Making Mathematical Induction Meaningful

Christian R. HirschDepartment of MathematicsWestern Michigan UniversityKalamazoo, Michigan 49008

Some work with mathematical induction is quite generally consideredto be a part of the content of the algebra courses in our secondaryschools. This is reasonable since mathematical induction is one ofthe main forms of proof used in mathematics, perhaps second onlyto indirect proof in the number of instances where it is applied. Itis safe to say, however, that many students continue to find mathemat-ical induction to be one of the most difficult topics in the algebraprogram. Though some of the newer texts give a more lucid explanationthan the standard presentation of "pre-reform" years, neverthelessthe mathematical and pedagogical approach is most traditional. Spe-cifically this usually involves postulating the Principle of MathematicalInduction; rationalizing its acceptance by appeal to models involvinga chain reaction phenomena (e.g. a row of dominoes standing onend) or to "an ordinary ladder with an indefinite number of steps";and finally providing several worked-out examples.Malcom (1974) has suggested that the Well-Ordering Property has

a distinct pedagogical advantage over the Principle of MathematicalInduction and perhaps should be considered as an alternative. Onthe other hand, recent research by Ward (1972) indicates that studentsperform significantly better using mathematical induction than theydo using the Well-Ordering Property. It would seem therefore thatan instructional approach which would make mathematical inductionmeaningful for our students may have the greatest payoff. The purposeof this present article is to describe a discovery approach which hasbeen successfully used with both eleventh and twelfth grade studentsto give meaning to the Principle of Mathematical Induction.

Mathematical induction is a tool for proving the validity of statementsconcerning the positive integers. Hence, it would seem the place tobegin is with a closer study of the set of positive integers and itsvarious subsets. A consideration of examples such as {2, 4, 6, 8,...}, {3, 5, 7, 11}, {18, 19, 20, 21, ...}, {832, 833, 834, 835, ...},{12, 15, 18, 21, ...}, {n\n is a positive integer and n2 > 50}, and{125, 17, 32, 16, 64} soon leads to the student discovery that onecharacteristic of these sets is that each has a "smallest number"or if ordered a "first member." The numbers being 2, 3, 18, 832,12, 8, 16 respectively. An attempt by the class to give meaning to"smallest number" or "first member" results in the formulation ofa definition similar to that given below.

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28 School Science and Mathematics

Definition a is a least member of a set S if and only if a E. S andfor each x G S, a ^ x.

Does every nonempty set of positive integers have a least member?An affirmative answer is contained in the following student generali-zation which is accepted without proof.

LEAST NUMBER PRINCIPLE Each nonempty set of positiveintegers has a least member.

Is this also a property of the positive real numbers? No, for example,the {x\x > 3} has no least member.There are also characteristics distinguishing the above sets of positive

integers. For example, most students immediately note that somesets are finite, whereas others are infinite. Upon a closer inspectionof the infinite sets students will observe that in the case of {18,19, 20, 21, ...}, {832, 833, 834, 835, ...} or {n\n is a positive integerand n2 > 50}, each element after the "first" is obtained by adding1 to the preceding element. We may think of each of these setsgenerating its successive members. We abstract this notion and identifysets possessing this property as hereditary sets.

Definition A set T of numbers is hereditary if and only if x 4- 1 G Twhenever x E T.

Here the terminology is descriptive and most students are attractedby it. Other examples of hereditary sets are0, {integers}, and {positiverational numbers}. Is {1,2, 3,4} or {integral multiples of 5} hereditary?While reinforcing these preliminary concepts with appropriate prac-

tice exercises we may simultaneously begin to motivate the Principleof Mathematical Induction. Convincing students of the need for sucha principle is relatively easy, but it is often either overlooked orbypassed in the cause of expediency. The following is a sample ofconjectures (some true, some false) that students can be guided todiscover and formulate.

1. Every set of n elements has 2" subsets.2. The number of dominoes in a set which runs from double zero to double n

(n + l)(n + 2)is ��������.

23. For each positive integer n, a � b is a factor of a" � £>".4. 10" > n\ for each positive integer n.5. If n =s 2 students are entered in a chess tournament and if each plays one

game with each of the other students, then the number of games played isn(n- 1)

2n(n - 3)

6. For n >. 3, the number of diagonals of an n-sided convex polygon is �����.2

Making Mathematical Induction Meaningful 29

7. For each positive integer n, 1 + 3 + 5 + ... + {In � 1) = n2.8. The maximum number of disjoint regions formed when n points on a circle

are joined in pairs is 2""’.9. If z = r(cos 6 4- ( sin 6) then for each positive integer n, z" = r"(cos /i6 4- i

sin n6). (DeMoivre’s Theorem).10. If a > -1, then for each positive integer n, (1 + n)"> 1 + na. (Bernoulli’s

inequality).

If students are given the opportunity to make discoveries, theywill soon come to appreciate the need for some method of verification.How would one verify a conjecture such as 10?

Let T be the set of all positive integers for which the conjectureis false. If TT" j3, then by the Least Number Principle, Thas a leastmember�say p. Since this conjecture is true for n = 1, it must bethe case that p ^ 2 or p - 1 ^ 1. Also since p - 1 < p, where pis the least integer for which the conjecture is false, it follows thatthe conjecture is true for p � 1.That is, if a > -1, then (1 + rt)p--’ ^ 1 + (p - \)a. Since a > -1,

1 + a> 0

so (1 -h a)(\ + fl)P-1 ^ (1 + a)[\ + (p - l)o].

Hence, (1 + a)11 ^ (1 + a)[\ + (p - l)a]

^ 1 + (p- \)a + a 4- (p - \)a2

s= 1 + (p - \)a + a

^ 1 + pa

that is, the conjecture holds for p. But this is a contradiction. Thusour assumption is false. T=0 and we conclude that the statementis indeed true for each positive integer.

Students will find it instructive to use this indirect method of proofto verify a number of their conjectures. It is not long however, beforesome students will suggest using the notion of a hereditary set toprove their conjectures directly. At this point the Principle of Mathe-matical Induction can be meaningfully introduced. Moreover, the priorexperience with indirect proof tends to place the proof of the followingtheorem within the reach of most students.

Notation Let N= {1, 2, 3, 4, ...} of positive integers.

Theorem 1 (PRINCIPLE OF MATHEMATICAL INDUCTION)Let P(n) be a statement that is defined for each positive integer

n and let S = {n E N|P(/i) is true}. If

DIES

30 School Science and Mathematics

and 2) S is hereditary

then S = N.

Proof: Let P(n) be a statement that is defined for each positive inte-ger n and let S = {n E N|P(n) is true}. Suppose that 1 E S and Sis hereditary. Assume, however, S ^ N. Let Tbe the set of all posi-tive integers which do not belong to S. Since by assumption T 7^ j3,it follows by the Least Number Principle that Thas a least member�say p. Since p ^ S and 1 E S it follows that p ^ 1. So p - 1 ^ 1.Since p is the least positive integer which does not belong to S andsince p � 1 < p it follows that p � 1 E S. But this being so; since Sis hereditary (p - 1) + 1 E S, that is, p G S. But this is a contra-diction. Therefore our assumption is false. Consequently, S = N.

Let us return now to Bernoulli’s inequality and use the Principleof Mathematical Induction to prove it directly. Students should beencouraged to note the similarities between the two techniques.Let S= {n E N|(l + ^)"^ 1 + na where a > -1}.

1 E S since (1 + a)! = 1 + a ^ 1 + 1 � a.

Assume now that k E S. Hence, it follows that

(1 + a)^ 1 + ka.

Since a > -1, 1 + a > 0 so (1 + a)(\ + a)k ^ (1 + a)(\ + to); thatis

(1 + fl)^1 ^ (1 + a)(l + ka)

^ 1 + ka + a + ka2

^ 1 + ka + a

^ 1 + (fc+ l)fl.

Thus k + 1 E S and therefore S is hereditary. Consequently, itfollows by Theorem 1 that S = N, that is the conjecture is true foreach positive integer n.

Conjectures 5 and 6 suggest the following theorem whose proofis entirely similar to that of Theorem 1.

Theorem 2 (SECOND PRINCIPLE OF MATHEMATICAL INDUC-TION)

Let P(n) be a statement that is defined for each positive integern ^ k, for some k E N and let S == {n E N|P(«) is true}. If

1) k E S

Making Mathematical Induction Meaningful 31

and 2) S is hereditary

then S= {n E N|n^ k}.

Though Theorem 1 is obviously a consequence of Theorem 2,pedagogically the above ordering is preferable. The specificity ofTheorem 1 minimizes the amount of guidance the teacher must givethe class as they construct its proof. Moreover, the students’ under-standing of the proof of Theorem 1 is enhanced by requiring themto later prove Theorem 2 individually on their own.Given a statement concerning the positive integers the student now

has two techniques for establishing its validity. Neither technique,however, will establish conjectures 4 or 8. The reader is encouragedto find appropriate counterexamples. That some of the student conjec-tures were false serves to emphasize the need for proof and deductivemethods.There are a number of consequences that follow from adopting

the above approach to mathematical induction.First, it is in keeping with the current emphasis now being placed

on mathematical structure, proof and unifying concepts in the second-ary curriculum.

Secondly, mathematical induction is given logical meaning whenit is related to the more intuitively acceptable Least Number Principle.Moreover, the preliminary work with hereditary sets eliminates thefrequently raised objection by students that in proof by mathematicalinduction one assumes the truth of the proposition to begin with.(Recall that the 0 is hereditary.)

Finally, and perhaps most importantly, this approach grows outof the students’ experiences and thus mathematical induction haspsychological meaning for them.

REFERENCES

MALCOM, PAUL S. ’The Well-Ordering Property as an Alternative to MathematicalInduction." School Science and Mathematics, Vol. 74 (April 1974), 277-279.

WARD, RONALD A. "A Comparative Study of the Ability of Fourth Year High SchoolMathematics Students to Use the Principle of Mathematical Induction and the WellOrdering Principle to Prove Conjectures." (Ph.D. dissertation, The Florida StateUniversity). Dissertation Abstracts, 32B(March 1972), 5337.

1976 SSMA CONVENTIONToledo, Ohio