Without associativity, what would multiplying/adding 3 or more digits mean?

[dvdrdiscs]

Say you have X*X*X*X. Subconsciously you multiply two numbers at a time, which is associativity. But say you do not have associativity, what would X*X*X*X mean?

 

[PlatinumGold]

huh??

3*3*3*3??

would be 81 right??

 

[dvdrdiscs]

Originally posted by: PlatinumGold
huh??

3*3*3*3??

would be 81 right??

Yeah but think about how you got 81. You did this:

(3x3) x 3 x 3
(9x3) x 3
27x3=81

You used the associativity rule. My question is, without such a rule, how would we reason the meaning of X*X*X*X, or in your case, 3x3x3x3?

 

[notfred]

It doesn't matter what the hell order you multiply them in, it's always X^4.

 

[Tom]

full frontal nudity

 

[PlatinumGold]

are you discussing the process or the result?? the result would be the same either way right?

 

[mugs]

Not sure what you're getting at here...

 

[PrincessGuard]

Without associativity, it wouldn't be multiplication.

 

[dvdrdiscs]

Originally posted by: notfred
It doesn't matter what the hell order you multiply them in, it's always X^4.

Yes, but what is X^4 without associativity?
With associativity, it works like this:

(X*X)*X*X
((X^2)*X)*X
((X^3)*X)
X^4

Now if you are not able to use associativity, how would you explain X*X*X*X?

 

[Amorphus]

Originally posted by: notfred
It doesn't matter what the hell order you multiply them in, it's always X^4.

but we would still think of it as X*X*X*X, if we had to do it Q) by hand! (gasp)

I suppose it would be rote, then, if we didn't have the 1-and-1 associativity. We are taught to multiply in pairs as children, which may be why.

 

[dvdrdiscs]

This is abstract algebra. Basically without associativity, we would be dead in the water if we try to reason how we can multiply 3 or more numbers. As a bonus question on my homework, I was asked to reason what would multiplying/adding more than 3 numbers mean without such a rule as associativity.

 

[notfred]

Ok, I guess you're right. We can't do that, we don't have a trinary multiplication operation in our math. We'd have to come up with a new multiplication operator.

 

[PlatinumGold]

5 fingers, 2 hands, 3 people no matter how you do it, it's still 30 fingers.

 

[dvdrdiscs]

Originally posted by: PlatinumGold
5 fingers, 2 hands, 3 people no matter how you do it, it's still 30 fingers.

And what rule gave you the ability to do so? Associativity! You can multiply 5 fingers by 2 hands first, or 2 hands by 3 people first. My question is, without such a rule, how can one solve such a problem?

 

[PlatinumGold]

Originally posted by: dvdrdiscs

Originally posted by: PlatinumGold
5 fingers, 2 hands, 3 people no matter how you do it, it's still 30 fingers.

And what rule gave you the ability to do so? Associativity! You can multiply 5 fingers by 2 hands first, or 2 hands by 3 people first. My question is, without such a rule, how can one solve such a problem?

no, you could also choose to just count the fingers, either way, it's 30 fingers. i know i tried it.

 

[DrPizza]

Here's your answer... multiplying is just a shortcut for adding.
If I had ten sevens in a row for you to add up
7 + 7 + 7 + 7 + 7 + 7 + 7 + 7 + 7 + 7, any sensible person would do multiplication 10 times 7 = 70.

But, what it means is add 7 ten times.

Now, arrange it like this:
7 + 7 + 7 + 7 + 7 + 7 + 7 + 7 + 7 + 7
+7 + 7 + 7 + 7 + 7 + 7 + 7 + 7 + 7 + 7
+7 + 7 + 7 + 7 + 7 + 7 + 7 + 7 + 7 + 7
+7 + 7 + 7 + 7 + 7 + 7 + 7 + 7 + 7 + 7
+7 + 7 + 7 + 7 + 7 + 7 + 7 + 7 + 7 + 7

You have 10 rows of 7 being added, and 5 columns being added. To find the total, any sensible person would do: 7 * 10 * 5

Now, start stacking up 7's vertically... make a cube out of them that's 8 layers high. To add up all the 7's in the cube, you could simply 7 * 10 * 5 * 8.

Thus, X*X*X*X means adding up x rows of x columns of x (whatever the equivalent for row/column is for height) of x.

Does that answer your question?


oh, but then again, it'd suck to add up those numbers without associativity

 

[dvdrdiscs]

Originally posted by: DrPizza
Here's your answer... multiplying is just a shortcut for adding.
If I had ten sevens in a row for you to add up
7 + 7 + 7 + 7 + 7 + 7 + 7 + 7 + 7 + 7, any sensible person would do multiplication 10 times 7 = 70.

But, what it means is add 7 ten times.

Now, arrange it like this:
7 + 7 + 7 + 7 + 7 + 7 + 7 + 7 + 7 + 7
+7 + 7 + 7 + 7 + 7 + 7 + 7 + 7 + 7 + 7
+7 + 7 + 7 + 7 + 7 + 7 + 7 + 7 + 7 + 7
+7 + 7 + 7 + 7 + 7 + 7 + 7 + 7 + 7 + 7
+7 + 7 + 7 + 7 + 7 + 7 + 7 + 7 + 7 + 7

You have 10 rows of 7 being added, and 5 columns being added. To find the total, any sensible person would do: 7 * 10 * 5

Now, start stacking up 7's vertically... make a cube out of them that's 8 layers high. To add up all the 7's in the cube, you could simply 7 * 10 * 5 * 8.

Thus, X*X*X*X means adding up x rows of x columns of x (whatever the equivalent for row/column is for height) of x.

Does that answer your question?


oh, but then again, it'd suck to add up those numbers without associativity

Well yeah you're still using associativity to add those #'s two at a time.

 

[DrPizza]

Hmmmmm...... use masses to represent the numbers... use a 7 gram mass for each of the 7's in my example above. Put them all on an electronic scale and you'll have the product or sum, or whatever you want to call it, all in one step.

ta daaaaaaa, no associativity to solve the problem.

edit: skip the electronic scale, who's to say there isn't associativity involved in the programming somewhere. Let's use a balance beam instead... One with only 1 long scale on it.