Survey of Mathematical IdeasMath 100Chapter 1

John Rosson

Thursday January 18

Chapter 1The Art of Problem Solving

1. Solving Problems by Inductive Reasoning

2. Number Patterns

3. Strategies for Problem Solving

4. Calculating, Estimating and Reading Graphs

Inductive ReasoningInductive reasoning is characterized by drawing a general conclusion (making a conjecture) from repeated observation of specific examples. The conjecture may or may not be true.

Specific General

Inductive Reasoning• Last year was a bad hurricane season and so was

this year. Therefore next year will be a bad hurricane season.

• In decimals, the number starts 3.1415. Therefore the next two digits in are 16.

• The sun has come up every day for as long as anyone remembers. Therefore it will come up tomorrow.

• The Cubs will not win the World Series.• Any reasoning of a general scientific or

philosophical law or principle directly from data or experience is inductive reasoning.

Inductive Reasoning; An ExampleThe following is the sequence of squares of whole numbers:

Look at consecutive differences:

Conjecture: Starting with 1 you get the sequence of squares by adding the sequence of odd numbers.

753116

5319

314

11

+++=++=

+==

This conjecture turns out to be true.

etc.

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4 −1 9 − 4 16 − 9 25 −16 36 − 25 49 − 36 64 − 49 81− 64 100 − 81K

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1 4 9 16 25 36 49 64 81 100L

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3 5 7 9 11 13 15 17 19 L

Inductive Reasoning; An ExampleA prime number is a positive whole number larger than 1 that is evenly divisible only by 1 and itself.

Here are the positive whole numbers with the primes highlighted.1, 2, 3, 4, 5, 6, 7, 8, …..

Conjecture: All odd numbers greater than 1 are prime.

This conjecture turns out to be false since the next odd number 9 is evenly divisible by 3 and therefore not prime.

Since 9 shows the conjecture to be false, 9 is called a counterexample to the conjecture.

Deductive ReasoningDeductive reasoning is characterized by applying general principles to specific examples.

SpecificGeneral

ConclusionPremise

Logical ArgumentCalculation

Proof

Deductive Reasoning.

• It is raining or cloudy. Therefore it is not sunny and dry.

• If a miracle happens the Cubs will win the World Series.

• All mathematical facts (theorems rather than conjectures) are the result of deductive reasoning.

• Any reasoning of a scientific or philosophical law or principle directly from first principles or assumptions.

Deductive Reasoning; An ExampleIn calculations the general principles are the rules of arithmetic and algebra.

Theorem: If then63 =x

Proof:

2

213

6

3

3

63

13

3

1

63

==⋅

=

⋅=⋅

=

xx

x

x

x Premise

Conclusion

Thus the theorem is true.

2=x

Deductive Reasoning; An Example

2

3h

Theorem: If the legs of a right triangle are 2 and 3 respectively then the hypotenuse .

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h= 13

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a2 + b2 = c2

32 + 22 = h2

9 + 4 = h2

13 = h2

± 13 = h

13 = h

Proof: Pythagorean Theorem

Premise

Conclusion

Inductive and Deductive ReasoningWe used inductive reasoning to get:

Conjecture: Starting with 1 you get the sequence of squares by adding the sequence of odd numbers. That is

)12(5312 −+⋅⋅⋅+++= nn

We now use deductive reasoning to get:

Theorem: For every positive whole number n

)12(5312 −+⋅⋅⋅+++= nn

Inductive and Deductive Reasoning

)12()32(531

)32(53112

)1)1(2(531)1(

11121

2

2

2

2

−+−+⋅⋅⋅+++=

−+⋅⋅⋅+++=+−

−−+⋅⋅⋅+++=−

=−⋅=

nnn

nnn

nn

Theorem: For every positive whole number n

)12(5312 −+⋅⋅⋅+++= nn

Proof:

Premise

Conclusion

Inductive and Deductive ReasoningSummary

• First use inductive reasoning to develop a conjecture.

• Second use deductive reasoning to validate or prove the conjecture.

Number Patterns: Successive Differences

Problem: Given a sequence of numbers, say 2, 57, 220, 575, 1230, 2317,….determine (a good guess for) the next number in the sequence.

This is an inductive form of argument since there is no guarantee what the next number will be.

Number Patterns: Successive Differences

The method of successive difference tries to find a pattern in a sequence by taking successive differences until a pattern is found and then working backwards.

2 57 220 575 1230 2317 ….55 163 355 655 1087 ….108 192 300 432 ….

84 108 132 ….24 24 ….

Number Patterns: Successive Differences

Fill in the obvious pattern and work backwards by adding.

2 57 220 575 1230 2317 399255 163 355 655 1087 1675

108 192 300 432 58884 108 132 156

24 24 24

The method of successive differences predicts 3992 to be the next number in the sequence.

Number Patterns: Successive Differences

The method of successive differences is not always helpful. Consider

1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 ….1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 ….

0 1 1 2 3 5 8 13 21 34 55 89 144 233 ….1 0 1 1 2 3 5 8 13 21 34 55 89 ….

Since the sequence reproduces itself after applying successive differences, the method can give us no simplification.

Assignments 1.2, 1.3, 1.4, 2.1, 2.2Read Section 1.2 Due January 23

Exercises p. 16 1-5, 15, 21, 33, 34

Read Section 1.3 Due January 23

Exercises p. 25 1, 10, 19, 21, 27, 40

Read Section 1.4 Due January 23

Exercises p. 35 10-15

Read Section 2.1 Due January 25

Exercises p. 54 1-8, 21-28, 33-40, 41-50, and 67-76

Read Section 2.2 Due January 25

Exercises p. 61 1-6, 23-42, 44, 49-54