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Binary is a counting system that uses only two numerals: 0 and 1. In the decimal system (with which most people are accustomed), you use ten numerals: 0–9. In an ordinary decimal number — such as 3,482 — the rightmost digit represents ones; the next digit to the left, tens; the next, hundreds; the next, thousands; and so on. These digits represent powers of ten: first 10⁰ (which is 1); next, 10¹ (10); then 10² (100); then 10³ (1,000); and so on.

In binary, you have only two numerals rather than ten, which is why binary numbers look somewhat monotonous, as in 110011, 101111, and 100001.

The positions in a binary number (called bits rather than digits) represent powers of two rather than powers of ten: 1, 2, 4, 8, 16, 32, and so on. To figure the decimal value of a binary number, you multiply each bit by its corresponding power of two and then add the results. The decimal value of binary 10111, for example, is calculated as follows:

 1 × 20 = 1 × 1 = 1

 1 × 21 = 1 × 2 = 2

 1 × 22 = 1 × 4 = 4

 0 × 23 = 0 × 8 = 0

 1 × 24 = 1 × 16 = 16

 Total = 1 + 2 + 4 + 0 + 16 = 23

Fortunately, converting a number between binary and decimal is something a computer is good at — so good, in fact, that you’re unlikely ever to need to do any conversions yourself. The point of learning binary is not to be able to look at a number such as 1110110110110 and say instantly, “Ah! Decimal 7,606!” (If you could do that, Piers Morgan would probably interview you, and they would even make a movie about you.)

Instead, the point is to have a basic understanding of how computers store information and — most important — to understand how the binary counting system works, which I describe in the following section.

Here are some of the more interesting characteristics of binary and how the system is similar to and differs from the decimal system:

 In decimal, the number of decimal places allotted for a number determines how large the number can be. If you allot six digits, for example, the largest number possible is 999,999. Because 0 is itself a number, however, a six-digit number can have any of 1 million different values.Similarly, the number of bits allotted for a binary number determines how large that number can be. If you allot eight bits, the largest value that number can store is 11111111, which happens to be 255 in decimal.

  To quickly figure how many different values you can store in a binary number of a given length, use the number of bits as an exponent of two. An eight-bit binary number, for example, can hold 28 values. Because 28 is 256, an eight-bit number can have any of 256 different values. This is why a byte — eight bits — can have 256 different values.

 This “powers of two” thing is why computers don’t use nice, even, round numbers in measuring such values as memory or disk space. A value of 1K, for example, is not an even 1,000 bytes: It’s actually 1,024 bytes because 1,024 is 210. Similarly, 1MB is not an even 1,000,000 bytes but instead 1,048,576 bytes, which happens to be 220.

One basic test of computer nerddom is knowing your powers of two because they play such an important role in binary numbers. Just for the fun of it, but not because you really need to know, Table 3-1 lists the powers of two up to 32.

TABLE 3-1 Powers of Two

Power	Bytes	Kilobytes	Power	Bytes	K, MB, or GB
21	2		217	131,072	128K
22	4		218	262,144	256K
23	8		219	524,288	512K
24	16		220	1,048,576	1MB
25	32		221	2,097,152	2MB
26	64		222	4,194,304	4MB
27	128		223	8,388,608	8MB
28	256		224	16,777,216	16MB
29	512		225	33,554,432	32MB
210	1,024	1K	226	67,108,864	64MB
211	2,048	2K	227	134,217,728	128MB
212	4,096	4K	228	268,435,456	256MB
213	8,192	8K	229	536,870,912	512MB
214	16,384	16K	230	1,073,741,824	1GB
215	32,768	32K	231	2,147,483,648	2GB
216	65,536	64K	232	4,294,967,296	4GB

Table 3-1 also shows the common shorthand notation for various powers of two. The abbreviation K represents 2¹⁰ (1,024). The M in MB stands for 2²⁰, or 1,024K, and the G in GB represents 2³⁰, which is 1,024MB. These shorthand notations don’t have anything to do with TCP/IP, but they’re commonly used for measuring computer disk and memory capacities, so I thought I’d throw them in at no charge because the table had extra room.