🚀 go-pugleaf

M. Klepcyk (mjk+@osu.edu) wrote:

>I have some other questions also.  I have to make a program that uses
>infinite series to approximate the value of SIN (a).  The line that my
>teacher gave me for this is: "sin(a)=a-a3/3!+a5/5!-a7/7!+a9/9!...."
>What does this translate to in the way of a formula?  I need to know
>this so I can write a procedure that will use the variable that will
>obviously be involved.

Come on, this is too easy.  You should work this one out
for yourself.

>Another problem concerns writing a procedure that will computer the
>binary XOR's of two decimal numbers.  I know how to get the binary
>values for each of the decimal numbers (using MOD & DIV), but how do I
>compare the two binary numbers when I'm finished with this?  Is there
>some kind of comparison command like with strings?

This, however, deserves more comment.  I've noticed that
students often get confused over what is meant by "binary number"
and "decimal number" in the context of programming.

The names "binary" and "decimal" refer to the _representation_
of a number, as distinct from the underlying object which is the
number itself.  You're probably aware that "101 binary" and
"5 decimal" are not two distinct numbers, but instead are two
alternative ways of writing the _same_ number.  It's a bit like
saying that "pomme" and "apple" are two ways of saying the
same thing: different languages are used, but despite the
different words we're still talking about the same fruit.

When manipulating numbers in a computer program, we have to be
aware of a number of different concepts:

 1. The number itself.  This is an abstract object which can
    be defined by, for example, a mathematician's set of axioms
    for arithmetic.  I can't write down an example, because as
    soon as I'd written it down I would have written a
    representation of the number, rather than the number itself.

 2. The representation of the number inside the computer's
    memory.  For most present-day computers this is a binary
    representation - or, more precisely, it is a collection 
    of electrical signals which in turn are representations of
    binary digits - but it doesn't have to be.  A hardware
    designer could, if he wished, build a computer in which
    ternary representations were used internally.

 3. The representation of the number in the source code of
    your computer program.  This depends on the rules of the
    programming language.  For most programming languages a
    decimal notation is used, but there are other possibilities.
    (Modula-2 also has a way to use octal or hexadecimal
    notation.)  The important point is that the representation
    doesn't have to be - and usually isn't - the same as the
    representation in 2 above.  The compiler takes care of
    translating the notation.

 4. The representation of the number in the external interface;
    for example, the way the number is typed at the keyboard, or the
    way it is printed on the screen.  This depends on the I/O
    routines you use.  The I/O routines most commonly used
    perform a translation between an external notation
    (usually decimal) and an internal representation (usually
    binary).  But that's not compulsory, you could write your
    own formatting procedures that used any base you liked.

This might all sound like quibbling, but it has a bearing on
your question.  Let's look at the question again.  You said
"compute the binary XORs of two decimal numbers".  Right there,
the problem is badly posed.  What it should say is something
like "compute the binary XORs of two numbers".  They're
numbers, not decimal numbers, and the distinction is important.
There's no such thing as a decimal number.  (Although,
informally, we often use the phrase "decimal number" as a
shorthand way of saying "decimal representation of a number"
- which in some ways is a pity, because it's this sloppy
terminology which is the root of the confusion.)

When you write a statement like "abc := 23", is abc a decimal
number?  Most decidedly not!  The character string "23" is the
decimal representation of a number, but that's where it ends.
The variable abc holds a number - not a decimal number - and,
depending on your computer hardware, the internal representation
is probably binary.

Now, this doesn't give an answer to your original question.
I don't want to give an answer, because I think it's
something that you have to work out for yourself.  But
I'll at least give you a hint: if you're thinking of using
DIV and MOD to convert between decimal and binary, then
you're completely on the wrong track.  You shouldn't have
to do any conversions.

--
Peter Moylan                           peter@ee.newcastle.edu.au
                  http://www.eng.newcastle.edu.au/ee/Moylan.html
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