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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 4·5m-1.
  • Negative powers of two look like positive powers of five and vice versa. For example, 2-4 = 0.0625 and 54 = 625.
  • Negative powers of two end with the digit 5.
  • The last m digits of the negative powers of two cycle with period 2m-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.)

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  1. 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.


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

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