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As a completely uneducated simpleton, it seems bizarre to me that addition and multiplication are considered "different" in the deeper explorations of math and

As a completely uneducated simpleton, it seems bizarre to me that addition and multiplication are considered "different" in the deeper explorations of math and number theory. It seems like multiplication is just an extension of addition. How many times do you want to add numbers together? The result is multiplication. Similarly, addition can be used to represent multiplication. You want to multiply, which can be represented as adding things a certain number of times. Of course, the conjecture introduces rules about prime numbers, and then says "ooo" now we see that prime numbers being added together (with rules of what kinds of prime numbers are allowed in the equation) results in "predictable" rates of incidence. I guess it's a little late to go back and be born again with a life of education centered around number theory so I could see and comprehend the complexity!

ColinWright 14y ago
This really deserves a longer and better answer, but I'm struggling to explain. It's a pretty deep conceptual thing, but I'll try. As a completely uneducated simpleton, it seems bizarre to me that addition and multiplication are considered "different" in the deeper explorations of math and number theory. It seems like multiplication is just an extension of addition. You're not alone in this, but as you go on in advanced math you find more and more that multiplication is not really repeated addition, it just happens to coincide with repeated addition when that makes sense. The problem/opportunity is that multiplication still makes sense when repeated addition doesn't. It might be easier to think of this with regard powers. People teach that A^5 is just AxAxAxAxA. You then deduce that A^a x A^b = A^(a+b). From that you start to assign meanings to things like A^0. And A^(-1). But what does it mean to multiply together -1 copies of a number? That doesn't make sense! And what about A^{\pi} ? How can you have a transcendental number of things multiplied together? It doesn't make sense! As you get deeper into math you need different definitions of powers, and of multiplication, and you find they they coincide with repeated multiplication and repeated addition, they may have originated with those ideas, but that's not really the best way to think about them, and it's not, in some sense, what they "are". A poor analogy might be this. To an outsider, Smalltalk and Haskell will kind of look the same. They're programming languages, they do the same things. But they are really very different animals. So multiplication is really a very different animal from repeated addition.
SatvikBeri 14y ago
The key behind Math is to generalize from very specific rules like the concept of multiplication as n copies of addition to very general rules based on behavior. For example, in Ring Theory multiplication and addition are just functions that satisfy specific properties. By focusing on the class of all functions that satisfy these properties, we're able to prove very general theorems that have much wider applications than if we just focused on addition-based multiplication. You see the same pattern in programming. Imagine if you had to have separate addition operators for doubles, floats, ints, and bigints. It's far more useful to have one function that only depends on the behavior of its input, rather than having its input be a very specific type of object.
andreasvc 14y ago
> It seems like multiplication is just an extension of addition. [...] > Similarly, addition can be used to represent multiplication. Yet, addition cannot be represented as multiplication (AFAICT), so there you have a difference.
ky3 14y ago
It seems like multiplication is just an extension of addition. How many times do you want to add numbers together? The result is multiplication. Similarly, addition can be used to represent multiplication. You want to multiply, which can be represented as adding things a certain number of times. If you're interested in this train of thought here's a resource you might want to chase up: http://en.wikipedia.org/wiki/Hyperoperation http://en.wikipedia.org/wiki/Hyperoperation As for the "bizarre" interplay between addition and multiplication, you might want to contemplate The Other FLT [1]. TOFLT can be understood with just high-school math but is a big gateway drug into big-ass number theory. In more ways than one! [1] http://en.wikipedia.org/wiki/Fermats_little_theorem http://en.wikipedia.org/wiki/Fermats_little_theorem

ColinWright 14y ago

This really deserves a longer and better answer, but I'm struggling to explain. It's a pretty deep conceptual thing, but I'll try. As a completely uneducated simpleton, it seems bizarre to me that addition and multiplication are considered "different" in the deeper explorations of math and number theory. It seems like multiplication is just an extension of addition. You're not alone in this, but as you go on in advanced math you find more and more that multiplication is not really repeated addition, it just happens to coincide with repeated addition when that makes sense. The problem/opportunity is that multiplication still makes sense when repeated addition doesn't. It might be easier to think of this with regard powers. People teach that A^5 is just AxAxAxAxA. You then deduce that A^a x A^b = A^(a+b). From that you start to assign meanings to things like A^0. And A^(-1). But what does it mean to multiply together -1 copies of a number? That doesn't make sense! And what about A^{\pi} ? How can you have a transcendental number of things multiplied together? It doesn't make sense! As you get deeper into math you need different definitions of powers, and of multiplication, and you find they they coincide with repeated multiplication and repeated addition, they may have originated with those ideas, but that's not really the best way to think about them, and it's not, in some sense, what they "are". A poor analogy might be this. To an outsider, Smalltalk and Haskell will kind of look the same. They're programming languages, they do the same things. But they are really very different animals. So multiplication is really a very different animal from repeated addition.

SatvikBeri 14y ago

The key behind Math is to generalize from very specific rules like the concept of multiplication as n copies of addition to very general rules based on behavior. For example, in Ring Theory multiplication and addition are just functions that satisfy specific properties. By focusing on the class of all functions that satisfy these properties, we're able to prove very general theorems that have much wider applications than if we just focused on addition-based multiplication. You see the same pattern in programming. Imagine if you had to have separate addition operators for doubles, floats, ints, and bigints. It's far more useful to have one function that only depends on the behavior of its input, rather than having its input be a very specific type of object.

andreasvc 14y ago

> It seems like multiplication is just an extension of addition. [...] > Similarly, addition can be used to represent multiplication. Yet, addition cannot be represented as multiplication (AFAICT), so there you have a difference.

ky3 14y ago

It seems like multiplication is just an extension of addition. How many times do you want to add numbers together? The result is multiplication. Similarly, addition can be used to represent multiplication. You want to multiply, which can be represented as adding things a certain number of times. If you're interested in this train of thought here's a resource you might want to chase up: http://en.wikipedia.org/wiki/Hyperoperation http://en.wikipedia.org/wiki/Hyperoperation As for the "bizarre" interplay between addition and multiplication, you might want to contemplate The Other FLT [1]. TOFLT can be understood with just high-school math but is a big gateway drug into big-ass number theory. In more ways than one! [1] http://en.wikipedia.org/wiki/Fermats_little_theorem http://en.wikipedia.org/wiki/Fermats_little_theorem