Survey of Mathematical Ideas
Math 100
Chapter 1
John Rosson
Thursday January 18
Chapter 1
The Art of Problem Solving
1.
2.
3.
4.
Solving Problems by Inductive Reasoning
Number Patterns
Strategies for Problem Solving
Calculating, Estimating and Reading
Graphs
Inductive Reasoning
Inductive 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 Example
The following is the sequence of squares of whole numbers:
1 4
9 16
25
36
49
64
81 100
Look at consecutive differences:
4 1 9  4 16  9 25 16

3 5
36  25
49  36 64  49 81  64 100  81
7 9 11 13 15 17 19
Conjecture: Starting with 1 you get the sequence of squares by
adding
the sequence of odd numbers.

1 1
4  1 3
9  1 3  5
16  1  3  5  7
This conjecture turns out to be true.
etc.
Inductive Reasoning; An Example
A 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 Reasoning
Deductive reasoning is characterized by applying general
principles to specific examples.
Specific
General
Premise
Logical Argument
Calculation
Proof
Conclusion
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 Example
In calculations the general principles are the rules of
arithmetic and algebra.
Theorem: If 3 x  6 then x  2
Proof:
3 x  6 Premise
1
1
 3x   6
3
3
3x 6

3
3
1 x  2
x  2 Conclusion
Thus the theorem is
true.
Deductive Reasoning; An Example
h
Theorem: If the legs of a right triangle are 2 and
3 respectively then the hypotenuse h  13 .
3
2
2
2
Proof: a  b  c
32  2 2  h 2
2
Pythagorean Theorem
Premise

9  4  h2
13  h 2
 13  h
13  h

Conclusion
Inductive and Deductive Reasoning
We used inductive reasoning to get:
Conjecture: Starting with 1 you get the sequence of squares by
adding the sequence of odd numbers. That is
n2  1  3  5      (2n 1)
We now use deductive reasoning to get:
Theorem: For every positive whole number n
n2  1  3  5      (2n 1)
Inductive and Deductive Reasoning
Theorem: For every positive whole number n
n2  1  3  5      (2n 1)
Proof:
12  2 1  1  1
(n  1) 2  1  3  5      (2(n  1)  1)
Premise
n 2  2n  1  1  3  5      (2n  3)
n 2  1  3  5      (2n  3)  (2n  1) Conclusion
Inductive and Deductive Reasoning
Summary
• 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 3992
55 163 355 655 1087 1675
108 192 300 432 588
84 108 132
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, 2.2
Read 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