Home » CEEFAX disks » telesoftware3.adl » 23_10_87/T\OSB02
23_10_87/T\OSB02
This website contains an archive of files for the Acorn Electron, BBC Micro, Acorn Archimedes, Commodore 16 and Commodore 64 computers, which Dominic Ford has rescued from his private collection of floppy disks and cassettes.
Some of these files were originally commercial releases in the 1980s and 1990s, but they are now widely available online. I assume that copyright over them is no longer being asserted. If you own the copyright and would like files to be removed, please contact me.
Tape/disk: | Home » CEEFAX disks » telesoftware3.adl |
Filename: | 23_10_87/T\OSB02 |
Read OK: | ✔ |
File size: | 1ED6 bytes |
Load address: | 0000 |
Exec address: | 0000 |
Duplicates
There is 1 duplicate copy of this file in the archive:
- CEEFAX disks » telesoftware3.adl » 01_11_87/T\OSB02
- CEEFAX disks » telesoftware3.adl » 23_10_87/T\OSB02
File contents
OSBITS - An Exploration of the BBC Micro at Machine Level by Programmer ............................................................ Part 2: Counting Two By Two Computers count in twos. Just like we have ten fingers and count in tens an electrical circuit in a computer is either on (called set) or off (called clear). An individual circuit represents a binary digit, called a bit, and the 6502 microprocessor inside the BBC Micro works with numbers made up of eight bits, each called a byte. These eight bits of a byte give us a method of representing a number by building it from powers of 2 .... like this Bit 7 6 5 4 3 2 1 0 Value 128 64 32 16 8 4 2 1 By having each bit of this byte either set (on) or clear (off) you can represent any number between 0 and 255 (which is 2^8 -1). This way the binary number 1001 equals 9 in decimal (2^3+2^0) and 101101 is equal to 45 (2^5+2^3+2^2+2^0). Several bytes can be strung together to make a word. For integers BBC BASIC has words that are 32 bits long. You might say that BBC BASIC calculates integers using 32 bit words while the 6502 microprocessor in it calculates in 8 bit words. There are microprocessors that work with 16 and even 32 bit words. (Hence all the talk of '16 bit technology' etc.) With a 6502 words are made by stringing bytes together with the so-called Least Significant Byte (LSB) in the lowest address and the Most significant Byte (MSB) in the highest. As an example, in memory it would look like this: Address Number stored in this byte &2173 13 &2172 5 &2171 126 &2170 99 So the value stored in the 32 bit (4 byte) word starting at address &2170 is 99 + 256*126 + 256*256*5 + 256*256*256*13. In BBC BASIC you would look at !&2170 to get this number. As you can see, just as each higher, and more significant, bit is 2 times the value of its neighbour so each more significant byte is 2^8 or 256 times the value of its neighbour. Two things to notice about this table. Firstly the lower memory addresses are at the bottom and secondly the addresses are given in hexadecimal (hex). The ampersand (&) tells us, and the computer, that the number is in hex. Both of these are conventions for the BBC Micro. Other computers may use other conventions. Hexadecimal is numbers with a base of 16 (decimal is base 10 and binary base 2). Hex gives us a neat shorthand for the values of bytes since one hex digit represents four bits. Four bits can hold any value between 0 and 1111 (binary) or 0 and 15 (decimal) which is between 0 and F in hex. A full byte, holding 255 in decimal is FF in hex. We use the numbers from 0 to 9 and the letters A to F as the 16 numbers in hex. Once we get past arithmetic you will find it increasingly useful to think in hex when writing assembler. That's how the numbers are written down, so what can we do with them? Addition is straightforward. Adding 1 to 0 or 0 to 1 produces 1 and adding 0 to 0 produces 0. Adding 1 to 1 produces 10 since 1 and 1 is 0 carry 1 and we add that carry to 0 to give the left hand 1. 10111010 11010111 --------- 110010001 Now our two numbers, 186 and 215, both fit in a byte because they are less than 256. But 186+215 does not fit and, as you can see, it has overflowed to the right. If our calculations only used one byte that overflow would be lost and 186+215 would produce only 145. This idea of carrying over from a byte addition and finally overflowing your word is very important. There is an additional complication caused by our need for negative numbers. The computer doesn't understand a negative number, a number is either there or it isn't, so we have to invent a convention. This uses the top bit of your word to indicate whether the number is negative or not. If we apply this to a byte you get this result: +ve numbers 0 to 127 are &00 to &7F (00000000 to 01111111) -ve numbers -1 to -128 are &FF to &80 (11111111 to 10000000) So negative numbers have their top bit set. Note that zero is positive. There is method in this seeming madness since addition automatically takes account of the signs of the numbers being added, and the same happens with subtraction. In fact the overflow out of the word helps with subtraction. Let's add -1 and -1, which gives -2: 11111111 &FF 11111111 &FF -------- --- 11111110 &FE There is a way of checking that you don't accidentally turn a byte negative by adding together two positive numbers both over 64, that is something called the overflow flag. A carry, which you can think of as being the '9th bit' of a byte, sets another flag called the carry flag. (More on flags in another module.) The relationship between a number and its negative is that each bit of the number is reversed (or inverted, 0 becomes 1 and 1 becomes 0) and then one is added to it. This means that 1 (00000001) becomes 11111110 + 1 = 11111111. When you invert the bits of a number you are said to be taking its complement and when you add one you get its 2's complement. Subtraction is carried out similarly to addition. If you take 0 from 0 you get 0, 0 from 1 is 1 and 1 from 1 is 0. To take 1 from 0 you borrow from the next highest bit and so take 1 from 10, which is 1. 10100101 - 01110110 -------- 00101111 It's more difficult to get the hang of binary subtraction than addition but, in the end, it doesn't really matter. Apart, that is, from the idea of borrowing since it is this that enables you to subtract numbers larger than a single byte. Carrying does this job in addition and in both cases the carry flag is used. To add two bytes (stored in the addresses labelled byte_1 and byte_2) and store the result at address byte_3 you do this: CLear the Carry flag CLC LoaD A with the first number LDA byte_1 ADd the second (with Carry) ADC byte_2 STore the result (from A) STA byte_3 A, the accumulator, is the part of the microprocessor where you do your arithmetic. ADC adds a number to the value in the accumulator and adds the value of the carry flag to that. The result is in the accumulator. To subtract 2 bytes you have to start by setting the carry flag: SEt Carry SEC LoaD A with the first number LDA byte_1 SuBtract the second (with Carry) SBC byte_2 STore the result (from A) STA byte_3 In both cases, if you are adding or subtracting numbers made up of many bytes, you only explicitly clear or set the carry at the beginning. After that it serves its proper purpose and carries across or borrows across if needed. Look at this weeks assembler program B/osb02 which adds and subtracts two four byte numbers. I have used bits of BASIC to enable you to get the numbers into and out of the machine but eventually we will use machine code to do this as well. We could extend that code to work with numbers of any size but BASIC's integers are only 4 bytes in size so we will stick to 4 bytes for the moment. Note that B/osb02 will give an incorrect result if you add together numbers such that their sum is greater than 2147483647 (and similarly for negative numbers). This is because the overflow between the top two bits is not handled correctly. BASIC would trap this as a Too Big error but this machine code does not. To do so uses another of the microprocessor's flags called the overflow flag. We'll look at the flags in more detail next week and use them to get the computer to make some decisions.
00000000 4f 53 42 49 54 53 20 2d 20 41 6e 20 45 78 70 6c |OSBITS - An Expl| 00000010 6f 72 61 74 69 6f 6e 20 6f 66 20 74 68 65 20 42 |oration of the B| 00000020 42 43 20 4d 69 63 72 6f 20 61 74 20 4d 61 63 68 |BC Micro at Mach| 00000030 69 6e 65 20 4c 65 76 65 6c 0d 0d 62 79 20 50 72 |ine Level..by Pr| 00000040 6f 67 72 61 6d 6d 65 72 0d 0d 2e 2e 2e 2e 2e 2e |ogrammer........| 00000050 2e 2e 2e 2e 2e 2e 2e 2e 2e 2e 2e 2e 2e 2e 2e 2e |................| * 00000080 2e 2e 2e 2e 2e 2e 0d 0d 0d 50 61 72 74 20 32 3a |.........Part 2:| 00000090 20 20 43 6f 75 6e 74 69 6e 67 20 54 77 6f 20 42 | Counting Two B| 000000a0 79 20 54 77 6f 0d 0d 0d 43 6f 6d 70 75 74 65 72 |y Two...Computer| 000000b0 73 20 63 6f 75 6e 74 20 69 6e 20 74 77 6f 73 2e |s count in twos.| 000000c0 20 20 4a 75 73 74 20 6c 69 6b 65 20 77 65 20 68 | Just like we h| 000000d0 61 76 65 20 74 65 6e 20 66 69 6e 67 65 72 73 20 |ave ten fingers | 000000e0 61 6e 64 0d 63 6f 75 6e 74 20 69 6e 20 74 65 6e |and.count in ten| 000000f0 73 20 61 6e 20 65 6c 65 63 74 72 69 63 61 6c 20 |s an electrical | 00000100 63 69 72 63 75 69 74 20 69 6e 20 61 20 63 6f 6d |circuit in a com| 00000110 70 75 74 65 72 20 69 73 20 65 69 74 68 65 72 0d |puter is either.| 00000120 6f 6e 20 28 63 61 6c 6c 65 64 20 73 65 74 29 20 |on (called set) | 00000130 6f 72 20 6f 66 66 20 28 63 61 6c 6c 65 64 20 63 |or off (called c| 00000140 6c 65 61 72 29 2e 20 20 41 6e 20 69 6e 64 69 76 |lear). An indiv| 00000150 69 64 75 61 6c 0d 63 69 72 63 75 69 74 20 72 65 |idual.circuit re| 00000160 70 72 65 73 65 6e 74 73 20 61 20 62 69 6e 61 72 |presents a binar| 00000170 79 20 64 69 67 69 74 2c 20 63 61 6c 6c 65 64 20 |y digit, called | 00000180 61 20 62 69 74 2c 20 61 6e 64 20 74 68 65 0d 36 |a bit, and the.6| 00000190 35 30 32 20 6d 69 63 72 6f 70 72 6f 63 65 73 73 |502 microprocess| 000001a0 6f 72 20 69 6e 73 69 64 65 20 74 68 65 20 42 42 |or inside the BB| 000001b0 43 20 4d 69 63 72 6f 20 77 6f 72 6b 73 20 77 69 |C Micro works wi| 000001c0 74 68 20 6e 75 6d 62 65 72 73 0d 6d 61 64 65 20 |th numbers.made | 000001d0 75 70 20 6f 66 20 65 69 67 68 74 20 62 69 74 73 |up of eight bits| 000001e0 2c 20 65 61 63 68 20 63 61 6c 6c 65 64 20 61 20 |, each called a | 000001f0 62 79 74 65 2e 0d 0d 54 68 65 73 65 20 65 69 67 |byte...These eig| 00000200 68 74 20 62 69 74 73 20 6f 66 20 61 20 62 79 74 |ht bits of a byt| 00000210 65 20 67 69 76 65 20 75 73 20 61 20 6d 65 74 68 |e give us a meth| 00000220 6f 64 20 6f 66 20 72 65 70 72 65 73 65 6e 74 69 |od of representi| 00000230 6e 67 0d 61 20 6e 75 6d 62 65 72 20 62 79 20 62 |ng.a number by b| 00000240 75 69 6c 64 69 6e 67 20 69 74 20 66 72 6f 6d 20 |uilding it from | 00000250 70 6f 77 65 72 73 20 6f 66 20 32 20 2e 2e 2e 2e |powers of 2 ....| 00000260 20 6c 69 6b 65 20 74 68 69 73 0d 0d 42 69 74 20 | like this..Bit | 00000270 20 20 20 20 20 37 20 20 20 20 20 20 36 20 20 20 | 7 6 | 00000280 20 20 20 35 20 20 20 20 20 20 34 20 20 20 20 20 | 5 4 | 00000290 20 33 20 20 20 20 20 20 32 20 20 20 20 20 20 31 | 3 2 1| 000002a0 20 20 20 20 20 20 30 0d 56 61 6c 75 65 20 20 31 | 0.Value 1| 000002b0 32 38 20 20 20 20 20 36 34 20 20 20 20 20 33 32 |28 64 32| 000002c0 20 20 20 20 20 31 36 20 20 20 20 20 20 38 20 20 | 16 8 | 000002d0 20 20 20 20 34 20 20 20 20 20 20 32 20 20 20 20 | 4 2 | 000002e0 20 20 31 0d 0d 42 79 20 68 61 76 69 6e 67 20 65 | 1..By having e| 000002f0 61 63 68 20 62 69 74 20 6f 66 20 74 68 69 73 20 |ach bit of this | 00000300 62 79 74 65 20 65 69 74 68 65 72 20 73 65 74 20 |byte either set | 00000310 28 6f 6e 29 20 6f 72 20 63 6c 65 61 72 0d 28 6f |(on) or clear.(o| 00000320 66 66 29 20 79 6f 75 20 63 61 6e 20 72 65 70 72 |ff) you can repr| 00000330 65 73 65 6e 74 20 61 6e 79 20 6e 75 6d 62 65 72 |esent any number| 00000340 20 62 65 74 77 65 65 6e 20 30 20 61 6e 64 20 32 | between 0 and 2| 00000350 35 35 20 28 77 68 69 63 68 0d 69 73 20 32 5e 38 |55 (which.is 2^8| 00000360 20 2d 31 29 2e 0d 0d 54 68 69 73 20 77 61 79 20 | -1)...This way | 00000370 74 68 65 20 62 69 6e 61 72 79 20 6e 75 6d 62 65 |the binary numbe| 00000380 72 20 31 30 30 31 20 65 71 75 61 6c 73 20 39 20 |r 1001 equals 9 | 00000390 69 6e 20 64 65 63 69 6d 61 6c 0d 28 32 5e 33 2b |in decimal.(2^3+| 000003a0 32 5e 30 29 20 61 6e 64 20 31 30 31 31 30 31 20 |2^0) and 101101 | 000003b0 69 73 20 65 71 75 61 6c 20 74 6f 20 34 35 20 28 |is equal to 45 (| 000003c0 32 5e 35 2b 32 5e 33 2b 32 5e 32 2b 32 5e 30 29 |2^5+2^3+2^2+2^0)| 000003d0 2e 0d 0d 53 65 76 65 72 61 6c 20 62 79 74 65 73 |...Several bytes| 000003e0 20 63 61 6e 20 62 65 20 73 74 72 75 6e 67 20 74 | can be strung t| 000003f0 6f 67 65 74 68 65 72 20 74 6f 20 6d 61 6b 65 20 |ogether to make | 00000400 61 20 77 6f 72 64 2e 20 20 46 6f 72 0d 69 6e 74 |a word. For.int| 00000410 65 67 65 72 73 20 42 42 43 20 42 41 53 49 43 20 |egers BBC BASIC | 00000420 68 61 73 20 77 6f 72 64 73 20 74 68 61 74 20 61 |has words that a| 00000430 72 65 20 33 32 20 62 69 74 73 20 6c 6f 6e 67 2e |re 32 bits long.| 00000440 20 20 59 6f 75 0d 6d 69 67 68 74 20 73 61 79 20 | You.might say | 00000450 74 68 61 74 20 42 42 43 20 42 41 53 49 43 20 63 |that BBC BASIC c| 00000460 61 6c 63 75 6c 61 74 65 73 20 69 6e 74 65 67 65 |alculates intege| 00000470 72 73 20 75 73 69 6e 67 20 33 32 20 62 69 74 0d |rs using 32 bit.| 00000480 77 6f 72 64 73 20 77 68 69 6c 65 20 74 68 65 20 |words while the | 00000490 36 35 30 32 20 6d 69 63 72 6f 70 72 6f 63 65 73 |6502 microproces| 000004a0 73 6f 72 20 69 6e 20 69 74 20 63 61 6c 63 75 6c |sor in it calcul| 000004b0 61 74 65 73 20 69 6e 20 38 0d 62 69 74 20 77 6f |ates in 8.bit wo| 000004c0 72 64 73 2e 20 20 54 68 65 72 65 20 61 72 65 20 |rds. There are | 000004d0 6d 69 63 72 6f 70 72 6f 63 65 73 73 6f 72 73 20 |microprocessors | 000004e0 74 68 61 74 20 77 6f 72 6b 20 77 69 74 68 20 31 |that work with 1| 000004f0 36 20 61 6e 64 0d 65 76 65 6e 20 33 32 20 62 69 |6 and.even 32 bi| 00000500 74 20 77 6f 72 64 73 2e 20 20 28 48 65 6e 63 65 |t words. (Hence| 00000510 20 61 6c 6c 20 74 68 65 20 74 61 6c 6b 20 6f 66 | all the talk of| 00000520 20 27 31 36 20 62 69 74 0d 74 65 63 68 6e 6f 6c | '16 bit.technol| 00000530 6f 67 79 27 20 65 74 63 2e 29 0d 0d 57 69 74 68 |ogy' etc.)..With| 00000540 20 61 20 36 35 30 32 20 77 6f 72 64 73 20 61 72 | a 6502 words ar| 00000550 65 20 6d 61 64 65 20 62 79 20 73 74 72 69 6e 67 |e made by string| 00000560 69 6e 67 20 62 79 74 65 73 20 74 6f 67 65 74 68 |ing bytes togeth| 00000570 65 72 20 77 69 74 68 0d 74 68 65 20 73 6f 2d 63 |er with.the so-c| 00000580 61 6c 6c 65 64 20 4c 65 61 73 74 20 53 69 67 6e |alled Least Sign| 00000590 69 66 69 63 61 6e 74 20 42 79 74 65 20 28 4c 53 |ificant Byte (LS| 000005a0 42 29 20 69 6e 20 74 68 65 20 6c 6f 77 65 73 74 |B) in the lowest| 000005b0 0d 61 64 64 72 65 73 73 20 61 6e 64 20 74 68 65 |.address and the| 000005c0 20 4d 6f 73 74 20 73 69 67 6e 69 66 69 63 61 6e | Most significan| 000005d0 74 20 42 79 74 65 20 28 4d 53 42 29 20 69 6e 20 |t Byte (MSB) in | 000005e0 74 68 65 20 68 69 67 68 65 73 74 2e 20 0d 41 73 |the highest. .As| 000005f0 20 61 6e 20 65 78 61 6d 70 6c 65 2c 20 69 6e 20 | an example, in | 00000600 6d 65 6d 6f 72 79 20 69 74 20 77 6f 75 6c 64 20 |memory it would | 00000610 6c 6f 6f 6b 20 6c 69 6b 65 20 74 68 69 73 3a 0d |look like this:.| 00000620 0d 20 20 20 20 20 20 41 64 64 72 65 73 73 20 20 |. Address | 00000630 20 20 20 20 20 20 20 20 20 4e 75 6d 62 65 72 20 | Number | 00000640 73 74 6f 72 65 64 20 69 6e 20 74 68 69 73 20 62 |stored in this b| 00000650 79 74 65 0d 20 20 20 20 20 20 20 26 32 31 37 33 |yte. &2173| 00000660 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 | | 00000670 20 20 20 20 20 31 33 0d 20 20 20 20 20 20 20 26 | 13. &| 00000680 32 31 37 32 20 20 20 20 20 20 20 20 20 20 20 20 |2172 | 00000690 20 20 20 20 20 20 20 20 20 20 35 0d 20 20 20 20 | 5. | 000006a0 20 20 20 26 32 31 37 31 20 20 20 20 20 20 20 20 | &2171 | 000006b0 20 20 20 20 20 20 20 20 20 20 20 20 31 32 36 0d | 126.| 000006c0 20 20 20 20 20 20 20 26 32 31 37 30 20 20 20 20 | &2170 | 000006d0 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 | | 000006e0 20 39 39 0d 0d 53 6f 20 74 68 65 20 76 61 6c 75 | 99..So the valu| 000006f0 65 20 73 74 6f 72 65 64 20 69 6e 20 74 68 65 20 |e stored in the | 00000700 33 32 20 62 69 74 20 28 34 20 62 79 74 65 29 20 |32 bit (4 byte) | 00000710 77 6f 72 64 20 73 74 61 72 74 69 6e 67 20 61 74 |word starting at| 00000720 0d 61 64 64 72 65 73 73 20 26 32 31 37 30 20 69 |.address &2170 i| 00000730 73 20 39 39 20 2b 20 32 35 36 2a 31 32 36 20 2b |s 99 + 256*126 +| 00000740 20 32 35 36 2a 32 35 36 2a 35 20 2b 20 32 35 36 | 256*256*5 + 256| 00000750 2a 32 35 36 2a 32 35 36 2a 31 33 2e 20 0d 49 6e |*256*256*13. .In| 00000760 20 42 42 43 20 42 41 53 49 43 20 79 6f 75 20 77 | BBC BASIC you w| 00000770 6f 75 6c 64 20 6c 6f 6f 6b 20 61 74 20 21 26 32 |ould look at !&2| 00000780 31 37 30 20 74 6f 20 67 65 74 20 74 68 69 73 20 |170 to get this | 00000790 6e 75 6d 62 65 72 2e 20 0d 41 73 20 79 6f 75 20 |number. .As you | 000007a0 63 61 6e 20 73 65 65 2c 20 6a 75 73 74 20 61 73 |can see, just as| 000007b0 20 65 61 63 68 20 68 69 67 68 65 72 2c 20 61 6e | each higher, an| 000007c0 64 20 6d 6f 72 65 20 73 69 67 6e 69 66 69 63 61 |d more significa| 000007d0 6e 74 2c 0d 62 69 74 20 69 73 20 32 20 74 69 6d |nt,.bit is 2 tim| 000007e0 65 73 20 74 68 65 20 76 61 6c 75 65 20 6f 66 20 |es the value of | 000007f0 69 74 73 20 6e 65 69 67 68 62 6f 75 72 20 73 6f |its neighbour so| 00000800 20 65 61 63 68 20 6d 6f 72 65 0d 73 69 67 6e 69 | each more.signi| 00000810 66 69 63 61 6e 74 20 62 79 74 65 20 69 73 20 32 |ficant byte is 2| 00000820 5e 38 20 6f 72 20 32 35 36 20 74 69 6d 65 73 20 |^8 or 256 times | 00000830 74 68 65 20 76 61 6c 75 65 20 6f 66 20 69 74 73 |the value of its| 00000840 0d 6e 65 69 67 68 62 6f 75 72 2e 0d 0d 54 77 6f |.neighbour...Two| 00000850 20 74 68 69 6e 67 73 20 74 6f 20 6e 6f 74 69 63 | things to notic| 00000860 65 20 61 62 6f 75 74 20 74 68 69 73 20 74 61 62 |e about this tab| 00000870 6c 65 2e 20 20 46 69 72 73 74 6c 79 20 74 68 65 |le. Firstly the| 00000880 20 6c 6f 77 65 72 0d 6d 65 6d 6f 72 79 20 61 64 | lower.memory ad| 00000890 64 72 65 73 73 65 73 20 61 72 65 20 61 74 20 74 |dresses are at t| 000008a0 68 65 20 62 6f 74 74 6f 6d 20 61 6e 64 20 73 65 |he bottom and se| 000008b0 63 6f 6e 64 6c 79 20 74 68 65 0d 61 64 64 72 65 |condly the.addre| 000008c0 73 73 65 73 20 61 72 65 20 67 69 76 65 6e 20 69 |sses are given i| 000008d0 6e 20 68 65 78 61 64 65 63 69 6d 61 6c 20 28 68 |n hexadecimal (h| 000008e0 65 78 29 2e 20 20 54 68 65 20 61 6d 70 65 72 73 |ex). The ampers| 000008f0 61 6e 64 20 28 26 29 0d 74 65 6c 6c 73 20 75 73 |and (&).tells us| 00000900 2c 20 61 6e 64 20 74 68 65 20 63 6f 6d 70 75 74 |, and the comput| 00000910 65 72 2c 20 74 68 61 74 20 74 68 65 20 6e 75 6d |er, that the num| 00000920 62 65 72 20 69 73 20 69 6e 20 68 65 78 2e 20 20 |ber is in hex. | 00000930 42 6f 74 68 0d 6f 66 20 74 68 65 73 65 20 61 72 |Both.of these ar| 00000940 65 20 63 6f 6e 76 65 6e 74 69 6f 6e 73 20 66 6f |e conventions fo| 00000950 72 20 74 68 65 20 42 42 43 20 4d 69 63 72 6f 2e |r the BBC Micro.| 00000960 20 4f 74 68 65 72 20 63 6f 6d 70 75 74 65 72 73 | Other computers| 00000970 0d 6d 61 79 20 75 73 65 20 6f 74 68 65 72 20 63 |.may use other c| 00000980 6f 6e 76 65 6e 74 69 6f 6e 73 2e 0d 0d 48 65 78 |onventions...Hex| 00000990 61 64 65 63 69 6d 61 6c 20 69 73 20 6e 75 6d 62 |adecimal is numb| 000009a0 65 72 73 20 77 69 74 68 20 61 20 62 61 73 65 20 |ers with a base | 000009b0 6f 66 20 31 36 20 28 64 65 63 69 6d 61 6c 20 69 |of 16 (decimal i| 000009c0 73 20 62 61 73 65 20 31 30 0d 61 6e 64 20 62 69 |s base 10.and bi| 000009d0 6e 61 72 79 20 62 61 73 65 20 32 29 2e 20 20 48 |nary base 2). H| 000009e0 65 78 20 67 69 76 65 73 20 75 73 20 61 20 6e 65 |ex gives us a ne| 000009f0 61 74 20 73 68 6f 72 74 68 61 6e 64 20 66 6f 72 |at shorthand for| 00000a00 20 74 68 65 0d 76 61 6c 75 65 73 20 6f 66 20 62 | the.values of b| 00000a10 79 74 65 73 20 73 69 6e 63 65 20 6f 6e 65 20 68 |ytes since one h| 00000a20 65 78 20 64 69 67 69 74 20 72 65 70 72 65 73 65 |ex digit represe| 00000a30 6e 74 73 20 66 6f 75 72 20 62 69 74 73 2e 0d 46 |nts four bits..F| 00000a40 6f 75 72 20 62 69 74 73 20 63 61 6e 20 68 6f 6c |our bits can hol| 00000a50 64 20 61 6e 79 20 76 61 6c 75 65 20 62 65 74 77 |d any value betw| 00000a60 65 65 6e 20 30 20 61 6e 64 20 31 31 31 31 20 28 |een 0 and 1111 (| 00000a70 62 69 6e 61 72 79 29 20 6f 72 0d 30 20 61 6e 64 |binary) or.0 and| 00000a80 20 31 35 20 28 64 65 63 69 6d 61 6c 29 20 77 68 | 15 (decimal) wh| 00000a90 69 63 68 20 69 73 20 62 65 74 77 65 65 6e 20 30 |ich is between 0| 00000aa0 20 61 6e 64 20 46 20 69 6e 20 68 65 78 2e 20 20 | and F in hex. | 00000ab0 41 20 66 75 6c 6c 0d 62 79 74 65 2c 20 68 6f 6c |A full.byte, hol| 00000ac0 64 69 6e 67 20 32 35 35 20 69 6e 20 64 65 63 69 |ding 255 in deci| 00000ad0 6d 61 6c 20 69 73 20 46 46 20 69 6e 20 68 65 78 |mal is FF in hex| 00000ae0 2e 20 20 57 65 20 75 73 65 20 74 68 65 0d 6e 75 |. We use the.nu| 00000af0 6d 62 65 72 73 20 66 72 6f 6d 20 30 20 74 6f 20 |mbers from 0 to | 00000b00 39 20 61 6e 64 20 74 68 65 20 6c 65 74 74 65 72 |9 and the letter| 00000b10 73 20 41 20 74 6f 20 46 20 61 73 20 74 68 65 20 |s A to F as the | 00000b20 31 36 20 6e 75 6d 62 65 72 73 0d 69 6e 20 68 65 |16 numbers.in he| 00000b30 78 2e 0d 0d 4f 6e 63 65 20 77 65 20 67 65 74 20 |x...Once we get | 00000b40 70 61 73 74 20 61 72 69 74 68 6d 65 74 69 63 20 |past arithmetic | 00000b50 79 6f 75 20 77 69 6c 6c 20 66 69 6e 64 20 69 74 |you will find it| 00000b60 20 69 6e 63 72 65 61 73 69 6e 67 6c 79 0d 75 73 | increasingly.us| 00000b70 65 66 75 6c 20 74 6f 20 74 68 69 6e 6b 20 69 6e |eful to think in| 00000b80 20 68 65 78 20 77 68 65 6e 20 77 72 69 74 69 6e | hex when writin| 00000b90 67 20 61 73 73 65 6d 62 6c 65 72 2e 0d 0d 54 68 |g assembler...Th| 00000ba0 61 74 27 73 20 68 6f 77 20 74 68 65 20 6e 75 6d |at's how the num| 00000bb0 62 65 72 73 20 61 72 65 20 77 72 69 74 74 65 6e |bers are written| 00000bc0 20 64 6f 77 6e 2c 20 73 6f 20 77 68 61 74 20 63 | down, so what c| 00000bd0 61 6e 20 77 65 20 64 6f 0d 77 69 74 68 20 74 68 |an we do.with th| 00000be0 65 6d 3f 0d 0d 41 64 64 69 74 69 6f 6e 20 69 73 |em?..Addition is| 00000bf0 20 73 74 72 61 69 67 68 74 66 6f 72 77 61 72 64 | straightforward| 00000c00 2e 20 20 41 64 64 69 6e 67 20 31 20 74 6f 20 30 |. Adding 1 to 0| 00000c10 20 6f 72 20 30 20 74 6f 20 31 0d 70 72 6f 64 75 | or 0 to 1.produ| 00000c20 63 65 73 20 31 20 61 6e 64 20 61 64 64 69 6e 67 |ces 1 and adding| 00000c30 20 30 20 74 6f 20 30 20 70 72 6f 64 75 63 65 73 | 0 to 0 produces| 00000c40 20 30 2e 20 20 41 64 64 69 6e 67 20 31 20 74 6f | 0. Adding 1 to| 00000c50 20 31 0d 70 72 6f 64 75 63 65 73 20 31 30 20 73 | 1.produces 10 s| 00000c60 69 6e 63 65 20 31 20 61 6e 64 20 31 20 69 73 20 |ince 1 and 1 is | 00000c70 30 20 63 61 72 72 79 20 31 20 61 6e 64 20 77 65 |0 carry 1 and we| 00000c80 20 61 64 64 20 74 68 61 74 20 63 61 72 72 79 0d | add that carry.| 00000c90 74 6f 20 30 20 74 6f 20 67 69 76 65 20 74 68 65 |to 0 to give the| 00000ca0 20 6c 65 66 74 20 68 61 6e 64 20 31 2e 0d 0d 20 | left hand 1... | 00000cb0 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 | | 00000cc0 20 20 20 31 30 31 31 31 30 31 30 0d 20 20 20 20 | 10111010. | 00000cd0 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 | | 00000ce0 31 31 30 31 30 31 31 31 0d 20 20 20 20 20 20 20 |11010111. | 00000cf0 20 20 20 20 20 20 20 20 20 20 20 20 2d 2d 2d 2d | ----| 00000d00 2d 2d 2d 2d 2d 0d 20 20 20 20 20 20 20 20 20 20 |-----. | 00000d10 20 20 20 20 20 20 20 20 20 31 31 30 30 31 30 30 | 1100100| 00000d20 30 31 0d 0d 4e 6f 77 20 6f 75 72 20 74 77 6f 20 |01..Now our two | 00000d30 6e 75 6d 62 65 72 73 2c 20 31 38 36 20 61 6e 64 |numbers, 186 and| 00000d40 20 32 31 35 2c 20 62 6f 74 68 20 66 69 74 20 69 | 215, both fit i| 00000d50 6e 20 61 20 62 79 74 65 20 62 65 63 61 75 73 65 |n a byte because| 00000d60 0d 74 68 65 79 20 61 72 65 20 6c 65 73 73 20 74 |.they are less t| 00000d70 68 61 6e 20 32 35 36 2e 20 20 42 75 74 20 31 38 |han 256. But 18| 00000d80 36 2b 32 31 35 20 64 6f 65 73 20 6e 6f 74 20 66 |6+215 does not f| 00000d90 69 74 20 61 6e 64 2c 20 61 73 0d 79 6f 75 20 63 |it and, as.you c| 00000da0 61 6e 20 73 65 65 2c 20 69 74 20 68 61 73 20 6f |an see, it has o| 00000db0 76 65 72 66 6c 6f 77 65 64 20 74 6f 20 74 68 65 |verflowed to the| 00000dc0 20 72 69 67 68 74 2e 20 20 49 66 20 6f 75 72 0d | right. If our.| 00000dd0 63 61 6c 63 75 6c 61 74 69 6f 6e 73 20 6f 6e 6c |calculations onl| 00000de0 79 20 75 73 65 64 20 6f 6e 65 20 62 79 74 65 20 |y used one byte | 00000df0 74 68 61 74 20 6f 76 65 72 66 6c 6f 77 20 77 6f |that overflow wo| 00000e00 75 6c 64 20 62 65 20 6c 6f 73 74 0d 61 6e 64 20 |uld be lost.and | 00000e10 31 38 36 2b 32 31 35 20 77 6f 75 6c 64 20 70 72 |186+215 would pr| 00000e20 6f 64 75 63 65 20 6f 6e 6c 79 20 31 34 35 2e 20 |oduce only 145. | 00000e30 20 54 68 69 73 20 69 64 65 61 20 6f 66 20 63 61 | This idea of ca| 00000e40 72 72 79 69 6e 67 0d 6f 76 65 72 20 66 72 6f 6d |rrying.over from| 00000e50 20 61 20 62 79 74 65 20 61 64 64 69 74 69 6f 6e | a byte addition| 00000e60 20 61 6e 64 20 66 69 6e 61 6c 6c 79 20 6f 76 65 | and finally ove| 00000e70 72 66 6c 6f 77 69 6e 67 20 79 6f 75 72 20 77 6f |rflowing your wo| 00000e80 72 64 0d 69 73 20 76 65 72 79 20 69 6d 70 6f 72 |rd.is very impor| 00000e90 74 61 6e 74 2e 0d 0d 54 68 65 72 65 20 69 73 20 |tant...There is | 00000ea0 61 6e 20 61 64 64 69 74 69 6f 6e 61 6c 20 63 6f |an additional co| 00000eb0 6d 70 6c 69 63 61 74 69 6f 6e 20 63 61 75 73 65 |mplication cause| 00000ec0 64 20 62 79 20 6f 75 72 20 6e 65 65 64 20 66 6f |d by our need fo| 00000ed0 72 0d 6e 65 67 61 74 69 76 65 20 6e 75 6d 62 65 |r.negative numbe| 00000ee0 72 73 2e 20 20 54 68 65 20 63 6f 6d 70 75 74 65 |rs. The compute| 00000ef0 72 20 64 6f 65 73 6e 27 74 20 75 6e 64 65 72 73 |r doesn't unders| 00000f00 74 61 6e 64 20 61 0d 6e 65 67 61 74 69 76 65 20 |tand a.negative | 00000f10 6e 75 6d 62 65 72 2c 20 61 20 6e 75 6d 62 65 72 |number, a number| 00000f20 20 69 73 20 65 69 74 68 65 72 20 74 68 65 72 65 | is either there| 00000f30 20 6f 72 20 69 74 20 69 73 6e 27 74 2c 20 73 6f | or it isn't, so| 00000f40 20 77 65 0d 68 61 76 65 20 74 6f 20 69 6e 76 65 | we.have to inve| 00000f50 6e 74 20 61 20 63 6f 6e 76 65 6e 74 69 6f 6e 2e |nt a convention.| 00000f60 20 20 54 68 69 73 20 75 73 65 73 20 74 68 65 20 | This uses the | 00000f70 74 6f 70 20 62 69 74 20 6f 66 20 79 6f 75 72 0d |top bit of your.| 00000f80 77 6f 72 64 20 74 6f 20 69 6e 64 69 63 61 74 65 |word to indicate| 00000f90 20 77 68 65 74 68 65 72 20 74 68 65 20 6e 75 6d | whether the num| 00000fa0 62 65 72 20 69 73 20 6e 65 67 61 74 69 76 65 20 |ber is negative | 00000fb0 6f 72 20 6e 6f 74 2e 20 20 49 66 0d 77 65 20 61 |or not. If.we a| 00000fc0 70 70 6c 79 20 74 68 69 73 20 74 6f 20 61 20 62 |pply this to a b| 00000fd0 79 74 65 20 79 6f 75 20 67 65 74 20 74 68 69 73 |yte you get this| 00000fe0 20 72 65 73 75 6c 74 3a 0d 0d 2b 76 65 20 6e 75 | result:..+ve nu| 00000ff0 6d 62 65 72 73 20 20 30 20 74 6f 20 20 31 32 37 |mbers 0 to 127| 00001000 20 61 72 65 20 26 30 30 20 74 6f 20 26 37 46 20 | are &00 to &7F | 00001010 28 30 30 30 30 30 30 30 30 20 74 6f 20 30 31 31 |(00000000 to 011| 00001020 31 31 31 31 31 29 0d 2d 76 65 20 6e 75 6d 62 65 |11111).-ve numbe| 00001030 72 73 20 2d 31 20 74 6f 20 2d 31 32 38 20 61 72 |rs -1 to -128 ar| 00001040 65 20 26 46 46 20 74 6f 20 26 38 30 20 28 31 31 |e &FF to &80 (11| 00001050 31 31 31 31 31 31 20 74 6f 20 31 30 30 30 30 30 |111111 to 100000| 00001060 30 30 29 0d 0d 53 6f 20 6e 65 67 61 74 69 76 65 |00)..So negative| 00001070 20 6e 75 6d 62 65 72 73 20 68 61 76 65 20 74 68 | numbers have th| 00001080 65 69 72 20 74 6f 70 20 62 69 74 20 73 65 74 2e |eir top bit set.| 00001090 20 20 4e 6f 74 65 20 74 68 61 74 0d 7a 65 72 6f | Note that.zero| 000010a0 20 69 73 20 70 6f 73 69 74 69 76 65 2e 20 20 54 | is positive. T| 000010b0 68 65 72 65 20 69 73 20 6d 65 74 68 6f 64 20 69 |here is method i| 000010c0 6e 20 74 68 69 73 20 73 65 65 6d 69 6e 67 20 6d |n this seeming m| 000010d0 61 64 6e 65 73 73 0d 73 69 6e 63 65 20 61 64 64 |adness.since add| 000010e0 69 74 69 6f 6e 20 61 75 74 6f 6d 61 74 69 63 61 |ition automatica| 000010f0 6c 6c 79 20 74 61 6b 65 73 20 61 63 63 6f 75 6e |lly takes accoun| 00001100 74 20 6f 66 20 74 68 65 20 73 69 67 6e 73 20 6f |t of the signs o| 00001110 66 0d 74 68 65 20 6e 75 6d 62 65 72 73 20 62 65 |f.the numbers be| 00001120 69 6e 67 20 61 64 64 65 64 2c 20 61 6e 64 20 74 |ing added, and t| 00001130 68 65 20 73 61 6d 65 20 68 61 70 70 65 6e 73 20 |he same happens | 00001140 77 69 74 68 0d 73 75 62 74 72 61 63 74 69 6f 6e |with.subtraction| 00001150 2e 20 20 49 6e 20 66 61 63 74 20 74 68 65 20 6f |. In fact the o| 00001160 76 65 72 66 6c 6f 77 20 6f 75 74 20 6f 66 20 74 |verflow out of t| 00001170 68 65 20 77 6f 72 64 20 68 65 6c 70 73 0d 77 69 |he word helps.wi| 00001180 74 68 20 73 75 62 74 72 61 63 74 69 6f 6e 2e 0d |th subtraction..| 00001190 0d 4c 65 74 27 73 20 61 64 64 20 2d 31 20 61 6e |.Let's add -1 an| 000011a0 64 20 2d 31 2c 20 77 68 69 63 68 20 67 69 76 65 |d -1, which give| 000011b0 73 20 2d 32 3a 0d 0d 20 20 20 20 20 20 20 20 20 |s -2:.. | 000011c0 20 20 20 20 20 20 20 20 20 20 20 20 31 31 31 31 | 1111| 000011d0 31 31 31 31 20 20 20 20 20 20 20 26 46 46 0d 20 |1111 &FF. | 000011e0 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 | | 000011f0 20 20 20 20 31 31 31 31 31 31 31 31 20 20 20 20 | 11111111 | 00001200 20 20 20 26 46 46 0d 20 20 20 20 20 20 20 20 20 | &FF. | 00001210 20 20 20 20 20 20 20 20 20 20 20 20 2d 2d 2d 2d | ----| 00001220 2d 2d 2d 2d 20 20 20 20 20 20 20 2d 2d 2d 0d 20 |---- ---. | 00001230 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 | | 00001240 20 20 20 20 31 31 31 31 31 31 31 30 20 20 20 20 | 11111110 | 00001250 20 20 20 26 46 45 0d 0d 54 68 65 72 65 20 69 73 | &FE..There is| 00001260 20 61 20 77 61 79 20 6f 66 20 63 68 65 63 6b 69 | a way of checki| 00001270 6e 67 20 74 68 61 74 20 79 6f 75 20 64 6f 6e 27 |ng that you don'| 00001280 74 20 61 63 63 69 64 65 6e 74 61 6c 6c 79 20 74 |t accidentally t| 00001290 75 72 6e 0d 61 20 62 79 74 65 20 6e 65 67 61 74 |urn.a byte negat| 000012a0 69 76 65 20 62 79 20 61 64 64 69 6e 67 20 74 6f |ive by adding to| 000012b0 67 65 74 68 65 72 20 74 77 6f 20 70 6f 73 69 74 |gether two posit| 000012c0 69 76 65 20 6e 75 6d 62 65 72 73 20 62 6f 74 68 |ive numbers both| 000012d0 0d 6f 76 65 72 20 36 34 2c 20 74 68 61 74 20 69 |.over 64, that i| 000012e0 73 20 73 6f 6d 65 74 68 69 6e 67 20 63 61 6c 6c |s something call| 000012f0 65 64 20 74 68 65 20 6f 76 65 72 66 6c 6f 77 20 |ed the overflow | 00001300 66 6c 61 67 2e 20 20 41 0d 63 61 72 72 79 2c 20 |flag. A.carry, | 00001310 77 68 69 63 68 20 79 6f 75 20 63 61 6e 20 74 68 |which you can th| 00001320 69 6e 6b 20 6f 66 20 61 73 20 62 65 69 6e 67 20 |ink of as being | 00001330 74 68 65 20 27 39 74 68 20 62 69 74 27 20 6f 66 |the '9th bit' of| 00001340 20 61 0d 62 79 74 65 2c 20 73 65 74 73 20 61 6e | a.byte, sets an| 00001350 6f 74 68 65 72 20 66 6c 61 67 20 63 61 6c 6c 65 |other flag calle| 00001360 64 20 74 68 65 20 63 61 72 72 79 20 66 6c 61 67 |d the carry flag| 00001370 2e 20 20 28 4d 6f 72 65 20 6f 6e 0d 66 6c 61 67 |. (More on.flag| 00001380 73 20 69 6e 20 61 6e 6f 74 68 65 72 20 6d 6f 64 |s in another mod| 00001390 75 6c 65 2e 29 0d 0d 54 68 65 20 72 65 6c 61 74 |ule.)..The relat| 000013a0 69 6f 6e 73 68 69 70 20 62 65 74 77 65 65 6e 20 |ionship between | 000013b0 61 20 6e 75 6d 62 65 72 20 61 6e 64 20 69 74 73 |a number and its| 000013c0 20 6e 65 67 61 74 69 76 65 20 69 73 20 74 68 61 | negative is tha| 000013d0 74 0d 65 61 63 68 20 62 69 74 20 6f 66 20 74 68 |t.each bit of th| 000013e0 65 20 6e 75 6d 62 65 72 20 69 73 20 72 65 76 65 |e number is reve| 000013f0 72 73 65 64 20 28 6f 72 20 69 6e 76 65 72 74 65 |rsed (or inverte| 00001400 64 2c 20 30 20 62 65 63 6f 6d 65 73 20 31 0d 61 |d, 0 becomes 1.a| 00001410 6e 64 20 31 20 62 65 63 6f 6d 65 73 20 30 29 20 |nd 1 becomes 0) | 00001420 61 6e 64 20 74 68 65 6e 20 6f 6e 65 20 69 73 20 |and then one is | 00001430 61 64 64 65 64 20 74 6f 20 69 74 2e 20 20 54 68 |added to it. Th| 00001440 69 73 20 6d 65 61 6e 73 0d 74 68 61 74 20 31 20 |is means.that 1 | 00001450 28 30 30 30 30 30 30 30 31 29 20 62 65 63 6f 6d |(00000001) becom| 00001460 65 73 20 31 31 31 31 31 31 31 30 20 2b 20 31 20 |es 11111110 + 1 | 00001470 3d 20 31 31 31 31 31 31 31 31 2e 20 20 57 68 65 |= 11111111. Whe| 00001480 6e 20 79 6f 75 0d 69 6e 76 65 72 74 20 74 68 65 |n you.invert the| 00001490 20 62 69 74 73 20 6f 66 20 61 20 6e 75 6d 62 65 | bits of a numbe| 000014a0 72 20 79 6f 75 20 61 72 65 20 73 61 69 64 20 74 |r you are said t| 000014b0 6f 20 62 65 20 74 61 6b 69 6e 67 20 69 74 73 0d |o be taking its.| 000014c0 63 6f 6d 70 6c 65 6d 65 6e 74 20 61 6e 64 20 77 |complement and w| 000014d0 68 65 6e 20 79 6f 75 20 61 64 64 20 6f 6e 65 20 |hen you add one | 000014e0 79 6f 75 20 67 65 74 20 69 74 73 20 32 27 73 20 |you get its 2's | 000014f0 63 6f 6d 70 6c 65 6d 65 6e 74 2e 0d 0d 53 75 62 |complement...Sub| 00001500 74 72 61 63 74 69 6f 6e 20 69 73 20 63 61 72 72 |traction is carr| 00001510 69 65 64 20 6f 75 74 20 73 69 6d 69 6c 61 72 6c |ied out similarl| 00001520 79 20 74 6f 20 61 64 64 69 74 69 6f 6e 2e 20 20 |y to addition. | 00001530 49 66 20 79 6f 75 0d 74 61 6b 65 20 30 20 66 72 |If you.take 0 fr| 00001540 6f 6d 20 30 20 79 6f 75 20 67 65 74 20 30 2c 20 |om 0 you get 0, | 00001550 30 20 66 72 6f 6d 20 31 20 69 73 20 31 20 61 6e |0 from 1 is 1 an| 00001560 64 20 31 20 66 72 6f 6d 20 31 20 69 73 20 30 2e |d 1 from 1 is 0.| 00001570 20 0d 54 6f 20 74 61 6b 65 20 31 20 66 72 6f 6d | .To take 1 from| 00001580 20 30 20 79 6f 75 20 62 6f 72 72 6f 77 20 66 72 | 0 you borrow fr| 00001590 6f 6d 20 74 68 65 20 6e 65 78 74 20 68 69 67 68 |om the next high| 000015a0 65 73 74 20 62 69 74 20 61 6e 64 20 73 6f 0d 74 |est bit and so.t| 000015b0 61 6b 65 20 31 20 66 72 6f 6d 20 31 30 2c 20 77 |ake 1 from 10, w| 000015c0 68 69 63 68 20 69 73 20 31 2e 0d 0d 20 20 20 20 |hich is 1... | 000015d0 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 | | 000015e0 20 20 31 30 31 30 30 31 30 31 0d 20 20 20 20 20 | 10100101. | 000015f0 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 2d | -| 00001600 20 30 31 31 31 30 31 31 30 0d 20 20 20 20 20 20 | 01110110. | 00001610 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 | | 00001620 2d 2d 2d 2d 2d 2d 2d 2d 0d 20 20 20 20 20 20 20 |--------. | 00001630 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 30 | 0| 00001640 30 31 30 31 31 31 31 0d 0d 49 74 27 73 20 6d 6f |0101111..It's mo| 00001650 72 65 20 64 69 66 66 69 63 75 6c 74 20 74 6f 20 |re difficult to | 00001660 67 65 74 20 74 68 65 20 68 61 6e 67 20 6f 66 20 |get the hang of | 00001670 62 69 6e 61 72 79 20 73 75 62 74 72 61 63 74 69 |binary subtracti| 00001680 6f 6e 0d 74 68 61 6e 20 61 64 64 69 74 69 6f 6e |on.than addition| 00001690 20 62 75 74 2c 20 69 6e 20 74 68 65 20 65 6e 64 | but, in the end| 000016a0 2c 20 69 74 20 64 6f 65 73 6e 27 74 20 72 65 61 |, it doesn't rea| 000016b0 6c 6c 79 20 6d 61 74 74 65 72 2e 20 0d 41 70 61 |lly matter. .Apa| 000016c0 72 74 2c 20 74 68 61 74 20 69 73 2c 20 66 72 6f |rt, that is, fro| 000016d0 6d 20 74 68 65 20 69 64 65 61 20 6f 66 20 62 6f |m the idea of bo| 000016e0 72 72 6f 77 69 6e 67 20 73 69 6e 63 65 20 69 74 |rrowing since it| 000016f0 20 69 73 20 74 68 69 73 0d 74 68 61 74 20 65 6e | is this.that en| 00001700 61 62 6c 65 73 20 79 6f 75 20 74 6f 20 73 75 62 |ables you to sub| 00001710 74 72 61 63 74 20 6e 75 6d 62 65 72 73 20 6c 61 |tract numbers la| 00001720 72 67 65 72 20 74 68 61 6e 20 61 20 73 69 6e 67 |rger than a sing| 00001730 6c 65 0d 62 79 74 65 2e 20 20 43 61 72 72 79 69 |le.byte. Carryi| 00001740 6e 67 20 64 6f 65 73 20 74 68 69 73 20 6a 6f 62 |ng does this job| 00001750 20 69 6e 20 61 64 64 69 74 69 6f 6e 20 61 6e 64 | in addition and| 00001760 20 69 6e 20 62 6f 74 68 20 63 61 73 65 73 0d 74 | in both cases.t| 00001770 68 65 20 63 61 72 72 79 20 66 6c 61 67 20 69 73 |he carry flag is| 00001780 20 75 73 65 64 2e 0d 0d 54 6f 20 61 64 64 20 74 | used...To add t| 00001790 77 6f 20 62 79 74 65 73 20 28 73 74 6f 72 65 64 |wo bytes (stored| 000017a0 20 69 6e 20 74 68 65 20 61 64 64 72 65 73 73 65 | in the addresse| 000017b0 73 20 6c 61 62 65 6c 6c 65 64 20 62 79 74 65 5f |s labelled byte_| 000017c0 31 20 61 6e 64 0d 62 79 74 65 5f 32 29 20 61 6e |1 and.byte_2) an| 000017d0 64 20 73 74 6f 72 65 20 74 68 65 20 72 65 73 75 |d store the resu| 000017e0 6c 74 20 61 74 20 61 64 64 72 65 73 73 20 62 79 |lt at address by| 000017f0 74 65 5f 33 20 79 6f 75 20 64 6f 20 74 68 69 73 |te_3 you do this| 00001800 3a 0d 0d 43 4c 65 61 72 20 74 68 65 20 43 61 72 |:..CLear the Car| 00001810 72 79 20 66 6c 61 67 20 20 20 20 20 20 20 20 20 |ry flag | 00001820 20 20 20 20 43 4c 43 0d 4c 6f 61 44 20 41 20 77 | CLC.LoaD A w| 00001830 69 74 68 20 74 68 65 20 66 69 72 73 74 20 6e 75 |ith the first nu| 00001840 6d 62 65 72 20 20 20 20 20 4c 44 41 20 62 79 74 |mber LDA byt| 00001850 65 5f 31 0d 41 44 64 20 74 68 65 20 73 65 63 6f |e_1.ADd the seco| 00001860 6e 64 20 28 77 69 74 68 20 43 61 72 72 79 29 20 |nd (with Carry) | 00001870 20 20 20 20 20 41 44 43 20 62 79 74 65 5f 32 0d | ADC byte_2.| 00001880 53 54 6f 72 65 20 74 68 65 20 72 65 73 75 6c 74 |STore the result| 00001890 20 28 66 72 6f 6d 20 41 29 20 20 20 20 20 20 20 | (from A) | 000018a0 20 53 54 41 20 62 79 74 65 5f 33 0d 0d 41 2c 20 | STA byte_3..A, | 000018b0 74 68 65 20 61 63 63 75 6d 75 6c 61 74 6f 72 2c |the accumulator,| 000018c0 20 69 73 20 74 68 65 20 70 61 72 74 20 6f 66 20 | is the part of | 000018d0 74 68 65 20 6d 69 63 72 6f 70 72 6f 63 65 73 73 |the microprocess| 000018e0 6f 72 20 77 68 65 72 65 0d 79 6f 75 20 64 6f 20 |or where.you do | 000018f0 79 6f 75 72 20 61 72 69 74 68 6d 65 74 69 63 2e |your arithmetic.| 00001900 20 20 41 44 43 20 61 64 64 73 20 61 20 6e 75 6d | ADC adds a num| 00001910 62 65 72 20 74 6f 20 74 68 65 20 76 61 6c 75 65 |ber to the value| 00001920 20 69 6e 0d 74 68 65 20 61 63 63 75 6d 75 6c 61 | in.the accumula| 00001930 74 6f 72 20 61 6e 64 20 61 64 64 73 20 74 68 65 |tor and adds the| 00001940 20 76 61 6c 75 65 20 6f 66 20 74 68 65 20 63 61 | value of the ca| 00001950 72 72 79 20 66 6c 61 67 20 74 6f 0d 74 68 61 74 |rry flag to.that| 00001960 2e 20 20 54 68 65 20 72 65 73 75 6c 74 20 69 73 |. The result is| 00001970 20 69 6e 20 74 68 65 20 61 63 63 75 6d 75 6c 61 | in the accumula| 00001980 74 6f 72 2e 0d 0d 54 6f 20 73 75 62 74 72 61 63 |tor...To subtrac| 00001990 74 20 32 20 62 79 74 65 73 20 79 6f 75 20 68 61 |t 2 bytes you ha| 000019a0 76 65 20 74 6f 20 73 74 61 72 74 20 62 79 20 73 |ve to start by s| 000019b0 65 74 74 69 6e 67 20 74 68 65 20 63 61 72 72 79 |etting the carry| 000019c0 0d 66 6c 61 67 3a 0d 0d 53 45 74 20 43 61 72 72 |.flag:..SEt Carr| 000019d0 79 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 |y | 000019e0 20 20 20 20 20 20 20 20 20 53 45 43 0d 4c 6f 61 | SEC.Loa| 000019f0 44 20 41 20 77 69 74 68 20 74 68 65 20 66 69 72 |D A with the fir| 00001a00 73 74 20 6e 75 6d 62 65 72 20 20 20 20 20 4c 44 |st number LD| 00001a10 41 20 62 79 74 65 5f 31 0d 53 75 42 74 72 61 63 |A byte_1.SuBtrac| 00001a20 74 20 74 68 65 20 73 65 63 6f 6e 64 20 28 77 69 |t the second (wi| 00001a30 74 68 20 43 61 72 72 79 29 20 53 42 43 20 62 79 |th Carry) SBC by| 00001a40 74 65 5f 32 0d 53 54 6f 72 65 20 74 68 65 20 72 |te_2.STore the r| 00001a50 65 73 75 6c 74 20 28 66 72 6f 6d 20 41 29 20 20 |esult (from A) | 00001a60 20 20 20 20 20 20 53 54 41 20 62 79 74 65 5f 33 | STA byte_3| 00001a70 0d 0d 49 6e 20 62 6f 74 68 20 63 61 73 65 73 2c |..In both cases,| 00001a80 20 69 66 20 79 6f 75 20 61 72 65 20 61 64 64 69 | if you are addi| 00001a90 6e 67 20 6f 72 20 73 75 62 74 72 61 63 74 69 6e |ng or subtractin| 00001aa0 67 20 6e 75 6d 62 65 72 73 20 6d 61 64 65 0d 75 |g numbers made.u| 00001ab0 70 20 6f 66 20 6d 61 6e 79 20 62 79 74 65 73 2c |p of many bytes,| 00001ac0 20 79 6f 75 20 6f 6e 6c 79 20 65 78 70 6c 69 63 | you only explic| 00001ad0 69 74 6c 79 20 63 6c 65 61 72 20 6f 72 20 73 65 |itly clear or se| 00001ae0 74 20 74 68 65 20 63 61 72 72 79 0d 61 74 20 74 |t the carry.at t| 00001af0 68 65 20 62 65 67 69 6e 6e 69 6e 67 2e 20 20 41 |he beginning. A| 00001b00 66 74 65 72 20 74 68 61 74 20 69 74 20 73 65 72 |fter that it ser| 00001b10 76 65 73 20 69 74 73 20 70 72 6f 70 65 72 20 70 |ves its proper p| 00001b20 75 72 70 6f 73 65 0d 61 6e 64 20 63 61 72 72 69 |urpose.and carri| 00001b30 65 73 20 61 63 72 6f 73 73 20 6f 72 20 62 6f 72 |es across or bor| 00001b40 72 6f 77 73 20 61 63 72 6f 73 73 20 69 66 20 6e |rows across if n| 00001b50 65 65 64 65 64 2e 20 20 4c 6f 6f 6b 20 61 74 0d |eeded. Look at.| 00001b60 74 68 69 73 20 77 65 65 6b 73 20 61 73 73 65 6d |this weeks assem| 00001b70 62 6c 65 72 20 70 72 6f 67 72 61 6d 20 42 2f 6f |bler program B/o| 00001b80 73 62 30 32 20 77 68 69 63 68 20 61 64 64 73 20 |sb02 which adds | 00001b90 61 6e 64 0d 73 75 62 74 72 61 63 74 73 20 74 77 |and.subtracts tw| 00001ba0 6f 20 66 6f 75 72 20 62 79 74 65 20 6e 75 6d 62 |o four byte numb| 00001bb0 65 72 73 2e 20 20 49 20 68 61 76 65 20 75 73 65 |ers. I have use| 00001bc0 64 20 62 69 74 73 20 6f 66 20 42 41 53 49 43 0d |d bits of BASIC.| 00001bd0 74 6f 20 65 6e 61 62 6c 65 20 79 6f 75 20 74 6f |to enable you to| 00001be0 20 67 65 74 20 74 68 65 20 6e 75 6d 62 65 72 73 | get the numbers| 00001bf0 20 69 6e 74 6f 20 61 6e 64 20 6f 75 74 20 6f 66 | into and out of| 00001c00 20 74 68 65 20 6d 61 63 68 69 6e 65 0d 62 75 74 | the machine.but| 00001c10 20 65 76 65 6e 74 75 61 6c 6c 79 20 77 65 20 77 | eventually we w| 00001c20 69 6c 6c 20 75 73 65 20 6d 61 63 68 69 6e 65 20 |ill use machine | 00001c30 63 6f 64 65 20 74 6f 20 64 6f 20 74 68 69 73 20 |code to do this | 00001c40 61 73 20 77 65 6c 6c 2e 20 0d 57 65 20 63 6f 75 |as well. .We cou| 00001c50 6c 64 20 65 78 74 65 6e 64 20 74 68 61 74 20 63 |ld extend that c| 00001c60 6f 64 65 20 74 6f 20 77 6f 72 6b 20 77 69 74 68 |ode to work with| 00001c70 20 6e 75 6d 62 65 72 73 20 6f 66 20 61 6e 79 20 | numbers of any | 00001c80 73 69 7a 65 0d 62 75 74 20 42 41 53 49 43 27 73 |size.but BASIC's| 00001c90 20 69 6e 74 65 67 65 72 73 20 61 72 65 20 6f 6e | integers are on| 00001ca0 6c 79 20 34 20 62 79 74 65 73 20 69 6e 20 73 69 |ly 4 bytes in si| 00001cb0 7a 65 20 73 6f 20 77 65 20 77 69 6c 6c 0d 73 74 |ze so we will.st| 00001cc0 69 63 6b 20 74 6f 20 34 20 62 79 74 65 73 20 66 |ick to 4 bytes f| 00001cd0 6f 72 20 74 68 65 20 6d 6f 6d 65 6e 74 2e 0d 0d |or the moment...| 00001ce0 4e 6f 74 65 20 74 68 61 74 20 42 2f 6f 73 62 30 |Note that B/osb0| 00001cf0 32 20 77 69 6c 6c 20 67 69 76 65 20 61 6e 20 69 |2 will give an i| 00001d00 6e 63 6f 72 72 65 63 74 20 72 65 73 75 6c 74 20 |ncorrect result | 00001d10 69 66 20 79 6f 75 20 61 64 64 0d 74 6f 67 65 74 |if you add.toget| 00001d20 68 65 72 20 6e 75 6d 62 65 72 73 20 73 75 63 68 |her numbers such| 00001d30 20 74 68 61 74 20 74 68 65 69 72 20 73 75 6d 20 | that their sum | 00001d40 69 73 20 67 72 65 61 74 65 72 20 74 68 61 6e 0d |is greater than.| 00001d50 32 31 34 37 34 38 33 36 34 37 20 28 61 6e 64 20 |2147483647 (and | 00001d60 73 69 6d 69 6c 61 72 6c 79 20 66 6f 72 20 6e 65 |similarly for ne| 00001d70 67 61 74 69 76 65 20 6e 75 6d 62 65 72 73 29 2e |gative numbers).| 00001d80 20 20 54 68 69 73 20 69 73 0d 62 65 63 61 75 73 | This is.becaus| 00001d90 65 20 74 68 65 20 6f 76 65 72 66 6c 6f 77 20 62 |e the overflow b| 00001da0 65 74 77 65 65 6e 20 74 68 65 20 74 6f 70 20 74 |etween the top t| 00001db0 77 6f 20 62 69 74 73 20 69 73 20 6e 6f 74 20 68 |wo bits is not h| 00001dc0 61 6e 64 6c 65 64 0d 63 6f 72 72 65 63 74 6c 79 |andled.correctly| 00001dd0 2e 20 20 42 41 53 49 43 20 77 6f 75 6c 64 20 74 |. BASIC would t| 00001de0 72 61 70 20 74 68 69 73 20 61 73 20 61 20 54 6f |rap this as a To| 00001df0 6f 20 42 69 67 20 65 72 72 6f 72 20 62 75 74 0d |o Big error but.| 00001e00 74 68 69 73 20 6d 61 63 68 69 6e 65 20 63 6f 64 |this machine cod| 00001e10 65 20 64 6f 65 73 20 6e 6f 74 2e 20 20 54 6f 20 |e does not. To | 00001e20 64 6f 20 73 6f 20 75 73 65 73 20 61 6e 6f 74 68 |do so uses anoth| 00001e30 65 72 20 6f 66 20 74 68 65 0d 6d 69 63 72 6f 70 |er of the.microp| 00001e40 72 6f 63 65 73 73 6f 72 27 73 20 66 6c 61 67 73 |rocessor's flags| 00001e50 20 63 61 6c 6c 65 64 20 74 68 65 20 6f 76 65 72 | called the over| 00001e60 66 6c 6f 77 20 66 6c 61 67 2e 0d 0d 57 65 27 6c |flow flag...We'l| 00001e70 6c 20 6c 6f 6f 6b 20 61 74 20 74 68 65 20 66 6c |l look at the fl| 00001e80 61 67 73 20 69 6e 20 6d 6f 72 65 20 64 65 74 61 |ags in more deta| 00001e90 69 6c 20 6e 65 78 74 20 77 65 65 6b 20 61 6e 64 |il next week and| 00001ea0 20 75 73 65 0d 74 68 65 6d 20 74 6f 20 67 65 74 | use.them to get| 00001eb0 20 74 68 65 20 63 6f 6d 70 75 74 65 72 20 74 6f | the computer to| 00001ec0 20 6d 61 6b 65 20 73 6f 6d 65 20 64 65 63 69 73 | make some decis| 00001ed0 69 6f 6e 73 2e 0d |ions..| 00001ed6