Articles About How Numbers Are Represented and Manipulated in Computers - Exploring Binary

When Floats Don’t Behave Like Floats

By Rick Regan March 30th, 2010

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.

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Floating-Point Error in the NPR Media Player

By Rick Regan October 31st, 2009

The NPR Media Player apparently uses floating-point numbers to represent timestamps, based on this image (click it to enlarge):

NaNs in NPR Media Player (thumbnail) — NaNs in NPR Media Player (click image to enlarge).

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Displaying the Raw Fields of a Floating-Point Number

By Rick Regan May 20th, 2009

A double-precision floating-point number is represented internally as 64 bits, divided into three fields: a sign field, an exponent field, and a fraction field. You don’t need to know this to use floating-point numbers, but knowing it can help you understand them. This article shows you how to access those fields in C code, and how to print them — in binary or hex.

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Converting Floating-Point Numbers to Binary Strings in C

By Rick Regan May 6th, 2009

If you want to print a floating-point number in binary using C code, you can’t use printf() — it has no format specifier for it. That’s why I wrote a program to do it, a program I describe in this article.

(If you’re wondering why you’d want to print a floating-point number in binary, I’ll tell you that too.)

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Base Conversion in PHP Using BCMath

By Rick Regan April 11th, 2009

PHP has a component called BCMath which does arbitrary-precision, decimal arithmetic. I used BCMath in my decimal/binary converter because:

Arbitrary-precision lets it operate on very large and very small numbers, numbers that can’t be represented in standard computer word sizes.
Decimal arithmetic lets it use the same algorithms I’d use to convert between decimal and binary by hand.

(If you’ve written a conversion routine in standard code, especially one to convert decimal fractions to binary, you’ll see the advantage of the second point.)

This article describes the implementation of my conversion routines with BCMath.

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Base Conversion In PHP Using Built-In Functions

By Rick Regan April 7th, 2009

The PHP programming language has many built-in functions for converting numbers from one base to another. In fact, it has so many functions that it can be hard to know which to use. Some functions have similar capabilities, and some work with parameters of different types. We’ll sort through the differences in this article, and explain the proper context in which to use each function.

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A Simple C Program That Prints 2,098 Powers of Two

By Rick Regan February 13th, 2009

To write a computer program to print the first 1000 nonnegative powers of two, do you think you’d need to use arbitrary precision arithmetic? After all, 2¹⁰⁰⁰ is a 302-digit number. How about printing the first 1000 negative powers of two? 2^-1000 weighs in at a whopping 1000 decimal places. It turns out all you need is standard double-precision floating-point arithmetic — and the right compiler!

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Print Precision of Floating-Point Integers Varies Too

By Rick Regan February 11th, 2009

Recently I showed that programming languages vary in how much precision they allow in printed floating-point fractions. Not only do they vary, but most don’t meet my standard — printing, to full precision, decimal values that have exact floating-point representations. Here I’ll present a similar study for floating-point integers, which had similar results.

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Print Precision of Dyadic Fractions Varies by Language

By Rick Regan January 30th, 2009

Interestingly, programming languages vary in how much precision they allow in printed floating-point fractions. You would think they’d all be the same, allowing you to print as many decimal places as you ask for. After all, a floating-point fraction is a dyadic fraction; it has as many decimal places as it has bits in its fractional representation.

Consider the dyadic fraction 5,404,319,552,844,595/2⁵³. Its decimal expansion is 0.59999999999999997779553950749686919152736663818359375, and its binary expansion is 0.10011001100110011001100110011001100110011001100110011. Both are 53 digits long. The ideal programming language lets you print all 53 decimal places, because all are meaningful. Unfortunately, many languages won’t let you do that; they typically cap the number of decimal places at between 15 and 17, which for our example might be 0.59999999999999998.

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Ten Ways to Check if an Integer Is a Power Of Two in C

By Rick Regan January 18th, 2009

To write a program to check if an integer is a power of two, you could follow two basic strategies: check the number based on its decimal value, or check it based on its binary representation. The former approach is more human-friendly but generally less efficient; the latter approach is more machine-friendly but generally more efficient. We will explore both approaches, comparing ten different but equivalent C functions.

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What Powers of Two Look Like Inside a Computer

By Rick Regan January 7th, 2009

A power of two, when expressed as a binary number, is easy to spot: it has one, and only one, 1 bit. For example, 1000, 10, and 0.001 are powers of two. Inside a computer, however, numbers are more generally represented in binary code, not as “pure” binary numbers. As a result, you may not be able to look at the binary representation of a number and tell at a glance whether it’s a power of two or not; it depends on how it’s encoded.

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