DumbQuestions
New member
- Joined
- Jan 18, 2024
- Messages
- 1
This isn't really advanced math, but I am curious about this. I am aware it's a very stupid question (check the handle 
), but are they really two different operations in rings? Isn't a ring really just rules we define on how they distribute, that addition is commutative, and that "both" operations are associative?
I suppose the counter would be, "Oh yeah, then write [imath]8 \times\ \frac{2}{7}[/imath] as iterative multiplication." Well, [imath]\frac{2}{7}+\frac{2}{7}+\frac{2}{7}+\frac{2}{7}+\frac{2}{7}+\frac{2}{7}+\frac{2}{7}+\frac{2}{7} = \frac{16}{7}.[/imath] I don't see why this doesn't apply to transcendentals or any other type of number as well, but I tend to be wrong about everything, so I'm probably wrong here too. I suppose it becomes wildly inconvenient to try to write multiplication as iterative addition with something like, say, [imath]pi \times\ r^2[/imath], but is it really? [imath]pi \times\ r^2 = pi + pi + pi +pi +...+ pi[/imath] until you do it [imath]r^2[/imath] times.
I had no trouble recognizing that subtraction is just addition by the additive inverse, and division is just multiplication by the multiplicative inverse, and it seems few would trouble me about saying that. After all, how else could we say the real numbers are a ring, because if subtraction wasn't addition, and division wasn't multiplication, then the real numbers would have four operations, not two.
But by similar reasoning, why can't I just say that rings only have one operation, and multiplication is just a special type of iterative addition? So what gives here? Is it mere convention? Convenience? Or am I missing a piece of the puzzle? (I am missing many pieces in general)
.
Disclosure: my math education peaked at introduction to abstract algebra. I have not completed any other upper division math course.
), but are they really two different operations in rings? Isn't a ring really just rules we define on how they distribute, that addition is commutative, and that "both" operations are associative?I suppose the counter would be, "Oh yeah, then write [imath]8 \times\ \frac{2}{7}[/imath] as iterative multiplication." Well, [imath]\frac{2}{7}+\frac{2}{7}+\frac{2}{7}+\frac{2}{7}+\frac{2}{7}+\frac{2}{7}+\frac{2}{7}+\frac{2}{7} = \frac{16}{7}.[/imath] I don't see why this doesn't apply to transcendentals or any other type of number as well, but I tend to be wrong about everything, so I'm probably wrong here too. I suppose it becomes wildly inconvenient to try to write multiplication as iterative addition with something like, say, [imath]pi \times\ r^2[/imath], but is it really? [imath]pi \times\ r^2 = pi + pi + pi +pi +...+ pi[/imath] until you do it [imath]r^2[/imath] times.
I had no trouble recognizing that subtraction is just addition by the additive inverse, and division is just multiplication by the multiplicative inverse, and it seems few would trouble me about saying that. After all, how else could we say the real numbers are a ring, because if subtraction wasn't addition, and division wasn't multiplication, then the real numbers would have four operations, not two.
But by similar reasoning, why can't I just say that rings only have one operation, and multiplication is just a special type of iterative addition? So what gives here? Is it mere convention? Convenience? Or am I missing a piece of the puzzle? (I am missing many pieces in general)
.
Disclosure: my math education peaked at introduction to abstract algebra. I have not completed any other upper division math course.