
Where the notation 0.a_{1}a_{2}a_{3}... MEANS:
0.a_{1}a_{2}a_{3}...\equiv\frac{a_{1}}{10}+\frac{a_{2}}{100}+\frac{a_{3}}{1000}++
3. Let's start!
a) We note that 3<4, so our first step is to multiply 3/4 with an appropriate representation of the number "1":
\frac{3}{4}=1*\frac{3}{4}=\frac{10}{10}*\frac{3}{4}=\frac{1}{10}*(\frac{30}{4})
Henceforth, we will work with the expression included in the parenthesis.
b)
We note that 30>4, and the greatest multiple of 4 which is less than 30, is 4*7=28.
Hence, we write 30=4*7+2
We therefore have the equality:
\frac{30}{4}=\frac{4*7+2}{4}
c)We now use the fact that we can split up a sum in the numerator into a sum of fractions:
\frac{4*7+2}{4}=\frac{4*7}{4}+\frac{2}{4}=7+\frac{2}{4}
The last step follows since 4 is a common factor in both the numerator and denominator in the first fraction.
d) Hence we have shown:
\frac{3}{4}=\frac{1}{10}*(7+\frac{2}{4})
This can be rewritten as:
\frac{3}{4}=\frac{7}{10}+\frac{1}{10}(\frac{2}{4})=0.7+\frac{1}{10}(\frac{2}{4})
e) We will now work with the parenthesized 2/4.
Since 2<4, we multiply 2/4 by an appropriate version of 1:
\frac{2}{4}=1*\frac{2}{4}=\frac{10}{10}*\frac{2}{4}=\frac{1}{10}*(\frac{20}{4})
f) We note that 20=5*4, so we have:
\frac{2}{4}=\frac{1}{10}*(\frac{5*4}{4})=\frac{1}{10}*(5)=\frac{5}{10}
g) We now look back at the equation in d):
\frac{3}{4}=0.7+\frac{1}{10}(\frac{2}{4})
With the result from f), we have:
\frac{3}{4}=0.7+\frac{1}{10}(\frac{5}{10})=0.7+\frac{5}{100}=0.7+0.05=0.75
And that's our result..
