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Beginning and Intermediate Algebra
Chapter 1 - The Real Number System and
Geometry
Section 1.1 Review of Fractions
Section 1.1 Review of Fractions
What is a fraction?
A fraction is a number in the form a/b
where b not equal to 0, a is called the
numerator, and b is the denominator.
Example 1
Factors
Ex. 2
A fraction is in lowest terms when the numerator and
denominator have no common
factors except 1. Before discussing how to write a
fraction in lowest terms, we need to
know about factors.
Consider the number 12.
product factor factor
12
= 3 x
4
3 and 4 are factors of 12. (When we use the term
factors, we mean natural numbers.)
Multiplying 3 and 4 results in 12. 12 is the product.
Prime Numbers
We can also write 12 as a product of prime numbers.
A prime number is a natural number whose only factors
are 1 and itself. (The factors are natural
numbers.)
Ex. 3 Definition
Is 7 a prime number?
Solution
Yes. The only way to write 7 as a product of natural
numbers is 1 x 7.
Composite Numbers
A composite number is a natural number with factors other
than 1 and itself. Therefore, if a natural
number is not prime, it is composite.
To perform various operations in arithmetic and algebra, it
is helpful to write a number as the product of its prime
factors. This is called finding the prime factorization of
a number. We can use a factor tree to help us find the
prime factorization of a number.
Ex. Write 12 as the product of its prime factors.
4
Solution
Use a factor tree.
12
3 4
4 is not prime, so break it down into the product of
two factors
2 2 Final factorization is 3 x 2 x 2
Another example of composite number
Prime factorization: 120 =
2*2*2*3*5
Writing a fraction in lowest terms
Let’s return to writing a fraction in lowest
terms.
Write the fraction in lowest terms.
a) 4/6
Solution
Write 4 and 6 as the product of their
primes, and divide out common factors.
Write 4 and 6 as the product of their prime
4 22
factors.

6
23
Divide out common factor.
4 22 2


6 23 3
Since 2 and 3 have no common factors
other than 1,
the fraction is in lowest terms
Multiply and Divide Fractions
Multiplying Fractions
To multiply fractions, we multiply the numerators and
multiply the denominators. That is,
Ex. 7
a c ac
 
if b and d  0
b d bd
Multiply. Write the answer in lowest terms.
3 7 3  7 21
 

21 and 32 have no common factors
8 4 8  4 32
so the answer is in lowest ter ms
Dividing Fractions
To divide fractions, we must define a reciprocal.
The reciprocal of a number a/b, is b/a since their product
is 1. That is, a nonzero number times its reciprocal
equals 1.
For example, the reciprocal of 5/9 is 9/5 since their product
is 1. inition
Division of fractions: Let a, b, c, and d represent numbers
so that b, c, and d do not equal zero. Then, a
c a d
b

d

b
To perform division involving fractions, multiply the first
fraction by the reciprocal of the
second.

c
Examples of dividing fractions
Adding and Subtracting Fractions
Adding and Subtracting FractionsDefinition
Example 9
Example of adding fractions with Mixed
Numbers
Ex. 10
Adding or Subtracting Fractions with Unlike
Denominators
Ex. 11
Obtaining an Equivalent Fraction
Ex. 12
Adding or Subtracting with Unlike
Denominators
Ex. 13
Adding or Subtracting with Unlike
Denominators
Section 1.2 Exponents and Order of Operations
Ex. 1
More examples of Exponents
Ex.2
Order of Operations
We will start this topic with an example
Ex. 3
The correct answer to the above problem
is 58.
Order of Operations Problems
Ex. 4
Section 1.3 Geometry Review- Definitions
More Geometry Definitions
More Geometry Definitions
Example of complement
Ex. 1
Vertical Angles and Parallel and
Perpendicular Lines
Equilateral, Isosceles, and Scalene
Triangles
Example of Triangle Problem
Ex. 2
Area, Perimeter and Circumference Formulas
Circle Formulas
Perimeter and Area Problems
Ex. 3
More perimeter and Area Problems
Ex. 4
Circumference and area of a circle solution
Perimeter and Area of Polygons
Ex. 5
Volume Formulas
Example of Finding Volumes
Solution to Volume Problem
Application (Oil Drum)
Ex. 7
Sect. 1.4 Sets of Numbers and Absolute Value
Sets of Numbers
Ex. 1
Ex. 2
Rational Numbers
Examples of Rational Numbers
Ex. 3
Rational Numbers
Irrational Numbers
Ex. 4
Real Numbers
Examples of Real Numbers
Ex. 5
Comparing Numbers Using Inequality Symbols
Examples of Inequalities
Ex. 6
Ex. 7
Additive Inverse
Example of Additive Inverse
Ex. 8
Absolute Value
Ex. 9
Sect. 1.5 Addition and Subtraction of Real
Numbers
Adding Numbers with the Same Sign
Ex. 2
Adding Real Numbers with Different Signs
Subtracting Real Numbers
Definition
Ex. 4
Solving Applied Problems Involving
Addition and Subtraction
Ex. 5
Applying the Order of Operations.
Ex. 6
Translating English Expressions to
Mathematical Expressions
Ex. 7
Solutions to Translating Expressions
Sect. 1.6 Multiplication and Division of Real
Numbers
Example of product of negative and
positive
Ex. 1
Multiplying Negative Real Numbers
Multiplying Real Numbers
Ex. 2
Ans. a.) 21 b.) -20 c.) 2/15 d.) -60
Evaluating Exponential Expressions
Ex. 3
Identifying the Base
A negative number to an even power is positive, as we
have seen. The above situation would mean that the
base is negative. What is the base?
Ex. 4
Dividing Signed Numbers
Ex. 5
Ans. a.) -8 b.) 1/16 c.) 6 d.) -3/8
Applying the Order of Operations to Real
Numbers
Ex. 6
Translate English Expressions to
Mathematical Expressions
Ex. 7
Ans.
Sect. 1.7 Algebraic Expressions and
Properties of Real Numbers
Algebraic Expressions
Evaluating Algebraic Expressions
Ex. 2
= 17
Another example
Ex. 3
Commutative Properties
Ex. 4
Associative Properties
Ex. 5
Another Example of Associative Property
Ex. 6
Identity and Inverse Properties
Inverse Properties
Example of Inverse Properties
Ex. 7
Distributive
Properties
Ex. 8
Another example of Using the Distributive
Property
Ex. 9