Back to building extensions of number systems. The last number system we looked at was the rational numbers. The key point from the digression about fields last time is that the rational numbers have a larger group of properties for addition and multiplication than the natural numbers or the integers.
A few weeks ago, I posted about rational numbers and least upper bounds. That's the next step in extending the rational numbers. As I previously discussed, the square root of 2, for example, is not a rational number, which means that not every subset of the rational numbers has a least upper bound.
The real numbers are defined as the extension of the rational numbers such that every subset of the real numbers has a least upper bound. Actually proving that the real numbers exist is hard (but it can be done), but we can also get away with just assuming that they exist and using them.
The real numbers share all of the properties for addition and multiplication as the rational numbers. (In other words, the real numbers are also a field.) They are also ordered.
We can define rational exponents of positive bases as real numbers. If r is a rational number, then r=m/n, where m and n are integers. Then if a is a real number greater than 0, ar=am/n=(am)1/n. am is a positive real number by closure of multiplication, and if b is a positive real number b1/n is a real number because of the least upper bound property. Therefore ar is a real number for any positive real number a and any rational number r.
We can also define ax for any positive real number a and any real number x. Define a set of numbers of every number ar, where r is any rational number less than x. Then that set has a least upper bound, and we can define ax as equal to that least upper bound.
I've been limiting exponents to positive bases (or 0, for positive exponents only). I will deal with that soon, but next up will be some digressions and rants.