How is division defined in mathematics?

[AndersHermansson]

Multiplication is defined as repeated addition.

3x5 = 5+5+5

How do we define 10/2?

 

[Pocketwatch]

Originally posted by AndersHermansson
Multiplication is defined as repeated addition.

3x5 = 5+5+5

How do we define 10/2?

Repeated subtraction?
10-2-2-2-2-2=0 5 equal parts.

10/3
10-3-3-3=1 3.3333333333333 parts. 3 and 1/3

 

[HallsofIvy]

You also posted this under "general mathematics" where Hurkyl pointed out, correctly, that multiplication is NOT defined as "repeated addition".

 

[Soroban]

Reminds me of a truly awful joke . . .

Q: How many times can you subtract 5 from 21
and what do you have left?

A: I can subtract 5 from 21 as many times as I like,
and I always have 16 left.

 

[Dissident Dan]

If A/B = C, then I would define C as the the number of times that you have to subtract B from a quantity that starts out at A until you get to 0. Hence, 10/5 = 2, because you have to subtract 5 twice from a quantity that starts out at 10 in order to get 0, and 0/0 is undefined, because you always have 0, no matter how many times you subtract, and 10/0 is undefined because there is no answer(There is no amount of times that you can subtract in order to arrive at zero--"no amount of times" is NOT the same as "zero times", because zero is an amount of times; you have an empty set, as opposed to a set with an element 0).

 

[hypnagogue]

Originally posted by Dissident Dan
If A/B = C, then I would define C as the the number of times that you have to subtract B from a quantity that starts out at A until you get to 0. Hence, 10/5 = 2, because you have to subtract 5 twice from a quantity that starts out at 10 in order to get 0, and 0/0 is undefined, because you always have 0, no matter how many times you subtract, and 10/0 is undefined because there is no answer(There is no amount of times that you can subtract in order to arrive at zero--"no amount of times" is NOT the same as "zero times", because zero is an amount of times; you have an empty set, as opposed to a set with an element 0).

What about something like 1/3? How many times do you have to subtract 3 from 1 to get 0? The answer is "1/3 of 3" times, but the answer here using the above formulation doesn't get us any closer to a meaningful answer than the initial question. It's circular.

 

[rolandmath]

we define division of two integer numbers a and b as follows a/b is
a=b(q)+r
for some integers q and 0<= r< b. Here q and r are uniquely deteremine. 28/5 is the same thing as 28=5(5)+3 If you want to get fancy schmancy. Take the integers Z and since Z is an integral domain define the field of rationals Q as all the numbers(quotients) that satisfy the following equation for x

xm=n

all solution are n/m where m and n are integers, here you have the field of quotients or Q