Reductio ad absurdum and indirect proof are different but
related procedures.
      Reductio ad absurdum shows the falsity of an assumption 
by deriving from it a manifest absurdity. "reduction
 to an absurdity" is a mathematical procedure but it 
has some resemblance of irony which is the favourite
procedure of the satirist. Irony adopts , to all appearance,
a certain opinion and stresses it and over-stresses it till it
leads to a manifest absurdity.
    Indirect proof establishes the truth of an assertion by 
showing the falsity of the opposite assumption. Thus
indirect proof has some resemblance to a politician's 
trick of establishing a candidate by demolishing the reputation
 of his opponent.
   Both "reductio ad absurdum" and indirect proof are
effective tools of discovery which present themselves naturally
 to an intent mind. Nevertheless, they are disliked
by a few philosophers and many beginners, which is 
understandable; satirical people and tricky politicians do
not appeal to everybody.  We shall first illustrate the
effectiveness of both procedures by examples and discuss
objections against them afterwards.
    1. Reductio ad absurdum. Write numbers using each
of the ten digits exactly once so that the sum of the 
numbers is exactly 100.
    We may learn something by trying to solve this puzzle
whose statement demands some elucidation.
  What is the unknown?   A set of numbers; and by
numbers here we mean here, of course, ordinary integers.
 What is given?  the number 100.
 What is the condition?  The condition as two parts
first writing the desired set of numbers, we must use
each of the ten digits, 0,1,2,3,4,5,6,7,7,8,8 and 9 . just
once. second, the sum of all numbers in the set must
be 100.
     Keep only a part of the condition, drop the other part.
the first part alone is easy to satisfy. Take the set 19,
28, 37, 46,50; each figure occurs just once; but of course
the second part of the condition is not satisfied; the sum
of these numbers is 180, not 100. We could however do
better. "try, try again." Yes,

      19 + 28 + 30 + 7 + 6 + 5 + 4 = 99

The first part of the condition is satisfied, and the second part
is almost satisfied; we have 99 instead of 100. Of 
course, we can easily satisfy the second part if we drop
the first; 
       19 + 28 + 31 + 7 + 6 + 5 + 4 = 100

The first part is not satisfied; the figure 1 occurs twice,
and o not at all; the other figures are all right. "Try, try
again."
     After a few unsuccessful trials, however, we may be led
to suspect that it is not possible to obtain 100 in the
manner required . Eventually the problem arises; Prove
that it is impossible to satisfy both parts of the proposed
condition at the same time

    Quite good students may find that this problem is 
above their heads. Yet the answer is easy enough if we 
have the right attitude.We have to examine the hypothetical
 situations in which both parts of the condition
are satisfied.
     We suspect that this situation cannot actually arise and
our suspicion, based on experience of our
unsuccessful trials has some foundation. Nevertheless, let us keep
an open mind and let us face the situation in which hypothetically
, supposed, allegedly both parts of the condition
are satisfied. Thus let us imagine a set of numbers
 whose sum is 100. They must be numbers with one
or two figures. There are ten figures, and these ten figures
must be all different, since each of the figures, 0, 1, 2, ..., 9
should occur just once, Thus, the sum of all ten
figures must be 
0 + 1 + 2 + 3 + 4 + 5 + 6 + 6 + 8 + 9 = 45

Some of these figures denote units and others tens. It
takes a little sagacity to hit upon the idea that the Sum
of the figures denoting the tens may be of some importance.
In fact, let t stand for this sum. Then, the sum of the 
remaining figures, denoting units is 45 -0 t. Therefor, 
the sumo f all numbers in the set must be

10t + (45 - t) = 100

We have here an equation to determine t. It is of the first 
degree and gives 

                t = 55/9

Now, there is something that is definitely wrong. The
value of t that we have found is not an integer and 
should be, of course, an integer. Starting from the 
supposition that both parts of proposed condition can
be simultaneously satisfied, we have been led to a manifest
 absurdity. How can we explain this? Our original 
supposition must be wrong; both parts of the condition 
c cannot be satisfied at the same time.  And so we have 
attained our goal, we have succeeded in proving that the 
two parts of the proposed condition are incompatible.
  j   Our reasoning is a typical  "Reductio ad Absurdum."
2.  Remarks Let us look back at the foregoing reasoning
and understand its general trend.
    We wish to prove that it is impossible to fulfil a certain
condition, that is, that the situation in which all
parts of the condition are simultaneously satisfied can
never arise.  Burt if we have proved nothing yet, we have
to face the possibility that the situation could arise. Only
by facing squarely the hypothetical situation and 
examining it closely can we hope to perceive some definitely 
wrong point in it. And we must lay our hand upon some
definitely wrong point if we wish to know conclusively
that the situation is impossible. Hence we can see that
the procedure that was successful in our example is reasonable
in general; we have to examine the hypothetical 
situation in which all parts of the condition are satisfied,
althoughsuch a situation appears extremely unlikely
    The more experienced reader may see here another
point .he main step of our procedure consisted in set
ting up an equation for t. Now, we could have arrived
at the same equation without suspecting that something
was wrong with the condition. If we wish to set up an 
equation, we have to expression mathematical language
that all parts of the condition are satisfied,although we
do not know yet whether it is actually possible to satisfy
all these parts simultaneously.
    Our procedure is"open-minded." We may hope to find
the unknown satisfying the condition, or we may hope to 
show that the condition cannot be satisfied. It matters
little in one respect; the investigation, if it is well conducted,
starts in both cases in the same way, examining
the hypothetical situation in which the condition is
fulfilled, and shows only in its later course which  hope 
is justified compare FIGURES, 
2.    Compare also PAPPUS ; an 
analysis which ends in disproving the proposed theorem,
or in showing that the proposed "problems to find" has no
solution, is actually a "Reductio ad absurdum."
3.    Indirect proof< The prime number,s, or primes, are 
the numbers 2, 3, 5, 7 ,11, 13, 17, 19, 23, 29, 31, 37, ...
which cannot be resolved into smaller factors, although
they are greater than 1. (The last clause excludes the
number 1 which, obviously, cannot be resolved into
smaller factors, but has a different nature and should
not be counted as a prime.)  The primes are the "ultimate-
elements" into which all integers greater than 1) 
can be decomposed. For instance,

        630 = 2 * 3 * 3 * 5 * 7

is decomposed into a product of five primes.
    Is the series of primes infinite or does it end somewhere?
 It is natural to suspect that the series of primes
never ends. If it ended somewhere, all integers could be
decomposed into a finite number of ultimate elements
and the world would appear "too poor" in a manner of 
speaking.  Thus arises the problem of proving the existence
 of an infinity of prime numbers.
    This problem is very different from elementary mathematical
problems of the usual kind and appears at first
inaccessible. Yet, as we said, it is extremely unlikely that
there should be a last prime, say P.  Why is it so unlikely?
    Let us ace squarely the unlikely situation in which
hypothetically, supposedly, allegedly, there is a last prime
P. Then we could write down the complete series of
[primes 2,3,5,7,11,...,P.  Why is this so unlikely? What 
is wrong with it?  Can we point out anything  that is 
definitely wrong?  indeed, we can.  We can construct the 
number

        Q = ) ( 2 * 3 * 5 * 5 * 11 * ... * P ) + 1

This number q is greater than P and therefor, allegedly, 
Q cannot be prime. Consequently, q must be divisible by 
a prime. Now all primes at our disposal are, supposedly,
the numbers, 2,3,5, ..., P but Q, divided by 
any of these numbers, leaves the rest 1; and so Q is not 
divisible by any of the primes  mentioned which 
are hypothetically, all the primes. Now, there is something
that is definitely wrong; Q must either be a prime or it
must be divisible by some prime. Starting from the sup
position that there is a last prime P we have been led to 
a manifest absurdity. How can we explain this? Our
original supposition must be wrong; there cannot be a 
last prime P. And so we have succeeded  in proving that
the series of prime numbers never ends.
     Our proof is a typical indirect proof. (it is a famous 
proof too, due to Euclid; see Proposition 20 of Book IX 
of the Elements.)
    W43e have established our theorem (that the series of 
primes ever ends) by disproving its contradictory opposite
(that the series of primes ends somewhere) which
we have disproved by reducing form it a manifest absurdity.
Thus we have combined indirect proof with
"Reductio ad absurdum'; this combination is also very
typical.
4.objections. The procedures we have been studying 
encountered considerable opposition. Many objections
have bee raised which are, possibly, only various forms
of the same fundamental objection. We discuss here a 
"practical " form of the objection, which is on our level,
To find a not-obvious proof is a considerable intellectual
achievement but to learn such a proof, or even to 
understand it throughly costs a certain  amount of
mental effort. Naturally enough, we wish to retain some
benefit from our effort, and, of course, what we retain
in our memory should be true and correct and not false
or absurd.
     But seems difficult to retain something true from a 
"Reductio ad Absurdum. " The procedure starts from a 
false assumption and devices from it consequences which
are equally, but perhaps more visibly, false till it reaches
a last consequence which is manifestly false. If we do not 
wish to store falsehoods in our memory we should forget 
everything as quickly as possible which is , however, not 
feasible because all points must be remembered sharply
and correctly during our study of proof.
    The objection to indirect proofs can be now stated
very briefly. Listening to such a proof, we are obliged to
focus our attention all the time upon a false assumption
which we should forget and not upon the true theorem
which we should retain.
    If we wish to judge correctly of the merits of these
objections, we should distinguished between two uses of 
the "Reductio ad Absurdum" as a tool of research and as
a means of exposition, and make the same distinction 
concerning the indirect proof.
    It must be confessed that "reductio ad abusrdum": as a means of
exposition is not an unmixed blessing. Such a 
"reductio" especially if it is a long, may become very painful 
indeed for the reader or listener. All the derivations
which we may examine in successes are correct but all the 
situations which we have to face are impossible. Even 
the verbal expression may become tedious if it insists, was 
it should, john emphasising that everything is based on an 
initial assumption; the words hypothetically " supposedly, 
"allegedly" must recur incessantly, or some
other device must be applied continually.  We wish to 
reject and forget the situation as impossible but we have
to retain and reexamine it as the basis for the next step
and this inner discord may become unbearable in the 
long run.
    It would be foolish to repudiate "reductio ad 
aburudum as a tool of discovery." It may present itself
naturally and bring  a decision when all other means
seem to be exausted as the foregoing example may
show. 
    We need some experience to perceive that there is no 
essential opposition between our two contentions. Experience
shows that usually there is little difficulty in 
converting an indirect proof into a direct proof, or in
rearranging a proof found by a long "reductio ad absurdum"
into a more pleasant form from which the
"reductio ad absurdum" may even completely disappear
for, after due preparation, it may be compressed into
a few striking sentences).
   In short,  we wish, to make full use of our capacities
we should be familiar both with "reductio ad absurdum"
and width indirect proof. When , however, we have succeeded
in deriving a result by either of these methods 
we should not fail to look back at the solution and ask; 
Can you derive the result differently?

   Let us illustrate by examples what we have said
5. Rearranging a reductio ad absurdum. We look back
at the reasoning presented under 1. The reductio ad 
abusrdum started from a situation which e, eventually
turned out to be impossible. Let us however care out a 
part of the argument which is independent of the initial
false assumption and contains positive information. Reconsidering
 what we have done, we may perceive that
this much is doubtless true ; if a set of numbers with one 
or two digits is written so that each of the ten figures
occurs just once, then the sum of the set is of the form
10t + (45-t)   = 9 (t + 5)

    Thus, this sum is divisible by 9.  The proposed puzzle 
demands however that this sum should be 100. Is this 
possible? No, it is not, since 010 is not divisible by 9.
    The "reductio ad absurdum" which led to the discovery
of the argument vanished from our new presentation.
b    By the way, a reader acquainted with the procedure 
of "casting out nines" can see now the whole argument 
at a glance.
     6. Converting an indirect proof We look back at the 
reasoning presented under 3. Reconsidering carefully
what we have done, we may find elements of the argument
which are independent of any false assumption, yet
the best clue comes from a reconsideration of the
meaning of the original problem itself.
    What do we mean by saying that the series of primes
never ends? Evidently, just this: when we have ascertained
any finite set of primes as 2,3,5,7,11,...P where P is the 
last prime hitherto found, there is always
one more prime. Thus what must we do to prove the 
existence of an infinitely of primes? We have to point out 
a way of finding a prime different from all primes
hitherto found. Thus our "problem to prove" is in act reduced
to a problem to find"; being 
     Having restated your original problem in this new form
we have taken the main step. It is relatively easy now to 
see how to use the essential past of our former argument
for the new purpose in fact the number
q = 2*#*5*7*11,,..P))+1

is certainly divisible by a prime. Let us take this is the
idea - any prime divisor of Q (for instance, the smallest one) 
for N ((of course if Q happens to be a prime, then N= Q) 
obviously Q divided by any of the primes, 2,3, 5, ...,P 
e,leaves the remainder 1 and therefor none of these numbers
can be N which is a divisor of Q but that
is all we need; N is a  prime, and different from all
hitherto found primes 2,3,5,7,11,...,P

this proof gives a definite procedure of prolonging 
again and again the series of primes, without limit. Nothing
is indirect in it, no impossible situation needs to be 
considered. Yet fundamentally, it is the same as our
former indirect proof which we have succeeded in 
converting.