Mathematics doesn’t describe reality, it describes certain logical constructions. sometimes that is useful in modeling things in the real world. The point of my example with a negative number of instances of 5 apples was to show the real world model – most of us have some sense of being in debt. That example is also pretty illustrative of what multiplication over integers is. If multiplication is to be consistent you have to have …, 10,5,0,-5, … for products of {…,2,1,0,-1,…} x 5.

by the way mathematicians don’t do subtraction, just addition. When you would write 5-5 = 0 a mathematician would write 5 + (-5) = 0. Rings don’t have multiplicative inverses so there is just addition and multiplication. Division is much like subtraction, a mathematician doesn’t do division instead she multiplies by the multiplicative inverse (need to have a field of numbers rather than a ring to have multiplicative inverses).

Formally the logic of these operations is described as mappings between sets ie addition : integers X integers -> integers … eg + (5,5) -> 10.

There was a lot of thought given to develop rigorous logic to underpin arithmetic back in the early part of the 20th century. Bertrand Russell being one of the logicians that worked on it … at some point I want to use Principia Mathematica as bathroom wallpaper. kind of geeky I know. What’s much more mind blowing is developing real numbers from the rational numbers. I can remember the first time I saw a diagonalization proof that there were more numbers in the unit interval than there are integers, and being really excited by its conceptual beauty and elegance. It was some time before I learned about Cauchy sequences and how to rigorously develop the real numbers logically.
I’m really pleased to have all these concepts readily available in wikipedia. All this stuff is really beautiful, and now it’s out there for anyone to see.

“tommy, go and gather all the apples for the tree our backyard, (5)M bring them to bill, he has promised to multiply your apples this year by the amount of apples he has produced this year and you can have whatever that number is because we gave him some of our crop last year buy buying the differance from allen

bills crop was sick this year and he produced zero apples

does that mean bill keeps the apples tommy grew because the multiple is zero and tommy goes home with nothing?

no, that’s not what it means, it means tommy goes home with five apples

so you see, my solution IS the solution, mathematical theory fails

But if you want to have consistent addition and multiplication you will have a*0=0*a = a for all a in the set you’re doing arithmetic over.

there can be addendums, for instance there is a leap year after all

I believe my explanation might indeed solve some problems

for instancI, suppose I am to multiply the number of one formula by the number of the second formula, that second formula yields zero

I believe if we are talking about numbers that were derived from real life, the answer would be the solution to the first formula, not zero

if the solution to the first number is zero then the solution to the second formula would be mute

therefore the way theory should read is;

“when using the concept of zero, it is important to determine at which point the zero occurs, for when it comes to multiplication and the concept of zero, the inverse does not yeild the same result”

there, that’s the way the theory should read

let’s not forget, math represents reality, I can indeed refuse to multiply my apples, that would indeed be zero times when we are talking about reality

that makes perfect sense, but multiplying is a verb

the process of multiplication is a verb, we take a product, (so to speak), and we we do something with that product

if we have zero products we can multiply that product as many times as we want, there would zero

if we have 5 apples then you are saying we cannot therefore multiply by zero (unless we substract those apples)

I am saying that if you want to say multiplying by zero is in fact subtracting the quotient then I understand your point

but then theory would have to say that and that is the point, theory does not say any multiplication process becomes subtraction

(or does that theory indeed exist?)

having zero instances of something seems to make no sense

that makes perfect sense, but multiplying is a verb

when I multiply there is no negative action, I already havce those apples, I can’t then claim to have zero instances of them

if you mean to say the action of multiplying those apples by zero means you do indeed get rid of the apples then I quite understand but that would mean in the case of multiplying by zero the action becomes subtraction not multiplication

Having something zero times means you do not have something.

I think you’re trying to get at this when you say “since the action cannot occur” – you’re confused by what the second multiplicand means when it happens to be zero. you get it when it is 1, 2, … and probably when it is -1 or 1/2, but zero is confusing because having zero instances of something seems to make no sense, whereas cutting 5 apples in half makes sense physically. Numbers are abstract logical entities, not apples. And arithmetic is built up in a way that is consistent according to logic. Arithmetic is also necessarily incomplete, but that’s a deeper topic.

lets try this another way – if you have 5 apples 2 times you have 10 apples.
If you have 5 apples 1 times you have 5 apples.
If you have 5 apples zero times you have zero apples
if you have 5 apples negative one times you owe 5 apples…

How many instances of 5 apples do you have?

Zero is a very important concept in arithmetic, and it made a lot of modern mathematics possible when it was borrowed/stolen from arabia.

There definitely are times when you do not have to have a factor that is zero to get a zero product, linear transformations of the real plane is the example here

But if you want to have consistent addition and multiplication you will have a*0=0*a = a for all a in the set you’re doing arithmetic over. In most fields a lay person would recognize as “numbers” you also have a*b = b* a, that is multiplication is commutative (or viewed as a group the set is abelian with respect to multiplication).

multiply by zero means you have zero instances of 5 apples, which is zero apples.

this might be a thearhetical principle but that principle is flawed since it ignores the premise:

I HAVE 5 apples, this is not a number on a piece of paper, those apples exist

I THEN do not multiply what I have, this IS zero times whether or not I can do this theorhetically. I multiply what I DO have zero times

I STILL have those apples

if the theory says otherwise then the theory is wrong and that is my point

for instance, if I HAVE 5 apples and therefor am not allowed to multiply by zero, since the action cannot occur 3 Still have those apples.

if theory says otherwise but the apples are still there then the theory has to be rewritten

forgot to mention I’ve got a small flock of chickens for meat and eggs. just about done processing this year’s meat birds, nice to have home raised organic chicken in the freezer!

You’re a bit confused about the definition of multiplication. In your example you “do not multiply” then say that is the same as “multiply by zero”. not the same thing – do not multiply indeed eaves you with 5 apples. multiply by zero means you have zero instances of 5 apples, which is zero apples. This area of number theory is usually developed in a first course in abstract algebra, specifically the definition of a Ring.

Integers aren’t a Field with respect to addition and multiplication – there are no multiplicative inverses in the integers. I’ve not thought about integers as a Ring for a while, but do recall that they are an abelian ring – that is both addition and multiplication are commutative in the integers.

It may be instructive to think about integers modulo some prime.

Thanks for the back up.

With gratitude,
Heather

I look forward to your presence in Congress.

With gratitude,
Heather