Trig is artificially hard in calculus. When you get to complex numbers it all gets easier. But they don't teach that until later.
One thing I learned at Berkeley was that there are two kinds of problems: linear problems and problems you can't solve. The trick (EE 120 Linear Systems) was always how to transform a complicated problem into a linear problem. Yeah we used complex numbers as part of the trick to get to linear problems.
Maybe Sheldon Axler will do Calculus Done Right.
I agree that calculus sans trig is a good idea if you aren't going to use complex numbers.
This needs to be mentioned more frequently to be honest. I still am unsure why the hell Gibbs vector calculus is taught when Clifford algebras are so beautiful.
I feel like we're still only scratching the surface with complex numbers. I didn't learn Euler's Identity nor most of the interesting parts of i until recently on Youtube.
We can use complex numbers to describe a 2-dimensional number-space with a single digit. How do we describe a 3-dimensional number-space with a single digit? What about higher dimensional number-space?
> We can use complex numbers to describe a 2-dimensional number-space with a single digit. How do we describe a 3-dimensional number-space with a single digit? What about higher dimensional number-space?
The complex numbers equips the vector space of 2-dimensional real numbers with a multiplication. This structure is known as an algebra.
The analogue for 4 dimensions is the Quaternion Algebra, which is no longer commutative (i.e. a * b != b * a). The elements of unit length correspond (essentially) to rotations in 3 dimensions and they are used for this in e.g. computer graphics.
The analogue for 8 dimensions is the Octonions which are no longer associative (i.e. (a * b) * c != a * (b * c)). The construction can be continued with a doubling of the dimension through the Cayley-Dickson construction, however these higher dimensional versions are even worse.
It is not possible to associate a multiplication to 3-dimensional real space which makes it an algebra. This fact is related to the classification of exceptional Lie groups. The reason is roughly that the units in an algebra form a group, but the 2-sphere is not a Lie group.
[technical stuff] An algebra is just a vector space V with a bilinear operation * :VxV->V. [/technical stuff] There are plenty of interesting and intuitive algebras other than the stuff you get from the Cayley-Dickson constructions.
Some interesting algebras:
* you can have some fun making the imaginary unit i square to 0 instead of -1. The resulting algebra is called the dual numbers, and has some surprising properties. You can gain a bit of understanding of it by using analogies with the complex numbers.
* You can make i square to +1 without being equal to +1 or -1. Again you can gain some understanding using analogies with the complex numbers. It's perhaps not the most useful examples because it's isomorphic to R \oplus R.
* Another family of examples, a very important one actually, is the algebra of 2x2 or 3x3 or nxn matrices over the real numbers or the complex numbers.
The first two examples are useful for understanding the general concept of a quotient ring. They're hardly exhaustive, but they are easy to picture.
Now I can return to your claim about CD: What makes the Cayley-Dickson family significant is that it produces all the division algebras over the real numbers, which are the algebras for which division by nonzero elements is always possible.
> The complex numbers equips the vector space of 2-dimensional real numbers with a multiplication. This structure is known as an algebra.
This is a very formal/technical and not very enlightening (and often quite misleading to novices) way of interpreting what the complex numbers “are”. It is basically just a declaration “we have this particular set of symbolic manipulations which we are making up; deal with it.” It’s also kind of ironic because historically the invention of linear algebra came out of attempts to generalize complex numbers.
For me, the key point is that the complex numbers are isomorphic to the transformations of rotation and scaling centered at a point in a plane, with composition of such transformations corresponding to complex multiplication. That is, a complex number can be associated 1:1 with such a transformation. So to understand complex arithmetic, what we really need to understand is the composition of plane rotation and scaling.
In other words, a complex number is not a vector (in the sense of the word “vector” used in physics; it is a “vector” in the sense of an abstract mathematical object satisfying certain axioms).
Personally I would say that the complex numbers “are” quotients of two-dimensional vectors (using the Clifford product, and with non-zero divisor) which include a given planar orientation (the bivector i, with i^2 = –1) but are otherwise unitless and need not be defined in terms of any particular basis though splitting them into the sum of a scalar and bivector part or into the product of a rotation and a scaling is often convenient. But doing this explanation justice requires spending a considerable amount of time explaining the difference between affine points and vectors, talking about how two-dimensional vectors work and how to think about vector multiplication, etc.
This makes generalization to multiplication of vectors in higher dimensional Euclidean (or pseudo-Euclidean) spaces very natural without bothering with any discussion of classifying Lie groups or whatever.
But you could also say that complex numbers are linear combinations of the two matrices [1 0; 0 1] and [0 –1; 1 0]; which when right-multiplied by arbitrary column vectors are respectively the identity and a 90° rotation.
Or you could say that complex numbers are the quotient ring R[X]/(X^2 + 1). [This is closest to the historical approach, where we just declare by fiat that √–1 will be a meaningful symbol.]
Or you could define complex numbers purely in terms of classical trigonometry and analytic geometry.
But it seems clear that the grandparent poster was using the wrong word (“digit”) and likely meant something like “we can conceive of a 2-dimensional quantity as a single number-like entity”.
One thing I learned at Berkeley was that there are two kinds of problems: linear problems and problems you can't solve. The trick (EE 120 Linear Systems) was always how to transform a complicated problem into a linear problem. Yeah we used complex numbers as part of the trick to get to linear problems.
Maybe Sheldon Axler will do Calculus Done Right.
I agree that calculus sans trig is a good idea if you aren't going to use complex numbers.