in which physical quantity we combine and which we multiply and divide
Answers
Answer:
Explanation:
Let’s say for a moment that I have 2 bananas and 3 apples. Now, I add them together (mix them). Now, how many bananas do I have? How many apples? I added the two of them, and yet you are still telling me that I have 2 bananas and 3 apples? Of course, I DO have 5 pieces of fruit, but that’s only because I allowed the type to be more generic than before.
However, I must state here that I indeed added them together into a single grouping. This you can see by the 5 pieces of fruit question. However, they did not lose their type of 2 bananas and 3 apples.
We do this with complex numbers too. If I add 3 to the imaginary number i, I get: 3 + i. Both are still there, and available to be removed individually from the addition. So, you very well can add things of different types, but you get a combination of those types. In other words, it doesn’t simplify like you are accustomed to seeing.
Multiplication is another combination with a different property. Let’s say I had 5 pea pods with 3 peas in each (hypothetical, I know, since getting that many pods with exactly 3 peas requires some effort). It means that the structure of the objects are still there together. Looking at this another way, it is like we deconstruct the peas from the pod, or we can construct 3 pea each into 5 pods. Almost like putting components together to build things up.
This kind of combining is different than the combining of addition.
So, what happens when you assemble the components of acceleration and mass together? You produce a force. Now, according to physics, you cannot really “assemble” mass and acceleration. However, you will note that you are combining the units in a way, similar to assembling:
kg⋅ms2=N
When you learn about how rotations are actually multiplications, you will learn that multiplication is a strange and funky form of combining. However, you are bringing up the exact reason that multiplication is not repeated addition. Repeat, multiplication is not repeated addition. That only works for natural numbers.
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I actually want to teach a bit of type theory/set theory with this as well, because it’s informative to see what’s happening, and because it’s fun to do.
In type theory, we have the notion of a sum type and a product type (analogies to addition and multiplication). If I have a type of something ‘a’ and a type of another thing ‘b’, I can sum them to be a type which either ‘a’ or ‘b’, called a⊕b
. This is not your mother or fathers addition. It literally means “either”, like that thing ‘c’ is “either an ‘a’ or a ‘b’”. Think of a boolean, which has the values “true” or “false”. Booleans are sum types, such that something can only either be “true” or “false” (other logic systems notwithstanding). So, I could say there are 2 possible values for Booleans: true + false = 2 choices. If I had 3 options, it could be leftdoor + middledoor + rightdoor = 3 choices.
The product type is a different kind of combination, again. You take and combine two things together and make it a new thing. Let’s say I had another set of choices:
color: red, blue, or green
object: ball, fidget spinner, or baton
The product would be all of the choices possible from combining the two objects together:
red ball, blue ball, green ball, red FS, blue FS, green FS, red baton, blue baton, green baton
This naturally leads to 3x3 = 9 choices.
Again, this looks like natural numbers, but we are talking about types of things. We can denote one object as red ×
baton in type theory, or the entire type as: color × object. This combination assembles red together with baton to create red baton, which is different than having either a baton and a ball together at your disposal as addition would do.