How do we define division?

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In summary, Multiplication is defined as repeated addition. For multiplication of two integer numbers, division is defined as repeated subtraction.
  • #1
AndersHermansson
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Multiplication is defined as repeated addition.

3x5 = 5+5+5

How do we define 10/2?
 
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  • #2
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
 
  • #3
You also posted this under "general mathematics" where Hurkyl pointed out, correctly, that multiplication is NOT defined as "repeated addition".
 
  • #4
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.
 
  • #5
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).
 
  • #6
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.
 
  • #7
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
 

1. What is division?

Division is a mathematical operation that involves separating a given quantity into equal parts or groups.

2. How is division different from multiplication?

Division is the inverse operation of multiplication, meaning it is the opposite process. While multiplication combines equal groups, division separates a quantity into equal groups.

3. What are the basic components of a division problem?

A division problem consists of a dividend (the number being divided), a divisor (the number by which the dividend is divided), and a quotient (the result of the division).

4. Can division result in a fraction or decimal?

Yes, division can result in a fraction or decimal if the dividend is not evenly divisible by the divisor. For example, 1 divided by 3 would result in the fraction 1/3 or the decimal 0.33333...

5. Are there any special rules or properties for division?

Yes, some special rules for division include the commutative property, which states that the order of the numbers in a division problem can be changed without affecting the result, and the identity property, which states that the quotient of any number divided by 1 is equal to the original number.

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