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Chapter 2. Numbers

John von Neumann (1951)

Anyone who considers arithmetical methods of producing random digits is, of course, in a state of sin.

2.0. Introduction

Numbers, the most basic data type of almost any programming language, can be surprisingly tricky. Random numbers, numbers with decimal points, series of numbers, and conversion between strings and numbers all pose trouble.

Perl works hard to make life easy for you, and the facilities it provides for manipulating numbers are no exception to that rule. If you treat a scalar value as a number, Perl converts it to one. This means that when you read ages from a file, extract digits from a string, or acquire numbers from any of the other myriad textual sources that Real Life pushes your way, you don't need to jump through the hoops created by other languages' cumbersome requirements to turn an ASCII string into a number.

Perl tries its best to interpret a string as a number when you use it as one (such as in a mathematical expression), but it has no direct way of reporting that a string doesn't represent a valid number. Perl quietly converts non-numeric strings to zero, and it will stop converting the string once it reaches a non-numeric character—so "A7" is still 0, and "7A" is just 7. (Note, however, that the -w flag will warn of such improper conversions.) Sometimes, such as when validating input, you need to know whether a string represents a valid number. We show you how in Recipe 2.1.

Recipe 2.15 shows how to get a number from strings containing hexadecimal, octal, or binary representations of numbers such as "0xff", "0377", and "0b10110". Perl automatically converts numeric literals of these non-decimal bases that occur in your program code (so $a = 3 + 0xff will set $a to 258) but not data read by that program (you can't read "ff" or even "0xff" into $b and then say $a = 3 + $b to make $a become 258).

As if integers weren't giving us enough grief, floating-point numbers can cause even more headaches. Internally, a computer represents numbers with decimal points as floating-point numbers in binary format. Floating-point numbers are not the same as real numbers; they are an approximation of real numbers, with limited precision. Although infinitely many real numbers exist, you only have finite space to represent them, usually about 64 bits or so. You have to cut corners to fit them all in.

When numbers are read from a file or appear as literals in your program, they are converted from their textual representation—which is always in base 10 for numbers with decimal points in them—into an internal, base-2 representation. The only fractional numbers that can be exactly represented using a finite number of digits in a particular numeric base are those that can be written as the sum of a finite number of fractions whose denominators are integral powers of that base.

For example, 0.13 is one tenth plus three one-hundredths. But that's in base-10 notation. In binary, something like 0.75 is exactly representable because it's the sum of one half plus one quarter, and 2 and 4 are both powers of two. But even so simple a number as one tenth, written as 0.1 in base-10 notation, cannot be rewritten as the sum of some set of halves, quarters, eighths, sixteenths, etc. That means that, just as one third can't be exactly represented as a non-repeating decimal number, one tenth can't be exactly represented as a non-repeating binary number. Your computer's internal binary representation of 0.1 isn't exactly 0.1; it's just an approximation!

$ perl -e 'printf "%.60f\n", 0.1'

Recipe 2.2 and Recipe 2.3 demonstrate how to make your computer's floating-point representations behave more like real numbers.

Recipe 2.4 gives three ways to perform one operation on each element of a set of consecutive integers. We show how to convert to and from Roman numerals in Recipe 2.5.

Random numbers are the topic of several recipes. Perl's rand function returns a floating-point value between 0 and 1, or between 0 and its argument. We show how to get random numbers in a given range, how to make random numbers more random, and how to make rand give a different sequence of random numbers each time you run your program.

We round out the chapter with recipes on trigonometry, logarithms, matrix multiplication, complex numbers, and the often-asked question: "How do you put commas in numbers?"

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