# Positive or negative remainder

Is 23 = 5(-4)-3 gives a remainder -3 when divided by 5 ? is this statement true ? some of my colleagues said that remainder cannot be negative numbers as definition but I am doubt that can -3 be a remainder too?

fresh_42
Mentor
Is 23 = 5(-4)-3 gives a remainder -3 when divided by 5 ? is this statement true ? some of my colleagues said that remainder cannot be negative numbers as definition but I am doubt that can -3 be a remainder too?
Usually we consider entire equivalence classes in such cases: Every single element of ##\{\ldots -13, -8, -3, 2, 7 , 12, \ldots\}## belongs to the same remainder of a division by ##5##. We then define all five possible classes
##\{\ldots -15, -10, -5, 0, 5, 10, \ldots\}##
##\{\ldots -14, -9, -4, 1, 6, 11, \ldots\}##
##\{\ldots -13, -8, -3, 2, 7, 12, \ldots\}##
##\{\ldots -12, -7, -2, 3, 8, 13, \ldots\}##
##\{\ldots -11, -6, -1, 4, 9, 14, \ldots\}##
as elements of a new set with five elements ##\{ \; \{\ldots -15, -10, -5, 0, 5, 10, \ldots\}\, , \, \{\ldots -14, -9, -4, 1, 6, 11, \ldots\}\, , \, \ldots \}##.

This notation is a bit nasty to handle, so we choose one representative out of every set. E.g. ##\{[-15],[-9],[12],[3],[-1]\}## could be chosen, but this is still a bit messy to do calculations with. So the most convenient representation is ##\{[0],[1],[2],[3],[4]\}## with the non-negative remainders smaller than ##5##. However, this is only a convention. ##-3## is a remainder, too, belonging to the class ##[2]##. So the answer to your questions is: The statement is true, as all integers are remainders.

mfb
Mentor
The remainder is usually required to be between 0 and N-1 inclusive. 23 and -2 (not -3) are in the same equivalence class. This can also be written as 23 = -2 mod 5.

The remainder is usually required to be between 0 and N-1 inclusive. 23 and -2 (not -3) are in the same equivalence class. This can also be written as 23 = -2 mod 5.
Sorry it should be -23 = 5(-4) - 3 , so in conclusion is, this statement true ?

jbriggs444
Homework Helper
Sorry it should be -23 = 5(-4) - 3 , so in conclusion is, this statement true ?
"-23 divided by 5 is -4 with a remainder of -3". I would consider that statement true.
"-23 divided by 5 is -5 with a remainder of 2". I would also consider that statement to be true.

The convention you use for integer division will determine which of those statements is conventional and which is unconventional.

In many programming languages, integer division follows a "truncate toward zero" convention. For instance, in Ada, -23/5 = -4. The "rem" operator then gives the remainder. So -23 rem 5 = -3.

If one adopts a convention that integer division (by a positive number) truncates toward negative infinity then one would get a different conventional remainder. -23/5 would be -5 and -23 mod 5 would be +2. The Ada "mod" operator uses this convention.

In mathematics, one typically adopts the line of reasoning given by @fresh_42 in post#2 above. The canonical exemplar in the equivalence class of possible remainders is normally the one in the range from 0 to divisor - 1.