In this post, I prove that $ |xy| = |x||y|$ in which $x$ is a variable for any real number and $y$ is a variable for any real number. This is done by approximately following the steps in reference [1]. Then I provide a lot of commentary because that is what I like to do. Proof: Note that $x$ can be zero or nonzero, and $y$ can be zero or nonzero–this leads to four cases. The cases involving zero are investigated first. If $x=0$, then $|xy| = |0| = 0 = |0||y| = |x||y|$. If $y=0$, then $|xy| = […]

Exponents

The following rules apply for exponents that are positive integers. The justification is sketched here.   These rules are also said to apply for exponents that are not positive integers–that is, zero and negative integers. If $ p$ and $ q$ are rational numbers, they are also said to apply. Finally, if $ p$ and $ q$ are irrational numbers, they are said to hold if $ x$ and $ y$ are positive–the other cases are not important for my purposes [1]. References [1] Konrad Knopp. Theory and Application of Infinite Series. Dover Publications. 1990.