Understanding Multiplication/Dot Product of Two Numbers/Two Vectors This might be a silly question, but I just realized I don't think I understand what multiplication really is. For example I know $3\cdot4=12$ because that's what you learn in school. If we go with the idea of multiplication as repeated addition, $3\cdot4$ means add $3$ to $0$ (the zero elements) $4$ times. So here we have multiplication defined in terms of addition and the zero/identity element for natural numbers.
My confusion is around numbers that have units. Say we have oranges, how does one interpret $3$ oranges $\cdot 4$ oranges?
Does adding a unit (oranges) to the number make it a vector and therefore the multiplication operator has a different meaning because for two vectors it's dot products? If so, why do we interpret the multiplication of a basis vector with itself to be 1?
 A: It may be hard to understand from an outside point of
view but mathematics is based on precise definitions
that are decided on by mathematicians among themselves.
Thus, for example, the dot product of a unit vector
with itself is $\,1\,$ as a consequence of the precise
definition of the dot product of vectors. However, in
order to multiply oranges by oranges there is no such
precise definition of the result. Of course, you or
anyone else are free to come up with a definition if
that is desired. This new hypothetical definition may
not be useful or well defined but it could be done.
Other than that, there is no currently meaningful way
to multiply two oranges, as far as I know.
As for the precise definition of dot product of two
vectors, such as $\,\vec{i}\cdot\vec{i}\,,$ this is
closely related to the definition
of vectors themselves as geometric objects (loosely
imagined as arrows in space). The origin of vectors
has a long and interesting history. Some of this is
in Michael J. Crowe's book, A History of Vector
Analysis and several other sources. More details
are not suitable for MSE.
A: I think learning a bit of Abstract Algebra (here's an introduction) will answer some of these questions. An operator (addition/multiplication/etc) and type of objects (integers/reals/vectors/etc) it acts on - come together. You need to define what a multiplication is for this type of objects. For instance:

*

*Multiplication of 2 integers can be represented with repeated addition as you mentioned

*(Scalar) Multiplication of a geometrical vector (an arrow) is stretching squeezing that vector. Imagine a physical stretching of an arrow without multiplication of its components. You just have an instruction to stretch 2x and that's what you do with a pen and paper.

You can come up with any mathematical operation which does craziest things. But most of such operations will be useless. E.g. they won't help finding unknown $x$ in the equation: $a*x=b$ where $*$ is your new mathematical operation, and $a, x, b$ are your new type of objects (I don't know.. let's say they are circles - you multiply circles). While existing, real maths creates these structures & operations in a way to be useful for some purpose.
Dot product is not a multiplication in the typical sense. E.g. you can't solve  $a\cdot x=b$. This operation has a completely different purpose - it's useful for different reasons. E.g. it can be used:

*

*To define length of a vector: $|a|=\sqrt{a\cdot a}$

*To find a unit vector (of length 1) that points to the same direction: $\frac{a}{|a|}$

*To find out if vectors are perpendicular (when $a\cdot b=0$)

*To find projections of one vector onto another ($b\cdot \frac{a}{|a|}$)

That's why Dot Products are useful - not because they can e.g. solve equations. Incidentally all this math can be generalized. Notice that e.g. functions like polynomials also have similar properties like geometrical vectors: you can multiply them by a scalar, you can add them. But then - what would be a length of such "vector"? What is a dot product? To answer these questions a list of requirements are introduced for such products taken from what we just saw like length of a unit vector $\frac{a}{|a|}$ is 1, and others. So if we could come up with a new operation that obeys these properties - we can use it the way we used dot products.
Such products are now called Inner Products and Dot Product is just one of the implementations. For functions for instance you could define a different implementation like $\int{f(x)f(x)dx}$ is going to be similar to $a \cdot a$. It's possible to define all sorts of implementations for Inner Product, but they still have to obey those roles. And now we can define what length is and it means to be perpendicular for functions and other entities that obey similar rules. And we can re-use all the maths from vectors with these entities - when used in such a way these entities and their respective operations are called Abstract Vectors.
Now units is about giving physical sense to numbers. Can you multiply oranges? Mathematically - the requirement is that you have right entities (e.g. integers) and appropriate operation (e.g. multiplication). But whether this has physical sense - that's not up to math to decide. If $orange^2$ (area of a field of oranges) makes sense - then go ahead and do the operation.
If you use oranges as one of the components of your vector e.g. then $|o|=\sqrt{o \cdot o}$ is still a length of a vector, so it gives you number of oranges in it (which you knew from the vector's component already).
