How can we define division without using addition?

[AndersHermansson]

Multiplication is defined as repeated addition.

3x5 = 5+5+5

How do we define 10/2?

 

[Robert Zaleski]

Repeated subtraction;

10-2=8-2=6-2=4-2=2-2=0
subtracted 5 times

 

[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.

 

[Doctor Luz]

Repeated adition is not satisfied enven with negative integers.

 

[Doc]

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.

 

[Doc]

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.

 

[Hurkyl]

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?

 

[Doc]

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.

 

[Hurkyl]

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.