Articles with the ‘Definition’ Tag

The Structure of Binary/Hexadecimal Palindromes

By Rick Regan (Published February 25th, 2010)

Binary/hexadecimal palindromes are integers that are palindromic in both binary and hexadecimal. Unlike binary/decimal palindromes, for example, they have a predictable structure. This means they can be generated directly, rather than searched for. So what is their structure?

Certainly they’re made up of the hexadecimal digits that are themselves palindromic in binary: 0, 6, 9, F; for example, F060F₁₆ = 11110000011000001111₂ and 9F9₁₆ = 100111111001₂. Each of these four hexadecimal digits maps neatly to a 4-digit binary palindrome, so any hexadecimal palindrome made from them is automatically palindromic in binary.

But there are other binary/hexadecimal palindromes, like 525₁₆ = 10100100101₂ and 70207₁₆ = 1110000001000000111₂, that contain hexadecimal digits that are not palindromic in binary. In this case, binary palindromes are produced with combinations of hexadecimal digits. It turns out there are a limited number of valid combinations, and that they’re localized — they span only two hexadecimal digits.

In this article, I’ll analyze binary/hexadecimal palindromes and describe their structure — a structure due to the relationship of the two bases, binary and hexadecimal.

Example Binary/Hexadecimal Palindromes

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!