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may be pushing when we should be pulling, and so on, and it does not work.
Tf we make œ exactly equal to œọ, we fnd that ¡§ should oscillate at an
#nfinite amplitude, which is, of course, impossible. 'Phe reason it does not is that
something goes wrong with the equation, there are some other frictional terms,
and other forces, which are not in (21.S) but which occur in the real world. So
the amplitude does not reach infinity for some reason; it may be that the spring
breaksl
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Algeobr«
22-1 Addition and multiplication
In our study of oscillating systems we shall have occasion to use one of the
mmost remarkable, almost astounding, formulas in all of mathematics. EFrom the
physicist's point of view we could bring forth this formula in two minutes or
so, and be done with it. But science is as much for intellectual enjoyment as
for practical utility, so instead of just spending a few minutes on this amazing
jewel, we shall surround the jewel by its proper setting in the grand design of
that branch of mathematics which is called elementary algebra.
Now you may ask, “What is mathematics doïng in a physics lecbure?” We
have several possible excuses: first, of course, mathematics is an important tool,
but that would only excuse us for giving the formula in two minutes. Ơn the
other hand, in theoretical physics we discover that all our laws can be written in
mathematical form; and that this has a certain simplicity and beauty about it.
So, ultimately, in order to understand nature it may be necessary to have a deeper
understanding of mathematical relationships. But the real reason is that the
subject is enjoyable, and although we humans cut nature up in different ways, and
we have diferent courses in diferent departments, such compartmentalization 1s
really artifcial, and we should take our intellectual pleasures where we fnd them.
Another reason for looking more carefully at algebra now, even though most
of us studied algebra in high school, is that that was the first time we studied it;
all the equations were unfamiliar, and it was hard work, just as physics 1s now.
lvery so often it is a great pleasure to look back to see what territory has been
covered, and what the great map or plan of the whole thing is. Perhaps some day
somebody in the Mathematics Department will present a lecture on mechanics In
such a way as to show what it was we were trying to learn in the physics coursel
The subject of algebra will not be developed from the point of view of a
mathematician, exactly, because the mathematicians are mainly interested in how
various mathematical facts are demonstrated, and how many assumptions are
absolutely required, and what is not required. They are not so interested in the
--- Trang 390 ---
result oŸ what they prove. For example, we may fñnd the Pythagorean theorem
quite interesting, that the sum of the squares of the sides of a right triangle 1s
equal to the square of the hypotenuse; that is an interesting fact, a curiously
simple thing, which may be appreciated without discussing the question of how
to prove it, or what axioms are required. So, in the same spirit, we shall describe
qualitatively, if we may put it that way, the system of elementary algebra. We
say clementaru algebra because there is a branch of mathematics called rmodern
algebra in which some of the rules such as œb = ba, are abandoned, and ït ¡s still
called algebra, but we shall not discuss that.
To discuss this subJect we start in the middle. We suppose that we already
know what integers are, what zero is, and what it means to increase a number
by one unit. You may say, “That is not in the middlel” But it is the middle from
a mathematical standpoint, because we could go even further back and describe
the theory of sets in order to đerzue some of these properties of integers. But we
are not goïing in that direction, the direction of mathematical philosophy and
mathematical logic, but rather in the other direction, from the assumption that
we know what integers are and we know how to count.
Tf we start with a certain number ø, an integer, and we count successively
one unit b times, the number we arrive at we call ø + 0, and that defines øddiion
Of integers.
Once we have defned addition, then we can consider this: if we start with
nothing and add ø to it, b times in succession, we call the result rmultiplication oŸ
integers; we call it b tữmes a.
Now we can also have a swccession. o0 tmultiplicœtions: 1Ÿ we start with 1 and
multiply by ø, b tỉimes in succession, we call that raising to œ pouer: aP.
Now as a consequence of these definitions it can be easily shown that all of
the following relationships are true:
(a) œ+b=b+a (b) a+(b+c)=(a+Ù)+ec
(c) ab= ba (d) a(b+c)= ab+ ac
(e)_ (ab)c= a(bc) () (œb)“= a°°
be _ „(b+c) be —_ „(be) (22.1)
(g) a a°=ø (h) (a) =4
(Œ) a+0=wø 0) a:l=a
(k) aøl=a
'These results are well known and we shall not belabor the point, we merely list
--- Trang 391 ---
them. Of course, 1 and 0 have special properties; for example, œ + 0 is ø, ø times
1= 4a, and ø to the frst power 1s ø.
In this discussion we must also assume a few other properties like continuity
and ordering, which are very hard to deñne; we will let the rigorous theory do it.
Purthermore, it is defnitely true that we have written down too many “rules”;
some of them may be deducible from the others, but we shall not worry about
such matters.
22-2 The inverse operations
In addition to the direct operations of addition, multiplication, and raising to
a power, we have also the Zøw0erse operations, which are defned as follows. Let us
assume that ø and é are given, and that we wish to fnd what values of b satisfy
such equations as ø-L = eœ, ab = c, b“ =c. Ifa+b = c, b1s defined as e— a, which
1s called subfraction. 'The operation called division is also clear: if ab = c, then
b = c/a defines division—a solution of the equation øb = e “backwards.” Now if
we have a power 0“ = cand we ask ourselves, “What is b?,” ït is called the ath roo‡
of c: b= c. Eor instance, if we ask ourselves the following question, “What
Integer, raised to the third power, equals 8?,” then the answer is called the cube
root of 8; ït is 2. Because b“ and a are not equal, there are #wo inverse problems
associated with powers, and the other inverse problem would be, “To what power
must we raise 2 to get 8?” Thịis is called taking the logarithm. Tf a° = e, we write
b = log„c. The fact that it has a cumbersome notation relative to the others does
not mean that it is any less elementary, at least applied to integers, than the other
processes. Although logarithms come late in an algebra class, in practice they are,
Of course, just as simple as roots; they are just a diferent kind of solution of an
algebraic equation. The direct and inverse operations are summarized as follows: