Articles with the ‘Binary integers’ Tag - Exploring Binary

The Structure of Binary/Hexadecimal Palindromes

By Rick Regan 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.

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Finding Numbers That Are Palindromic In Multiple Bases

By Rick Regan September 1st, 2009

A palindromic number, or number palindrome, is a number like 74347, which is the same written forward and backward.

A number can be palindromic in any base, not just decimal. For example, 101101 is a palindrome in binary. A number can also be palindromic in more than one base, like decimal 719848917, which is 101010111010000000010111010101 in binary and 5272002725 in octal.

An efficient way to find palindromes in a single base is to generate them, iterating through each integer and constructing palindromes from them. An efficient way to find numbers that are palindromic in multiple bases is to take a palindrome in one base and test if it’s a palindrome in one or more additional bases.

In this article, I’ll show you C code I wrote that finds multi-base numeric palindromes. I used this code to generate tables of numbers that are palindromic in decimal and binary, decimal and hexadecimal, and decimal and octal. I also used this code to solve Euler problem 36, which asks for the sum of all numbers, less than one million, that are palindromic in decimal and binary.

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Ten Ways to Check if an Integer Is a Power Of Two in C

By Rick Regan January 18th, 2009

To write a program to check if an integer is a power of two, you could follow two basic strategies: check the number based on its decimal value, or check it based on its binary representation. The former approach is more human-friendly but generally less efficient; the latter approach is more machine-friendly but generally more efficient. We will explore both approaches, comparing ten different but equivalent C functions.

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What Powers of Two Look Like Inside a Computer

By Rick Regan January 7th, 2009

A power of two, when expressed as a binary number, is easy to spot: it has one, and only one, 1 bit. For example, 1000, 10, and 0.001 are powers of two. Inside a computer, however, numbers are more generally represented in binary code, not as “pure” binary numbers. As a result, you may not be able to look at the binary representation of a number and tell at a glance whether it’s a power of two or not; it depends on how it’s encoded.

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