Barbie Goes Binary
In case you haven’t heard, Mattel® has created Computer Engineer Barbie®, based on popular vote. Here’s the laptop she is holding:
Binary Code on Barbie's Laptop
It spells “Barbie” — repeatedly, in ASCII code.
Latest Articles
When Doubles Don’t Behave Like Doubles
In my article “When Floats Don’t Behave Like Floats” I explained how calculations involving single-precision floating-point variables may be done, under the covers, in double or extended precision. This leads to anomalies in expected results, which I demonstrated with two C programs — compiled with Microsoft Visual C++ and run on a 32-bit Intel Core Duo processor.
In this article, I’ll do a similar analysis for double-precision floating-point variables, showing how similar anomalies arise when extended precision calculations are done. I modified my two example programs to use doubles instead of floats. Interestingly, the doubles version of program 2 does not exhibit the anomaly. I’ll explain.
When Floats Don’t Behave Like Floats
These two programs — compiled with Microsoft Visual C++ and run on a 32-bit Intel Core Duo processor — demonstrate an anomaly that occurs when using single-precision floating point variables:
Program 1
#include "stdio.h"
int main (void)
{
float f1 = 0.1f, f2 = 3.0f, f3;
f3 = f1 * f2;
if (f3 != f1 * f2)
printf("Not equal\n");
}
Prints “Not equal”.
Program 2
#include "stdio.h"
int main (void)
{
float f1 = 0.7f, f2 = 10.0f, f3;
int i1, i2;
f3 = f1 * f2;
i1 = (int)f3;
i2 = (int)(f1 * f2);
if (i1 != i2)
printf("Not equal\n");
}
Prints “Not equal”.
In each case, f3 and f1 * f2 differ. But why? I’ll explain what’s going on.
Ending Digits of Powers of Five Form a Binary Tree
In my article “Patterns in the Last Digits of the Positive Powers of Five” I showed that the cycles of ending digits of the positive powers of five could be represented with a binary tree:
Binary Tree Showing Nested Ending Digit Patterns of the Positive Powers of Five
The tree layout shows that certain pairs of ending digits are related, and that these pairs differ by five in their starting digits. I will show why this is true.
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.
How to Install and Run GMP on Windows Using MPIR
To perform arbitrary-precision arithmetic in C and C++ programs on Windows, I use GMP. In particular, I use MPIR, a Windows port of GMP. MPIR is a simple alternative to using GMP under Cygwin or MinGW.
I will show you how to install MPIR in Microsoft Visual C++ as a static, 32-bit library. I will also show you how to install the optional C++ interface — also as a static, 32-bit library. I will provide two example C programs that call the GMP integer and floating-point functions, and two equivalent C++ programs — programs that use the same GMP functions, only indirectly through the C++ interface.
How to Install and Run PARI/GP Calculator on Windows
PARI/GP calculator, or gp for short, is an arbitrary-precision calculator (among other things) that I use frequently in my study of binary numbers. Here are instructions for installing and running it on Windows.
The Structure of Binary/Hexadecimal Palindromes
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
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.
Cycle Length of Powers of Five Mod Powers of Ten
In my article “Patterns in the Last Digits of the Positive Powers of Five” I noted that the positive powers of five modulo 10m cycle with period 2m-2, m ≥ 2, starting at 5m. In this article, I’ll present my proof, which has two parts:
- Part 1 shows that the powers of five mod 2m cycle with period 2m-2, m ≥ 2, starting at 50.
- Part 2 shows that the powers of five mod 10m cycle with the same period as the powers of five mod 2m, starting at 5m.
The highlight of my proof is in part 1, where I derive a formula to show that the period, or order, of 5 mod 2m is 2m-2. While it is in general not possible to derive a formula for the order of a number, I’ll show it is possible for the powers of five mod 2m — due to a hidden, binary structure I’ve uncovered.
Binary Dates in 2010 and 2011
People have been tweeting about the upcoming dates that look like binary numbers. 10/10/10 seems to be a favorite, both because of its symmetry and because 101010 = 42 in decimal (you know, the answer to the ultimate question of life, the universe, and everything). Here are the nine dates in 2010, interpreted as binary numbers, and with their decimal equivalents:
Patterns in the Last Digits of the Positive Powers of Five
The positive powers of five — 5, 25, 125, 625, 3125, 15625, … — have a compact, repeating pattern in their ending m digits, in the powers of five from 5m on. For example: starting with 5, their last digit is always 5; starting with 25, their last two digits are always 25; starting with 125, their last three digits alternate between 125 and 625. These cycles come in lengths of powers of two.
Cycles in the Ending Digits of the Powers of Five
I will show you why these cycles exist, how they are expressed mathematically, and how to visualize them.
