1. Real Numbers, Algebra, and Problem Solving
Chapter 1
Real Numbers, Algebra,
and Problem Solving

2. Real Number Sets Rational Numbers Irrational Numbers
Discuss the difference between equal and approx (including notation)

3. Rational Numbers
Integers
Whole Numbers
Natural Numbers

4. Real Numbers

5. Scientific Notation Standard Form Scientific Notation 345,000,000
1,234,000
4.5 x 104
3.7 x 10-3

6. Computing in Scientific Notation
Evaluate each expression
(4.5 x 104)(2.7 x 102)
(1.7 x 10-2)(3.1 x 105)
(6.3 x 104)
(2.1 x 102)
Discuss calculator E

7. Computing with a calculator
Use a calculator to evaluate each expression. Round to the nearest thousandth a. b. c. d.
e. 6! – e2

8. Axioms for real numbers
Closure
For every real number a and b,
a + b is a real number
ab is a real number
Commutative
a + b = b + a
ab = ba
Associative
For any numbers a, b and c,
a + (b + c) = (a + b) + c
a(bc) = (ab)c
Distributive Property of Multiplication wrt Addition
For every real number a, b, and c,
a(b + c) = ab + ac
Identity
For any number a,
a + 0 = a
1a = a
Inverse
For each number a, there is one and only one inverse, b, such that a + b = 0
For each nonzero number a, there is one and only one inverse, b, such that ab = 1

9. Properties of equality
For any real numbers a, b, and c,
Reflexive Property
a = a
Symmetric Property
If a = b, then b = a
Transitive Property
If a = b and b = c, then a = c

10. Theorems Multiplicative Property of 0 Additive Inverses of Products
For any real number a, a(0) = 0
Additive Inverses of Products
For any real numbers a and b,
-(ab) = a(-b)
Products of Additive Inverses
(-a)(-b) = ab