Showing n! > 2n When n Is A Power of Two

Which is bigger, 64! or 264? 64! is, because it follows from a proof by induction for any integer n greater than or equal to 4. It’s also easy to just reason that 64! is bigger: 264 is 64 factors of 2, whereas 64! has 64 factors, except all but one of them (1) are 2 or greater.

When I saw this problem though I wondered if I could solve it in another way: Could the factors of two alone in 64! be greater than 264? As it turns out, almost.

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Converting a Bicimal to a Fraction (Series Method)

I’ve shown you two ways to convert a bicimal to a fraction: the subtraction method and the direct method. In this article, I will show you a third method — a common method I call the series method — that uses the formula for infinite geometric series to create the fraction.

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1,073,741,823 Grains of Rice

In the children’s book “One Grain of Rice: A Mathematical Folktale” a girl uses her knowledge of exponential growth to trick a greedy king into turning over his stockpile of rice. Hidden in the story are mathematical concepts related to doubling: powers of two, geometric sequences, geometric series, and exponents. I will analyze the story from this perspective, and then discuss my experience reading it to first and third grade students.

One Grain of Rice Book Cover
Front Cover of the Book “One Grain of Rice”

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Counting Binary/Hexadecimal Palindromes

In my article “Counting Binary and Hexadecimal Palindromes” I derived formulas for counting binary palindromes and hexadecimal palindromes. For each type of palindrome, I derived two pairs of formulas: one pair to count n-digit palindromes, and one pair to count palindromes of n digits or less.

In this article, I will derive similar formulas to count binary/hexadecimal palindromes — multi-base palindromes I’ve shown to have an algorithmically defined structure.

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Counting Binary and Hexadecimal Palindromes

How many nonzero, n-digit, decimal number palindromes are there? These two formulas give the answer:

  • When n is even: 9·10n/2-1
  • When n is odd: 9·10(n+1)/2-1

How many nonzero, decimal number palindromes are there, consisting of n-digits or less? These two formulas give the answer:

  • When n is even: 2(10n/2 – 1)
  • When n is odd: 11·10(n-1)/2 – 2

So for example, there are 900 5-digit decimal palindromes, 9,000 8-digit decimal palindromes, 1,098 decimal palindromes of 5 digits or less, and 19,998 decimal palindromes of 8 digits or less.

In this article, I will derive similar formulas to count binary and hexadecimal number palindromes.

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Exploring Binary Numbers With PARI/GP Calculator

PARI/GP is an open source computer algebra system I use frequently in my study of binary numbers. It doesn’t manipulate binary numbers directly — input, and most output, is in decimal — so I use it mainly to do the next best thing: calculate with powers of two. Calculations with powers of two are, indirectly, calculations with binary numbers.

PARI/GP is a sophisticated tool, with several components — yet it’s easy to install and use. I use its command shell in particular, the PARI/GP calculator, or gp for short. I will show you how to use simple gp commands to explore binary numbers.

PARI/GP Calculator (Example Calculations) (My Setup on Windows)
PARI/GP Calculator (Sample of Calculations Used to Explore Binary Numbers)

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Elements of Binary in the NCAA Basketball Tournament

If you’re a sports fan, you think of basketball when you see this:

Diagram of the NCAA Basketball Tournament
Basketball or Binary?

If you’re like me, you also think of powers of two, binary trees, logarithms, laws of exponents, geometric sequences, geometric series, and Bernoulli trials — in short, the elements of binary numbers, binary code, and binary logic.

I hope you’re thinking: “wow — all that math stuff I hated actually has a practical use!”

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About the Header Image

Header image thumbnail

This image is designed to remind you of space exploration (imagine a photo taken from the surface of the moon). The symbols “floating in space” are exemplary of the topics to be explored on this site. The black and white imagery is also symbolic of binary encoding, which is characterized by two opposing “states.”

(A reader proposed an alternate view of the image — that of a chalkboard full of mathematical symbols!)

What do the symbols mean?

Here is a brief explanation of each of the sixteen symbols in the image, from left to right:

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