Articles with the ‘Infinity’ Tag

Exploring Binary Numbers With PARI/GP Calculator

By Rick Regan (Published August 15th, 2009)

PARI/GP is an open source computer algebra system I use frequently in my study of binary numbers. It doesn’t manipulate binary numbers directly — input, and most output, is in decimal — so I use it mainly to do the next best thing: calculate with powers of two. Calculations with powers of two are, indirectly, calculations with binary numbers.

PARI/GP is a sophisticated tool, with several components — yet it’s easy to install and use. I use its command shell in particular, the PARI/GP calculator, or gp for short. I will show you how to use simple gp commands to explore binary numbers.

PARI/GP Calculator (Example Calculations) (My Setup on Windows)

PARI/GP Calculator (Sample of Calculations Used to Explore Binary Numbers)

The Powers of Two

By Rick Regan (Published November 14th, 2008)

The following infinite set of numbers is known as the powers of two:

$\mbox{\footnotesize{\left\{ {\ldots, \frac{1}{64}, \frac{1}{32}, \frac{1}{16}, \frac{1}{8}, \frac{1}{4}, \frac{1}{2}, 1, 2, 4, 8, 16, 32, 64, \ldots} \right\}}}$ .

Why are they called powers of two? What is the pattern you see? How is the set described mathematically? What are the set’s components? We will answer those questions in this article.

A Standard Definition of The Powers of Two

By Rick Regan (Published November 14th, 2008)

What is a power of two exactly? Is 2⁰ a power of two? Is 2^-1 a power of two? How about $\mbox{\footnotesize{2^{\frac{1}{2}}}}$ or $\mbox{\footnotesize{2^{\pi}}}$ ? It depends on how you define it; there are several definitions from which you could choose. Let’s see if we can sort them out and propose a standard definition, or at least a standard definition for our use.

Nonstandard Names for The Powers of Two

By Rick Regan (Published November 14th, 2008)

Now that you know how the powers of two are named, lets look at other, nonstandard ways to name them. You will see these names on the internet as well as in books. We will not use them on this site other than in this article, and we only discuss them here to make you aware of their use. As a by-product of this discussion, you may gain some insight into the nature of the powers of two. But beware — you may become confused as well!