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Hexadecimal as Shorthand for Binary
The number 218 expressed in binary is 11011010B. Expressed in hex, however, the same value is quite compact: DAH. The two hex digits comprising DAH merit a closer look. AH (or 0AH as your assembler will require it for reasons I explain later) represents 10 decimal. Converting any number to binary simply involves detecting the powers of two within it. The largest power of 2 within 10 decimal is 8. Jot down a 1 digit and subtract 8 from 10. What's left is 2. Now, 4 is a power of 2, but there is no 4 hiding within 2, so we put a 0 to the right of the 1. The next smallest power of 2 is 2, and there is a 2 in 2. Jot down another 1 to the right of the 0. Two from 2 is 0, so there are no 1s left in the number. Jot down a final 0 to the right of the rest to represent the 1s column. What you have is this:
1 0 1 0
Look back at the binary equivalent of 218: 11011010. The last four digits are 1010—the binary equivalent of 0AH.The same will work for the upper half of DAH. If you work out the binary equivalence for 0DH as we just did (and it would be good mental exercise), it is 1101. Look at the binary equivalent of 218 this way:
218 decimal
1101 1010 binary
D A hex
It should be dawning on you that you can convert long strings of binary 1s and 0s into more compact hex format by converting every four binary digits (starting from the right, not from the left!) into a single hex digit.As an example, here is a 32-bit binary number that is not the least bit remarkable:
11110000000000001111101001101110
This is a pretty obnoxious collection of bits to remember or manipulate, so let's split it up into groups of four from the right:
1111 0000 0000 0000 1111 1010 0110 1110
Each of these groups of four binary digits can be represented by a single hexadecimal digit. Do the conversion now. What you should get is the following:
1111 0000 0000 0000 1111 1010 0110 1110
F 0 0 0 F A 6 E
In other words, the hex equivalent of that mouthful is
F000FA6E
In use, of course, you would append the H on the end, and also put a 0 at the beginning, so in any kind of assembly language work the number would actually be written 0F000FA6EH.This is still a good-sized number, but unless you're doing things like counting hard drive space or other high-value things, such 32-bit numbers are the largest quantities you would typically encounter in journeyman-level assembly language programming.Suddenly, this business starts looking a little more graspable.Hexadecimal is the programmer's shorthand for the computer's binary numbers.
This is why I said earlier that computers use base 2 (binary) and base 16 (hexadecimal) both at the same time in a rather schizoid fashion. What I didn't say is that the computer isn't really the schizoid one; you are. At their very hearts (as I explain in Chapter 3) computers use only binary. Hex is a means by which you and I make dealing with the computer easier. Fortunately, every four binary digits may be represented by a hex digit, so the correspondence is clean and comprehensible.
Prepare to Compute
Everything up to this point has been necessary groundwork. I've explained conceptually what computers do and have given you the tools to understand the slightly alien numbers they use. But I've said nothing so far about what computers actually are, and it's well past time. We return to hexadecimal numbers again and again in this book; I've said nothing thus far about hex multiplication or bit-banging. The reason is plain: Before you can bang a bit, you must know where the bits live. So, let's lift the hood and see if we can catch a few in action.
