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(a) addition (a) subtraction
a+b=ec b=c-—-q
(b)_ multiplication (b) division
qb=ec b = c/a
22.2
(c) power (c)_ root (22.2)
b“=ec b— ức
(d) power (d) logarithm
œ°=€ b =log„e
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Now here ¡is the idea. 'These relationships, or rules, are correct for integers,
since they follow from the definitions of addition, multiplication, and raising to a
power. IỨe are goïng ‡o điscuss tohether or noÈ tue can broaden the cÏass oƒ objects
thích a, Ð, and c represent‡ so that the uuilÏ obeu these sœme rules, although the
processes for ø + ð, and so on, will not be defnable in terms of the direct action
of adding 1, for instance, or successive multiplications by integers.
22-3 Abstraction and generalization
'When we try to solve simple algebraic equations using all these defnitions,
we soon discover some insoluble problems, such as the following. Suppose that
we try to solve the equation ö = 3— 5. That means, according to our def-
inition of subtraction, that we must fnd a number which, when added to 5ð,
gives 3. And of course there 2s no such number, because we consider only
positive Integers; this is an insoluble problem. However, the plan, the great
idea, 1s this: œbsfraclon and generalzalion. From the whole structure of al-
gebra, rules plus integers, we abstract the original defñnitions of addition and
multiplication, bu we leave the rules (22.1) and (22.2), and assume these to
be true ?w general on a wider class oŸ numbers, even though they are originally
derived on a smaller class. “Thus, rather than using integers symbolically to
defñne the rules, we use the rules as the defñnition of the symbols, which then
represent a more general kind of number. As an example, by working with
the rules alone we can show that 3 — 5 =0-—2. In facÿ we can show that
one can make øil subtractions, provided we defne a whole set of new num-
bers: 0— 1,0 —2,0—3,0— 4, and so on, called the megøf2ue ?mtegers. Then
we may use all the other rules, like ø(b + e) = øb + ac and so forth, to ñnd
what the rules are for multiplying negative numbers, and we will discover, in
fact, that all of the rules can be maintained with negative as well as positive
1ntegers.
So we have increased the range of objects over which the rules work, but the
mmeaning of the symbols is difÑerent.
One cannot say, for instance, that —2 times 5 really means to add ð together
successively —2 times. hat means nothing. But nevertheless everything will
work out all right according to the rules.
An interesting problem comes up in taking powers. Suppose that we wish
to discover what a(3~5) means. We know only that 3 — 5 is a solution of the
problem, (3 — 5) +5 = 3. Knowing that, we know that a(3~5)að = a3. Therefore
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a(—~5) = a3/a5, by the definition of division. With a little more work, this can
be reduced to 1/a2. So we find that the negative powers are the reciprocals of
the positive powers, but 1/42 is a meaningless symbol, because if ø is a positive
or negative integer, the square of it is greater than 1, and we do not yet know
what we mean by 1 divided by a number greater than 1l
Onwardl The great plan is to continue the process of generalization; whenever
we fnd another problem that we cannot solve we extend our realm of numbers.
Consider division: we cannot find a number which is an integer, even a negative
integer, which is equal to the result of dividing 3 by 5. But if we suppose that all
tractional numbers also satisfy the rules, then we can talk about multiplying and
adding fractions, and everything works as well as it did before.
Take another example of powers: what is a3/5? We know only that (3/5)5 = 3,
since that was the defnition of 3/5. So we know also that (a(3/5))5 = ạ(3/5)(5) =
a3, because this is one of the rules. Then by the defnition of roots we fñnd that
a\3/5) — a3,
In this way, then, we can delñne what we mean by putting fractions in the
various symbols, by using the rules themselves to help us determine the defnition——
1t is not arbitrary. It is a remarkable fact that all the rules still work for positive
and negative integers, as well as for fractionsl
We go on in the process of generalization. Are there any other equations
we cannot solve? Yes, there are. EFor example, it is impossible to solve this
cquation: b= 21⁄2 = v2. It is impossible to ñnd a number which is rational (a
fraction) whose square is equal to 2. Ib is very easy Íor us in modern days to
answer this question. We know the decimal system, and so we have no difliculty
in appreciating the meaning of an unending decimal as a type of approximation
to the square root of 2. Historically, this idea presented great dificulty to the
Greeks. To really delne ørecisel what is meant here requires that we add some
substance of continuity and ordering, and it is, in fact, quite the most dificult
step In the processes of generalization Just at this point. It was made, formally
and rigorously, by Dedekind. However, without worrying about the mathematical
rigor of the thing, it is quite easy to understand that what we mean is that we are
going to find a whole sequence of approximate fractions, perfect fractions (because
any decimal, when stopped somewhere, is oŸ course rational), which Just keeps
on going, getting closer and closer to the desired result. That is good enough
for what we wish to discuss, and it permits us to involve ourselves in irrational
numbers, and to calculate things like the square root of 2 to any accuracy that
we desire, with enough work.
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22-4 Approximating irrational numbers
The next problem comes with what happens with the irrational powers.
Suppose that we want to defne, for instance, 10Y2. In principle, the answer 1s
simple enough. lIÝ we approximate the square root of 2 to a certain number of
decimal places, then the power is rational, and we can take the approximate root,
using the above method, and get an øpprozimation to 10Y2. Then we may run it
up a few more decimal places (it is again rational), take the appropriate root,
this time a much higher root because there is a much bigger denominator in the
fraction, and get a better approximation. OÝ course we are going to geÈ some
enormously high roots involved here, and the work is quite difcult. How can we
cope with this problem?
In the computations of square roots, cube roots, and other small roots, there
1s an arithmetical process available by which we can get one decimal place after
another. But the amount of labor needed to calculate irrational powers and
the logarithms that go with them (the inverse problem) is so great that there
1s no simple arithmetical process we can use. Therefore tables have been built