Last time, we looked a little bit at the ordered pair representation for complex numbers. An obvious followup is to ask whether we can represent numbers as ordered triples or ordered tuples in general.
The answer is that of course we can. A euclidean space is the set of all ordered tuples of real numbers where the tuples all have the same number of members. Okay, that probably made no sense. Choose some natural number k. Choose k real numbers, x1,x2,…,xk. Then x=(x1,x2,…,xk) is a vector in euclidean k-space. We can write that x is a member of ℜk.
It should be clear that the real numbers are equivalent to ℜ1 and the complex numbers are equivalent to ℜ2. Every point in the interior of a cube can be represented as a number in ℜ3, so that's not too hard to visualize. Higher dimensions are hard to visualize, but fortunately they are not too hard to work with mathematically.
Addition and multiplication by real numbers are defined in euclidean spaces. Addition is defined to be consistent with addition of real numbers or complex numbers, just by adding the components in each direction. Multiplication by a real number multiplies each component by that number, so it is again consistent with multiplication of real numbers and complex numbers. Visually, multiplication by a real number changes a vector's length, but it does not change the direction the vector points in.
Multiplication of two vectors by each other to produce another vector of the same dimension is not defined. (Therefore, euclidean spaces are not fields.) Depending on what you are doing with them, it may be better to consider real numbers and complex numbers as numbers rather than as points in euclidean space, because the numbers can be multiplied by the points cannot be.
However, multiplying two vectors to produce a real number is defined. This is called the dot product or inner product, and is defined by x⋅y=∑xiyi. The norm of a vector is defined by ||x||=(x⋅x)1/2. The important thing here is that the norm of a vector is equivalent to the absolute value of a complex or real number.