Explanation:
"properly" is very vague.
as your can imagine, particularly in math, where many things are commutative or have other degrees of freedom what do when in what sequence, you will find particularly for the little techniques many different approaches. and most of them are correct.
particularly today I keep scratching my head about what i think is immensely complicated but teachers swear they are helping the students.
anyway, this is how I do divisions by using an example :
73496 ÷ 17
the "window" I use to look at the first number is a long as the second number. in this case 2 digits.
step 1a :
as mentioned, we look at the 2 left most digits (73) and see how often 17 fits inside that as a whole :
73 ÷ 17 = 4
4×17 = 68, so we have a remainder of 73-68 = 5.
I write it like this :
73|496 ÷ 17 = 4
05 (4×7 = 28 and 5 is then 33, carry over 3,
4×1 = 4 plus carry 3 = 7, 0 difference)
step 1b :
pull the next position down
734|96 ÷ 17 = 4
054
step 2a :
we look at the bottom number and divide this by 17.
54 ÷ 17 = 3 and a remainder of 3.
734|96 ÷ 17 = 43
054
03 (3×7 = 21 and 3 is then 24, carry over 2,
3×1 = 3 plus carry 2 = 5, 0 difference)
step 2b :
pull the next position down
7349|6 ÷ 17 = 43
054
039
step 3a :
we look at the bottom number and divide this by 17.
39 ÷ 17 = 2 and a remainder of 5.
734|96 ÷ 17 = 432
054
039
05 (2×7 = 14 and 5 is then 19, carry over 1,
2×1 = 2 plus carry 1 = 3, 0 difference)
step 3b :
pull the next position down
73496| ÷ 17 = 432
054
039
056
step 4a :
we look at the bottom number and divide this by 17.
56 ÷ 17 = 3 and a remainder of 5.
73496| ÷ 17 = 4323
054
039
056
05 (3×7 = 21 and 5 is then 26, carry over 2,
3×1 = 3 plus carry 2 = 5, 0 difference)
step 4b :
pull the next position down. but we have reached the last position before the decimal point. from that moment on we create result numbers after the decimal point.
and because the left number did not have any explicit digits after the decimal point, all the numbers we are pulling down now are 0.
73496.0| ÷ 17 = 4323.
054
039
056
050
step 5a :
we look at the bottom number and divide this by 17.
50 ÷ 17 = 2 and a remainder of 16.
73496.0| ÷ 17 = 4323.2
054
039
056
050
16 (2×7 = 14 and 6 is then 20, carry over 2,
2×1 = 2 plus carry 2 = 4, 1 difference)
step 5b :
pull the next position down. because the left number did not have any explicit digits after the decimal point, all the numbers we are pulling down now are 0.
73496.00| ÷ 17 = 4323.2
054
039
056
050
160
step 6a :
we look at the bottom number and divide this by 17.
160 ÷ 17 = 9 and a remainder of 7.
73496.00| ÷ 17 = 4323.29
054
039
056
050
160
07 (2×7 = 14 and 6 is then 20, carry over 2,
2×1 = 2 plus carry 2 = 4, 1 difference)
step 6b :
pull the next position down. because the left number did not have any explicit digits after the decimal point, all the numbers we are pulling down now are 0.
73496.000| ÷ 17 = 4323.29
054
039
056
050
160
070
step 7a :
we look at the bottom number and divide this by 17.
70 ÷ 17 = 4 and a remainder of 2.
73496.00| ÷ 17 = 4323.294
054
039
056
050
160
070
02 (4×7 = 28 and 2 is then 30, carry over 3,
4×1 = 4 plus carry 3 = 7, 0 difference)
step 7b :
pull the next position down. because the left number did not have any explicit digits after the decimal point, all the numbers we are pulling down now are 0.
73496.0000| ÷ 17 = 4323.294
054
039
056
050
160
070
020
and so on ...