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This comparison is true if f1 and f3 are within some small delta from each other, which should still be true even if you take some small round-off error into account.

Short circuits and C++

The& & and || perform what is called short-circuit evaluation. Consider the following:

condition1 && condition2

If condition1 is not true, the result is not true, no matter what the value of condition2. (For example, condition2 could be true or false without changing the result.) The same situation occurs in the following:

condition1 || condition2

If condition1 is true, the result is true, no matter what the value of condition2.

To save time, C++ (wisely) cuts to the chase and evaluates condition1 first. C++ does not evaluate condition2 if condition1 is false (in the case of & &) or condition1 is true (in the case of ||). This is known as short circuit evaluation.

Expressing Binary Numbers

C++ variables are stored internally as so-called binary numbers. Binary num­ bers are stored as a sequence of 1 and 0 values known as bits. Most of the time, you don’t really need to deal with which particular bits you use to rep­ resent numbers. Sometimes, however, it’s actually practical and convenient to tinker with numbers at the bit level — so C++ provides a set of operators for that purpose.

Fortunately, you won’t have to deal too often with C++ variables at the bit level, so it’s pretty safe to consider the remainder of this chapter a Deep Techie excursion.

The so-called bitwise logical operators operate on their arguments at the bit level. To understand how they work, let’s first examine how computers store variables.

54	Part I: Introduction to C++ Programming

The decimal number system

The numbers we’ve been familiar with from the time we could first count on our fingers are known as decimal numbers because they’re based on the number 10. (Coincidence? I don’t think so . . .) In general, the programmer expresses C++ variables as decimal numbers. Thus you could specify the value of var as (say) 123 — but consider the implications.

A number such as 123 refers to 1 * 100 + 2 * 10 + 3 * 1. Each of these base numbers — 100, 10, and 1 — are powers of 10.

123 = 1 * 100 + 2 * 10 + 3 * 1

Expressed in a slightly different (but equivalent) way, 123 looks like this:

123 = 1 * 102 + 2 * 101 + 3 * 100

Remember that any number to the zero power is 1.

Other number systems

Well, okay, using 10 as the basis (or base) of our counting system probably stems from those 10 human fingers, the original counting tools. An alterna­ tive base for a counting system could just as easily have been 20 (maybe the inventor of base 10 had shoes on at the time).

If our numbering scheme had been invented by dogs, it might well be based on 8 (one digit of each paw is out of sight on the back part of the leg). Mathemat­ ically, such an octal system would have worked just as well:

12310 = 1 * 82 + 7 * 81 + 3 * 80 = 1738

The small 10 and 8 here refer to the numbering system, 10 for decimal (base 10) and 8 for octal (base 8). A counting system may use any positive base.

The binary number system

Computers have essentially two fingers. (Maybe that’s why computers are so stupid: without an opposing thumb, they can’t grasp anything. And then again, maybe not.) Computers prefer counting using base 2. The number 12310 would be expressed this way:

Chapter 4: Performing Logical Operations

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12310 = 0*128 + 1*64 + 1*32 + 1*16 + 1*8 + 0*4 +1*2 + 1*1 = 011110112

Computer convention expresses binary numbers by using 4, 8, 16, 32 or even 64 binary digits even if the leading digits are zero. This is also because of the way computers are built internally.

Because the term digit refers to a multiple of ten, a binary digit is called a bit (an abbreviation of binary digit). Eight bits make up a byte. (Calling a binary digit a byte-it didn’t seem like a good idea.) A short word is two bytes; a long word is four bytes.

With such a small base, you have to use a large number of bits to express numbers. Human beings don’t want the hassle of using an expression such as 011110112 to express such a mundane value as 12310. Programmers prefer to express numbers by using an even number of bits. The octal system — which is based on 3 bits — has been almost completely replaced by the hexadeci­ mal system, which is based on 4-bit digits.

Hexadecimal uses the same digits for the numbers 0 through 9. For the digits between 9 and 16, hexadecimal uses the first six letters of the alphabet: A for 10, B for 11, etc. Thus, 12310 becomes 7B16, like this:

123 = 7 * 161 + B (i.e. 11) * 160 = 7B16

Programmers prefer to express hexadecimal numbers in 2, 4, or 8 hexadeci­ mal digits even when the leading digit in each case is 0.

Finally, who wants to express a hexadecimal number such as 7B16 by using a subscript? Terminals don’t even support subscripts. Even on a word proces­ sor such as the one I’m using now, it’s a drag to change fonts to and from sub­ script mode just to type two lousy digits. Therefore, programmers (no fools they) use the convention of beginning a hexadecimal number with a 0x. (Why? Well, the reason for such a strange convention goes back to the early days of C, in a galaxy far, far, away . . . never mind.) Thus, 7B becomes 0x7B. Using this convention, the hexadecimal number 0x7B is equal to 123 decimal while 0x123 hexadecimal is equal to 291 decimal.

You can use all the mathematical operators on hexadecimal numbers, in the same way you’d apply them to decimal numbers. (Well, okay, most of us can’t perform a multiplication such as 0xC * 0xE in our heads, but that has more to do with the multiplication tables we learned in school than it has to do with any limitation in the number system.)