# Topics

Copyright © 2008-2014 Exploring Binary

http://www.exploringbinary.com/topics/

Many of my articles address common topics; I’ve grouped them here for your convenience.

## Properties of the Powers of Two

Binary numbers are made of powers of two; these articles discuss their properties:

- How to Find the Last Digits of a Positive Power of Two
- Patterns in the Last Digits of the Positive Powers of Two
- Cycle Length of Powers of Two Mod Powers of Ten
- Seeing Powers of Five in Powers of Two and Vice Versa
- Patterns in the Last Digits of the Positive Powers of Five
- Cycle Length of Powers of Five Mod Powers of Ten
- Ending Digits of Powers of Five Form a Binary Tree

## Properties of Binary Numbers

These articles discuss properties of binary numerals:

## Binary Arithmetic

These articles discuss binary arithmetic:

## Bicimals

These articles discuss bicimals, the binary equivalent of decimals:

- Bicimals
- Converting a Bicimal to a Fraction (Subtraction Method)
- Converting a Bicimal to a Fraction (Direct Method)
- Converting a Bicimal to a Fraction (Series Method)
- “0.1 Repeating” In Binary Equals 1

## Binary Palindromes

These articles discuss binary palindromes (numbers like 1001001):

- Finding Numbers That Are Palindromic In Multiple Bases
- Counting Binary and Hexadecimal Palindromes
- The Structure of Binary/Hexadecimal Palindromes
- Counting Binary/Hexadecimal Palindromes
- In Search of Decimal/Binary/Hexadecimal Palindromes

## Visualizing Binary Numbers

These articles discuss ways to visualize binary numbers:

## Correctly Rounded Decimal to Floating-Point Conversion

These articles discuss conversion of decimal strings to floating-point binary numbers:

- Quick and Dirty Decimal to Floating-Point Conversion
- Decimal to Floating-Point Needs Arbitrary Precision
- Incorrectly Rounded Conversions in Visual C++
- Incorrectly Rounded Conversions in GCC and GLIBC
- Correctly Rounded Conversions in GCC and GLIBC
- GCC Conversions Are Incorrect, Architecture or Otherwise
- real.c Rounding Is Perfect (GCC Now Converts Correctly)
- GLIBC strtod() Incorrectly Converts 2^-1075
- Visual C++ and GLIBC strtod() Ignore Rounding Mode
- Correct Decimal To Floating-Point Using Big Integers
- How strtod() Works (and Sometimes Doesn’t)
- How GLIBC’s strtod() Works
- How GCC Converts Decimal Literals to Floating-Point
- Fast Path Decimal to Floating-Point Conversion
- strtod()’s Initial Decimal to Floating-Point Approximation
- Using Integers to Check a Floating-Point Approximation
- Adjusting the Floating-Point Approximation in strtod()
- Bigcomp: Deciding Truncated, Near Halfway Conversions
- Properties of the Correction Loop in David Gay’s strtod()
- Incorrect Directed Conversions in David Gay’s strtod()
- A Bug in the Bigcomp Function of David Gay’s strtod()
- Gay’s strtod() Returns Zero For Inputs Just Above 2^-1075
- A Better Way to Convert Integers in David Gay’s strtod()
- Double Rounding Errors in Floating-Point Conversions
- GCC Avoids Double Rounding Errors With Round-To-Odd
- Incorrect Decimal to Floating-Point Conversion In SQLite
- PHP Hangs On Numeric Value 2.2250738585072011e-308
- Why “Volatile” Fixes the 2.2250738585072011e-308 Bug
- A Better Fix for the PHP 2.2250738585072011e-308 Bug
- Java Hangs When Converting 2.2250738585072012e-308
- A Closer Look at the Java 2.2250738585072012e-308 Bug
- Incorrectly Rounded Subnormal Conversions in Java
- Nondeterministic Floating-Point Conversions in Java

## Correctly Rounded Floating-Point to Decimal Conversion

These articles discuss conversion of floating-point binary numbers to decimal strings:

- Quick and Dirty Floating-Point to Decimal Conversion
- Incorrect Floating-Point to Decimal Conversions
- Inconsistent Rounding of Printed Floating-Point Numbers
- Fifteen Digits Don’t Round-Trip Through SQLite Reals
- 15-Digit Quick and Dirty Conversions Don’t Round-Trip
- The Shortest Decimal String That Round-Trips: Examples
- Incorrect Round-Trip Conversions in Visual C++

## Printing the Contents of a Floating-Point Variable

These articles show different ways to display the exact contents of an IEEE 754 floating-point variable:

October 6th, 2013 at 10:18 am

Thx for interesting informations. Can you add :

* finding repeating paterns in binary ( for example :

1/6 = 0.0(01)= 0.00101010101010101010…

where () show repeating pattern

* expand converter to use also such notation. For example 0.((001)^{88}010)

see for more info :

http://www.hindawi.com/journals/mpe/2013/105283/ref/

TIA

October 6th, 2013 at 12:01 pm

@Adam,

These two online converters handle repeating binary; they may be helpful to you:

http://www.knowledgedoor.com/2/calculators/convert_a_number_with_a_repeating_fractional_part.html

http://www.knowledgedoor.com/2/calculators/convert_a_number_with_a_mixed_fractional_part.html