 Network Administration: Understanding Binary - dummies

Before you can understand the details of how IP addressing works, you need to understand how the binary numbering system works because binary is the basis of IP addressing.

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 10 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
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. 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.

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 nerd-dom 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, the table below lists the powers of two up to 32.

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