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Because computers don't natively understand the decimal system (or base 10 numbering), the first topic we need to cover is basic binary math Computers natively understand the binary system (or base 2 numbering) To use decimal numbering, they need to do a conversion For this reason, IP addresses are based on binary numbering Acquiring a basic understanding of binary numbering is easy First, let me show you the significance of placement Let's take a decimal number, say 762 Is this number two hundred sixty-seven, seven hundred sixty-two, or fifteen (7 + 6 + 2) Obviously, the answer is seven hundred sixty-two, but how did you know that By the placement of the numbers The seven was in the hundreds place, the six was in the tens place, and the two was in the ones place This is the same way binary works; the places just have different values In decimal, each place to the left is a power of ten greater than the one directly to the right of it (in other words,

inData 233 indenting code 124 index file 242 indexing tool 242 inetd 36 inheritance 175, 178 multiple 178 single 178 vs. delegation 196 init 222 init process 36 initializing data members 222 initToDefaults 222 Inkwell 281 installing programs from packages 37 installing programs from source code 37 instance creating 215 instance method 177 Integer 255 Integrated Development Environment (IDE) 58 CodeWarrior 60 MPW 59, 283 THINK C 59 THINK Pascal 59 interactive computing 326 interactive editor 112 interapplication communication (IAC) 318 Interface Builder 61, 81, 100, 133 AppleScript Studio 128 building program interface 210 connecting outlet to interface 105 creating interface objects 103 linking interface to code 104 Nib file 101 palette 81 palette window 105 testing interface 107 interface component adding 103 aligning 213 linking to code 104 interface definition 175 interface design 209

10 is ten times as much as 01); but in binary, each place to the left is a power of two greater This is probably best shown in Figure 6-1

FIGURE 8-8

Figure 6-1: Comparison of decimal and binary place values Binary also uses only two numbers for each place: one and zero This is because, in a numbering system based on powers of two, there can be only two values for each digit; so an example of a binary number is 100010100 Now, because most of us were probably taught to understand numbering using the decimal system, to understand binary, we will need to convert it to decimal This, again, isn't that difficult First, let's examine how to establish what the decimal representation of a number actually is In other words, how do we know 762 is seven hundred sixty-two This is fairly basic We know that anything in the hundreds place is that number multiplied by 100 This gives us 700 And anything in the tens place is multiplied by 10 This gives us 60 Finally, anything in the ones place is multiplied by 1 This gives us 2 If we add them all together (700 + 60 + 2), we have the full number, 762 Binary is the same way If we have the binary number 1100, we know, based on the place values, that anything in the eights place is multiplied by 8 (1 x 8 = 8), anything in the fours place is multiplied by 4 (1 x 4 = 4), anything in the twos place is multiplied by 2 (0 x 2 = 0), and anything in the ones place is multiplied by 1 (0 x 1 = 0) Now we add them all together (8 + 4 + 0 + 0), and we get 12 So 1100 in binary equals 12 in decimal

Here's another important point to keep in mind: any numbering scheme is infinite because numbers never stop Therefore, we need to know how to figure out the place values for any number This is also pretty easy In binary, every time you go one place to the left, you simply multiply by 2 This is shown in Figure 6-2

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Figure 6-2: Example of multiplying by 2 to find unknown place values Finally, let me show you some shortcuts to binary conversions First, if you are trying to figure out how many values (numbers) are in a given number of binary digits (bits), you can simply multiply the value of the most significant bit (MSB) the last one to the left by 2 (LSB stands for least significant bit, or "littlest one") This tells you how many individual numbers are represented by that number of bits An example is shown in Figure 6-3