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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:

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.

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*
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000019e0  20 20 20 20 20 20 20 20  20 53 45 43 0d 4c 6f 61  |         SEC.Loa|
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00001a70  0d 0d 49 6e 20 62 6f 74  68 20 63 61 73 65 73 2c  |..In both cases,|
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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|
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00001b20  75 72 70 6f 73 65 0d 61  6e 64 20 63 61 72 72 69  |urpose.and carri|
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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 |
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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|
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00001c40  61 73 20 77 65 6c 6c 2e  20 0d 57 65 20 63 6f 75  |as well. .We cou|
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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 |
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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|
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00001db0  77 6f 20 62 69 74 73 20  69 73 20 6e 6f 74 20 68  |wo bits is not h|
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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
23_10_87/T\OSB02.m0
23_10_87/T\OSB02.m1
23_10_87/T\OSB02.m2
23_10_87/T\OSB02.m4
23_10_87/T\OSB02.m5

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