Articles from November, 2008

How to Read a Binary Clock

By Rick Regan (Published November 23rd, 2008)

What is a binary clock? Before doing a web search I would have guessed this:

My Fake Binary Alarm Clock (time is 6:43 PM).

6:43 PM on my fake binary alarm clock (courtesy photo-editing software).

In other words, a regular digital clock, except with binary numerals instead of decimal numerals. But as far as I know, a clock like this doesn’t exist. If you search for “binary clock,” you get a clock of a different design, one like this:

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.

How Do You Pronounce “Binary”?

By Rick Regan (Published November 14th, 2008)

According to Merriam-Webster, there are two ways to pronounce binary:

“bye’ nuh ree” (rhymes with “winery”)
“bye’ nairy” (flows like the words “bye Mary”)

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!

How to Check If a Number Is a Power of Two

By Rick Regan (Published November 14th, 2008)

How can you tell if a number is a power of two?

That’s easy if it’s in the form 2ⁿ, where n is an integer. For example, 2¹², 2⁰, and 2^-37 are powers of two. That is by definition. But what about arbitrary positive numbers like 16,392, 524,288, or 0.00390625? Are they powers of two? Here’s how to tell — if they can be simplified to the form 2ⁿ, they are; if they can’t, they’re not.

Composing Powers of Two Using The Laws of Exponents

By Rick Regan (Published November 14th, 2008)

Powers of two can be combined, under the laws of exponents, to create other powers of two. Under these rules, you can multiply powers of two, divide powers of two, or raise a power of two to a power and still get another power of two. You can combine these rules to create complicated expressions, expressions that result in a single power of two. For example,

$\mbox{\footnotesize{\displaystyle {{\left(\frac{2^4 \cdot 2^3}{2^5}\right)}^3 = \: 2^6}}}$ .

The laws of exponents apply generally to any base; two is no different. But since we’re interested in powers of two, we’ll couch them in terms of powers of two. Once we explain the laws in this way, you’ll understand the math behind the example above.