The Euclidian Division is the way integers divide together without introducing the fractions.
It deals with problems of sharing quantities of objects that can not be individually shared.
For instance, if we want to share 13 marbles between 3 children equally, then each child will receive 4 marbles, and it will remain 1 marble unshared.
This is written 13 ÷ 3 = 4, remains 1, that is equivalent to 13 = 4 x 3 + 1.
The definition of the Euclidian division is so: a ÷ b, with a and b (relative) integers, b≠0, is the 2 integers q and r so that a = q x b + r and 0 ≤ r < |b| the absolute value of b.
q is called the quotient and r the remainder, and they are uniquely defined by the fact that
a = q x b + r and 0 ≤ r < |b|.
The division of positive integers may be done manually or… with a calculator or a computer!
To do a ÷ b (say 354 ÷ 35) with a calculator, do the following way:
- display a divided by b ('result' = 354/35 ~ 10.114)
- the integer part is the quotient q of the Euclidian division (q=10)
- subtract q to the result ('result1' = 'result' - 10 ~ 0.114)
- multiply the last result by b to obtain the rest r (r = 'result1' x 35 = 4)
The last result is an integer, because it is r = (a/b-q) x b = (a/b)xb - q x b = a - q x b (distribute x on - and simplify the fraction by b).
An application of that is the convert minutes in hours and minutes, days in years and days.
For instance, 1,000 minutes is: 1,000 ÷ 60 = 16 (the integer part of 1,000/60), remains 40
((1,000/60 - 16) x 60), so that 1,000 minutes is 16 hours and 40 minutes.
And 1,000 days is (neglecting the leap years): 1,000 ÷ 365 = 2 (the integer part of 1,000/365), remains 240 ((1,000/365 -2) x 365), so that 1,000 days is 2 years and 240 days (8 months of about 30 days).