How can we define division without using addition?

In summary, multiplication is defined as repeated addition, where the product is the result of adding a number to itself a certain number of times. Division is defined as multiplication by a multiplicative inverse, which is a number that, when multiplied by the original number, results in 1. However, this definition does not extend to all quantities, as seen in the example of 3.7 * 4.1. It is also important to note that repeated addition does not always work, as seen in the example of -1 * -1. Furthermore, it is possible to perform multiplication without using addition, by using only the increment operation.
  • #1
AndersHermansson
61
0
Multiplication is defined as repeated addition.

3x5 = 5+5+5

How do we define 10/2?
 
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  • #2
Repeated subtraction;

10-2=8-2=6-2=4-2=2-2=0
subtracted 5 times
 
  • #3
Multiplication is generally defined as satisfying the particular axioms. When multiplying integers, it reduces to "repeated addition", but "repeated addition" doesn't extend to quantities like 3.7 * 4.1.

Division is generally defined as multiplication by a multiplicative inverse.
 
  • #4
Repeated adition is not satisfied enven with negative integers.
 
  • #5
Originally posted by Hurkyl
Multiplication is generally defined as satisfying the particular axioms. When multiplying integers, it reduces to "repeated addition", but "repeated addition" doesn't extend to quantities like 3.7 * 4.1.

Division is generally defined as multiplication by a multiplicative inverse.

Not sure here, but how can you define division when using the word INVERSE? INVERSE as in RECIPROCAL means DIVIDING into ONE.

Why CAN'T you think of the product of 3.7 x 4.1 being arrived at by successive addition?

3.7 + 3.7 + 3.7 + 3.7 = 14.8 (Meaning 3.7 x 4)

Now add 3.7 to itself one tenth of a time. (Yeah right!)

In other words, divide by the reciprocal:

3.7 divided by 10. I came up with 10 using successive subtraction of .1 off of 1. (Reciprocal, remember?)

3.7 - 10 = -6.3 OOPS, doesn't work, already below zero. Answer WILL BE less than one. Cannot do conventional successive subtraction.

So the answer is zero with a remainder of 3.7. OR, fractionally stated 3.7 tenths. That should be legal, I have not yet multiplied or divided in the traditional sense. And since the 'divisor' is 10 and the remainder is 3.7, with no quotient this part of the answer is 3.7/10.

SO, let's add 3.7/10 to the first part of the answer which was 14.8.

3.7/10 + 14.8/1

-or-

3.7/10 + 148/10

(Came up with 148 and 10 by successive addition.)

Answer is: 151.7/10

Spoken "151 point 7 tenths"

Divide 151.7 by 10 using successive subtraction and you get an answer of:

15 with a remainder of 1.7


Once again, since the 'divisor' is 10 and the remainder is 1.7, the remainder turns into 1.7/10 as a fraction or through successive addition of both the numerator and denominator, 17 hundredths.

Answer is: 15 and 17 hundredths, or 15.17.

Yeah, I know it seems trivial and stupid, but it IS how some machines do math.
 
  • #6
Originally posted by Doctor Luz
Repeated adition is not satisfied enven with negative integers.

It kind of does work. Take for instance money owed. A debit of $20 (-20) multiplied by 4 is a debit of $80 or, -80.
 
  • #7
Not sure here, but how can you define division when using the word INVERSE? INVERSE as in RECIPROCAL means DIVIDING into ONE.


Definition: y is a multiplicative inverse of x iff y * x = x * y = 1

Compare with inverses of functions; a function g is a function of f if f.g = g.f = i (where i is the identity function and . means function composition)

Definition: for nonzero y, (x / y) is defined to be (x * z) where z is the unique multiplicative inverse of y.

That is how you define division using the word inverse.


Of course, from here, it's a trivial exercise from here to show that (1/x) is the multiplicative inverse of x.



And incidentally, you did not arrive at 3.7 * 4.1 with repeated addition; you added 3.7 a few times then used a distinct operation.

It kind of does work.

What about -1 * -1?
 
  • #8
Originally posted by Hurkyl
Definition: y is a multiplicative inverse of x iff y * x = x * y = 1

Compare with inverses of functions; a function g is a function of f if f.g = g.f = i (where i is the identity function and . means function composition)

Definition: for nonzero y, (x / y) is defined to be (x * z) where z is the unique multiplicative inverse of y.

That is how you define division using the word inverse.


Of course, from here, it's a trivial exercise from here to show that (1/x) is the multiplicative inverse of x.



And incidentally, you did not arrive at 3.7 * 4.1 with repeated addition; you added 3.7 a few times then used a distinct operation.



What about -1 * -1?

Read what I said. I said "It kind of does work." I didn't say it always works. And yeah, technically I did arrive at the answer with successive addition AND another operation. Actually several other operations. But the point was, the answer for 3.7 x 4.1 was arrived at using only addition. Technically subtraction too, but that can also be argued. Is subtraction simply the addition of a negative? Don't answer that, I'm through arguing.
 
  • #9
The point is, you did not arrive at 3.7*4.1 by adding 3.7 4.1 times. My apologies if my point was not clear from the context.

Incidentally, if you want to have some fun, you technically don't even need addition to perform multiplication; you can do it all in terms of the increment operation. :smile:
 

1. What is the definition of division?

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

2. How is division represented in mathematical notation?

Division is typically represented by the symbol "÷" or "/", and can also be expressed as a fraction or with the words "divided by". For example, 10 ÷ 2 = 5, 10/2 = 5, and 10 divided by 2 = 5.

3. What is the relationship between division and multiplication?

Division is the inverse operation of multiplication. This means that dividing a number by another number is the same as multiplying that number by the reciprocal of the second number. For example, 10 ÷ 2 is the same as 10 x 1/2.

4. Are there any rules or properties of division?

Yes, there are several rules and properties that apply to division, including the commutative property, associative property, and distributive property. These properties dictate the order in which numbers can be divided and how they can be grouped together in a division problem.

5. Can division ever result in a fraction or decimal?

Yes, division can result in a fraction or decimal if the dividend (the number being divided) is not evenly divisible by the divisor (the number doing the dividing). In these cases, the resulting quotient will be a fraction or decimal that represents the remainder of the division.

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