id="ulink_12fb1b36-4cc4-5c6b-a143-d2413ed258dd">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.
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 |
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