What's the rule about using 'ones' in algebra?

Can you use ones [1's] indescriminately in algebra? If not why not...

Dear Donald G. Shead:

"Donald G. Shead" <[Only registered users see links. ]> wrote in message
news:QWnbb.4796$[Only registered users see links. ].prodigy.co m...

They're used in mechanics quite a bit. Dimensional analysis, you know.

David A. Smith

In sci.physics, Donald G. Shead
<[Only registered users see links. ]>
wrote
on Sun, 21 Sep 2003 20:33:52 GMT
<QWnbb.4796$[Only registered users see links. ].prodigy.com> :

Depends.

In standard algebra 1 is the multiplicative identity. That is
to say, a * 1 = 1 * a = a, for all a. Also, a * 1/a = 1 for
all a != 0 (since a * 0 = 0 for all a, there's a little problem
with "1/0").

Of course a + 1 != a. But a lot of people use multiplication by
1 to solve certain problems.

For instance, how does one integrate 1 / x(x+1) ? Turns out
that one's equal to

((x+1) - x) / (x(x+1))

or (x+1) / x(x+1) - x / (x(x+1))

We can now take out the 1's (actually, a/a):

((x+1) / (x+1)) * (1 / x) - ((x / x) * (1 / (x+1)))

and we ultimately get:

1/x - 1/(x+1)

Integrating both pieces separately, we get log(x) - log(x+1) + C,
or log(x/(x+1)) + C. A bit weird-looking since log(x/(x+1))
is always negative for x > 0, but remember that log((x+1)/(x+2))
is greater than log(x/(x+1)); it's a strictly monotonic function
that asymptotically approaches the x axis. Besides, the integration
constant C can be pretty much any value one wants. :-)

I'm not sure I can think of another problem along these lines,
as it's after midnight, but you hopefully get the idea. Note
that one has to be a little careful; 1/0 has no meaning, and
0/0 can be anything at all, so it has no meaning either unless
very carefully set up using limits. For example,
lim (x->0) (x/x) = 1. However, lim(x->0) (2x/x) = 2. At x=0,
both x/x and 2x/x become 0/0.

So multiplying by 1 may introduce additional singularities over
the complex plane -- mostly because x/x has a singularity
already (namely, the point 0 + 0i).

It just takes a little care in problem setup. :-)

--
#191, [Only registered users see links. ]
It's still legal to go .sigless.

"The Ghost In The Machine" <[Only registered users see links. ]> wrote in
message news:[Only registered users see links. ]...

Thanks, _really_ Ghouly: What you say doesn't sound like a lot of
ghoulishness (pun intended) but it's beyond me. I appreciate your response.

Many years ago I overheard a H.S. algebra teacher tell his class _something
to the effect_ that the number one [1] was such that it didn't change the
value (of an equation) when it was inserted in an equation. That made a
lasting impression, since I didn't understand then, and don't now:

In particular: Writing that acceleration [a] is _inversely_ proportional to
the mass [m] of a body, as [a is proportional to 1/m]. That somehow doesn't
look 'copesthetic' to me; especially if 'm' is a variable.

I thought the rule might be simple enough for me to understand; but
apparently it's not.

Thanks again Ghoully(;^)

In article <wRBbb.6129$[Only registered users see links. ].prodigy.com>,
"Donald G. Shead" <[Only registered users see links. ]> wrote:

response.

HINT: Copy the damned lines down on paper without using slashes
to indicate divisions; instead use straight lines.

_something
to
doesn't

And has nothing to do with multiply by one.


You never really wanted an answer. Did you?

Well, perhaps a lurker benefited form the time and effort
Ghost took to answer your question.

<snip snot>

/BAH

Subtract a hundred and four for e-mail.

<[Only registered users see links. ]> wrote in message news:bkmttr$k4$[Only registered users see links. ].rcn.net...
Cut<
Huh? Read the question again; that wasn't it was it? It isn't all _I_ meant
anyway goofus(;^!

[Only registered users see links. ] wrote:






It seems likely that Donnie is a lot brighter than folks
here realize, and that he's terribly bored.

First of all you have to understand that the rules of algebra are human
inventions (to quote Kroneker: "The integer numbers were created by dear
God, everything else is human-made"). The reason that "1" is a special
number is simply because it was defined that way; there is nothing mystical
about it (any Pythagoreans alive still?). The real basis for algebra are
operations (addition, subtraction, etc.); which in turn are a type of
psychological activity.

Let me see if I can give you an idea of what is all about.

-An operation is just the act of assigning two numbers within a set a third:
e.g 2+3=5. Two numbers 2 and 3 are assigned (or mapped) the 5 (Never mind
how the corresponding numbers are assigned.)

-There is this set of numbers called real numbers in which we can carry out
the two operations of addition and multiplication.

-In the addition there is a special number: the "0". Special because every
number in the set is assigned itself under addition with 0, that is: x +
0=x, for x being any number.

-In the addition we say that a number is the inverse of another if adding
them together we get the special number "0"; i.e.: x+y=0, here y is the
additive inverse of x and vice-versa.

-In the multiplication there is a special number: the "1". Special because
every number in the set is assigned itself under multiplication with 1, that
is: x * 1=x, for x being any number.

-In the multiplication we say that a number is the inverse of another if
multiplying them together we get the special number "1"; i.e.: x*y=1, here y
is the multiplicative inverse of x and vice-versa.

-In the set of real numbers, every number has an additive inverse. In the
set of real numbers every number with the exception of the 0 has also a
multiplicative inverse.

Now, when we say 1/m we mean the multiplicative inverse of m, not m itself.
It is just written in a funny way to remind us that m*1/m=1; i.e. m
multiplied times its inverse is 1, which is the way we define the inverse.

What I have outlined here is just the way mathematicians go about putting
definition upon definition, cementing it with logic to achieve this
beautiful cathedral that lots of people behold with dread: modern
mathematics. Let me just add that by taking the above definitions and some
others we arrive at what mathematicians call a field (Britannica Enc:
"Broadly speaking, a field is an algebraic system consisting of elements
that are commonly called numbers, in which the four familiar operations of
addition, subtraction, multiplication and division are universally defined
and have all their usual properties"). The whole thing is very pretty and it
is amazing that it is useful at all (but you'll have to consult Kant on
that).

First of all you have to understand that the rules of algebra are human
inventions (to quote Kroneker: "The integer numbers were created by dear
God, everything else is human-made"). The reason that "1" is a special
number is simply because it was defined that way; there is nothing mystical
about it (any Pythagoreans alive still?). The real basis for algebra are
operations (addition, subtraction, etc.); which in turn are a type of
psychological activity.

Let me see if I can give you an idea of what is all about.

-An operation is just the act of assigning two numbers within a set a third:
e.g 2+3=5. Two numbers 2 and 3 are assigned (or mapped) the 5 (Never mind
how the corresponding numbers are assigned.)

-There is this set of numbers called real numbers in which we can carry out
the two operations of addition and multiplication.

-In the addition there is a special number: the "0". Special because every
number in the set is assigned itself under addition with 0, that is: x +
0=x, for x being any number.

-In the addition we say that a number is the inverse of another if adding
them together we get the special number "0"; i.e.: x+y=0, here y is the
additive inverse of x and vice-versa.

-In the multiplication there is a special number: the "1". Special because
every number in the set is assigned itself under multiplication with 1, that
is: x * 1=x, for x being any number.

-In the multiplication we say that a number is the inverse of another if
multiplying them together we get the special number "1"; i.e.: x*y=1, here y
is the multiplicative inverse of x and vice-versa.

-In the set of real numbers, every number has an additive inverse. In the
set of real numbers every number with the exception of the 0 has also a
multiplicative inverse.

Now, when we say 1/m we mean the multiplicative inverse of m, not m itself.
It is just written in a funny way to remind us that m*1/m=1; i.e. m
multiplied times its inverse is 1, which is the way we define the inverse.

What I have outlined here is just the way mathematicians go about putting
definition upon definition, cementing it with logic to achieve this
beautiful cathedral that lots of people behold with dread: modern
mathematics. Let me just add that by taking the above definitions and some
others we arrive at what mathematicians call a field (Britannica Enc:
"Broadly speaking, a field is an algebraic system consisting of elements
that are commonly called numbers, in which the four familiar operations of
addition, subtraction, multiplication and division are universally defined
and have all their usual properties"). The whole thing is very pretty and it
is amazing that it is useful at all (but you'll have to consult Kant on
that).