## Exploring the Math of Binary Numbers: guest post

[ Editor's note: Rick Regan took me up on my offer to promote web sites I like. Rick has a nice blog about binary numbers. I've enjoyed thinking about binary numbers in ways I hadn't considered before. I highly recommend Rick's blog. Here is a guest article from Rick about his blog. ]

## Exploring the Math of Binary Numbers

Hello readers of Wild About Math! My name is Rick Regan and I'm the author of a blog called Exploring Binary. On Exploring Binary, I write a lot about binary numbers. Binary numbers exist in two worlds: inside computers and inside mathematics. I study them in both contexts, but here I'll just highlight some of what I've written about their mathematical properties.

## Digit Properties of Binary Numbers

Here are some properties of the binary representation of certain decimal numbers:

- A nonnegative power of ten has a binary representation with trailing digits that match that power of ten. For example, 1000 decimal is 1111101000 in binary.
- A decimal integer consisting of n digits of 9s has a binary representation with n digits of trailing 1s. For example, 999 decimal is 1111100111 in binary.

## Binary Palindromes

Binary numbers can be palindromes, just like decimal numbers. For example, 1001001 is a binary palindrome. Here are some facts about binary palindromes:

- Binary palindromes can be generated and counted like decimal palindromes, using similar algorithms and formulas.
- Some numbers are palindromic in
*both*binary and decimal; for example, 11101111110111 base 2 = 15351 base 10. - Some numbers are palindromic in binary, decimal, and octal; for example, 1001001001 base 2 = 585 base 10 = 1111 base 8.
- It is unknown whether there are any numbers that are palindromic in binary, decimal, and
*hexadecimal*.

## Digit Properties of the Powers of Two

Binary numbers are composed of powers of two: binary integers are made of nonnegative powers of two, and binary fractions are made of negative powers of two. Here are some properties of their digits:

- The last digit of the positive powers of two cycles through the digits 2, 4, 8, 6.
- The last m digits of the positive powers of two cycle with period 4·5
^{m-1}. - Negative powers of two look like positive powers of five and vice versa. For example, 2
^{-4}= 0.0625 and 5^{4}= 625. - Negative powers of two end with the digit 5.
- The last m digits of the negative powers of two cycle with period 2
^{m-2}.

The details behind all these properties -- including proofs -- can be found in this list of articles on Exploring Binary.

(Also, be sure to check out these articles on Wild About Math -- both are related to binary numbers: Easy and fun Math trick and Impressive Math magic with 16 index cards.)

JonathanMay 15th, 2010 - 06:09

Divisibility in binary is different. Anything that ends in 0 is even. In 2 zeroes is a multiple of 4. In 3 zeroes a multiple of 8, and in n zeroes a multiple of 2^n.

Divisibility by 11 (binary), that is 3 in decimal, follows the same rule as division by 11 in decimal – add and subtract alternate digits, and if the result is divisible by 11 (binary), then so is the original number.

Example:

11010011011101

1-1+0-1+0-0+1-1+0-1+1-1+0-1

Since the result is -3 (decimal) or -11 (binary) the original number is a multiple of 3 (11 binary)