Download Chapter 1: Mathematical Processes

Chapter 1: Mathematical Processes
1.2 Reasoning Mathematically
1.2.1. Types of Reasoning
1.2.1.1. Several different types of reasoning
1.2.1.1.1. inductive
1.2.1.1.2. deductive
1.2.1.1.3. proportional
1.2.1.1.4. spatial
1.2.1.2. Sometimes used individually
1.2.1.3. Sometimes used in combination
1.2.2. Inductive Reasoning and Patterns
1.2.2.1. Description of inductive reasoning: Inductive reasoning involves the use of
information from specific examples to draw a general conclusion. The general
conclusion drawn is called a generalization.
1.2.2.2. An observed pattern from a finite number of trials is concluded to always work
that way
1.2.2.3. Procedure for using the inductive reasoning process (p. 15)
1.2.2.3.1. Check several examples of a possible relationship
1.2.2.3.2. Observe that the relationship is true for every example you checked
1.2.2.3.3. Conclude that the relationship is probably true for all other examples and
state a generalization
1.2.2.4. Description of counterexample: A counter example is an example that shows a
generalization to be false.
1.2.2.5. While an INFINITE number of examples CANNOT PROVE a generalization is
true, ONE counterexample CAN PROVE a generalization is false.
1.2.2.6. Patterns
1.2.2.6.1. Mathematics sometimes defined as study of patterns
1.2.2.6.2. sometimes order not important to see pattern
1.2.2.6.3. sometimes order is critical to finding pattern
1.2.2.7. Sequences
1.2.2.7.1. Sequence: a pattern involving an ordered arrangement of numbers,
geometric figures, letters, or other entities
1.2.2.7.2. Terms of a sequence: the individual numbers, geometric figures, or letters
that make up the sequence. Terms are usually separated by a comma
1.2.2.7.3. Numerical sequences are classified by the relationship between consecutive
terms
1.2.2.7.4. Arithmetic sequence: a numerical sequence where each successive term is
obtained by adding a fixed number to the preceding term
1.2.2.7.5. Common difference: the fixed number used to find terms in an arithmetic
sequence
1.2.2.7.6. Geometric sequence: a numerical sequence where each successive term is
obtained by multiplying a fixed number by the preceding term
1.2.2.7.7. Common ratio: the fixed number used to find terms in a geometric sequence
1.2.2.7.8. There are many sequences that are not arithmetic nor geometric
1.2.2.7.8.1.
1, 3, 6, 10, … (triangular numbers)
1.2.2.7.8.2.
2, 6, 12, 20, … (rectangular numbers)
1.2.2.7.8.3.
1, 4, 9, 16, … (square numbers)
1.2.2.7.9. Generalizing an arithmetic sequence
1.2.2.7.9.1.
a, a + d, a + 2d, a + 3d, …
Term #
1
2
3
4
5
6
7
…
n
Term
a
a+d
a + 2d
a + 3d
a + 4d
a + 5d
a + 6d
…
a + (n - 1)d
Term #
1
2
3
4
5
6
7
…
n
Term
a
ar
ar^2
ar^3
ar^4
ar^5
ar^6
…
ar^(n-1)
1.2.2.7.9.2.
1.2.2.7.9.3.
The nth term of an arithmetic sequence is a + (n – 1)d
1.2.2.7.10. Generalizing a geometric sequence
1.2.2.7.10.1.
a, ar, ar2, ar3, …
1.2.2.7.10.2.
1.2.2.7.10.3.
The nth term of an geometric sequence is ar(n – 1)
1.2.3. Deductive Reasoning
1.2.3.1. used for drawing logical conclusions
1.2.3.2. presenting convincing arguments
1.2.3.3. presenting proofs
1.2.3.4. Statements and Negations
1.2.3.4.1. Statement: A sentence that is either true or false but not both.
1.2.3.4.2. Negation: a statement that has the opposite truth value – negation of p is “not
p” written ~p. If p is true, then ~p is false. If p is false, then ~p is true.
1.2.3.5. If-then statements
1.2.3.5.1. Called conditional statements
1.2.3.5.2. If part is called the hypothesis or antecedent
1.2.3.5.3. Then part is called the conclusion or the consequent
1.2.3.6. Rules of logic
1.2.3.6.1. Rule A: used when a conditional and its antecedent are both true p. 27
1.2.3.6.2. Rule B: used when a conditional is true and its consequent is false p. 27
1.2.3.6.3. Logic Rule A – affirming the hypothesis or affirming the antecedent
1.2.3.6.4. Logic Rule B – denying the conclusion or denying the consequent
1.2.3.6.5. other rules of logic are also used
1.2.3.6.6. Description of deductive reasoning: Deductive reasoning involves drawing
conclusions from given true statements using rules of logic
1.2.3.6.7. Procedure for using the deductive reasoning process
1.2.3.6.7.1.
Start with a true statement, often in if-then form
1.2.3.6.7.2.
Note given information about the truth or falsity of the hypothesis or
the conclusion (antecedent or consequent)
1.2.3.6.7.3.
Use a rule of logic to determine the truth or falsity of the hypothesis
or the conclusion (antecedent or consequent)
1.2.3.6.8. Sometimes rules are used in sequence
1.2.3.7. Rules for Conditional Statements
1.2.3.7.1. Definition of Converse Statement: The converse of a conditional is formed
by interchanging the hypothesis and conclusion. That is, the converse of p → q
is q → p.
1.2.3.7.2. Definition of the Inverse of a Statement: The inverse of a conditional
statement is formed by negating both the hypothesis and the conclusion. That
is, the inverse of p → q is ~p → ~q.
1.2.3.7.3. Definition of Contrapositive Statement: The contrapositive of a conditional
is formed by interchanging the hypothesis and conclusion, then negating both.
That is, the converse of p → q is ~q → ~p.
1.2.4. Problems and Exercises p. 35-39
1.2.4.1. Home work: 4-7, 9, 11-13, 18, 21, 25, 27, 29-31, 35-37, 41, 43, 44, 46, 48-51, 531.2.5. 62 odd, 72