In abstract algebra, especially in body theory, field extension is one of the basic tools for describing the structure and properties of a body. Body enlargement - Wikipedia
I donāt want to be mathematically rigorous, so Iāll write softly: when you have a set and there are laws that are established within that set, you can create a larger set by adding things that are not in that set and still trying to maintain the laws as much as possible. This is often beneficial.
For example, when I was small, I had a set of numbers ā1, 2, 3, ā¦ā even vaguely, and the law of addition was established within that set. If we insert the law of subtraction here, we get, āHow about two minus three?ā I canāt!ā It became āI canāt!
To this set of numbers, we add a ānegative numberā, a number that is ānon-existentā in the sense that -1 apples are not real. In doing so, the law of subtraction becomes a simple definition that is always viable.
Similarly, by adding the concept of imaginary numbers, which are not real, it becomes possible to rotate by multiplication and describe oscillating phenomena more easily.
I began to think of a āvectorā with N numbers in a row instead of just one. This was also a non-existent number in the sense that āthere are not (1, 2, 3) apples,ā but it was convenient for expressing things like speed.
Think of a āmatrixā aligned with NM or, more generalizing, NMLā¦ and consider tensors lined up with āMā, the original ordinary numbers being āscalarsā and āzero-order tensorsā, vectors being āfirst-order tensorsā, and matrices being āsecond-order tensorsā.
Various other extensions have been made, such as adding āinfinitesimals,ā adding āinfinity,ā making functions a type of number, and so on. The most useful of these extensions are often used. It is interesting to note that the concept of numbers is not an innate one, but rather an object that is created by people and evaluated according to its usefulness, just like a program.
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