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Text Box: Math 1314
College Algebra

LO-1
Chapters 0.1, 0.2


Skills to Master

·       Determine the real number sets to which a given number belongs

·       Name and use the properties of real numbers

·       Graph subsets of the real numbers

·       Use inequality symbols

·       Use interval notation and convert between interval notation, inequalities, and graphs for a given subset of the real numbers

·       Write an expression containing natural number exponents as one without exponents

·       Know and use the rules of exponents

·       Know and use the order of operations

·       Evaluate expressions

·       Write numbers in scientific notation and convert between decimal notation and scientific notation


 

Number Sets


Natural Numbers {1, 2, 3, …}

Whole Numbers {0, 1, 2, …}

Integers {…, -2, -1, 0, 1, 2, …}

Rational Numbers

Irrational Numbers  is not rational}

Real Numbers {x: x is rational or x is irrational}

 

PRACTICE

Classify the following numbers:

4_________________________

-7_________________________

3/5________________________

1.4________________________

0__________________________

π_________________________


 

Number Properties

The following properties are assumed to be true for all real numbers:

·         Addition properties:

o   Closure: adding any two real numbers results in a real number

o   Associativity: for any real numbers a, b, and c,

o   Commutativity: for any real numbers a and b,

o   Identity property of zero: for any real number a,

o   Additive inverse property: for any real number , there exists a unique  such that

·         Multiplication properties:

o   Closure: multiplying any two real numbers results in a real number

o   Associativity: for any real numbers a, b, and c,

o   Commutativity: for any real numbers a and b,

o   Identity property of one: for any real number a,

o   Multiplicative inverse property: for any real number , there exists a unique  such that

·         Distributive property:

o   For any real numbers a, b, and c,  and

 


 

Examples:

Name the number property:


_____________________

_____________________

 

_________________________

 

_________________________


Inequalities

Symbol

Read as

Examples

“not equal to”

 

“is less than”

 

“is greater than”

“is less than or equal to

“is greater than or equal to”

“is approximately equal to”

 

Intervals and Interval Notation

For any real numbers a and b:


 is equivalent to

 is equivalent to

 is equivalent to

 is equivalent to

 is equivalent to

 is equivalent to

 is equivalent to

 is equivalent to



 

Natural Number Exponents

Addition of two or more of the same number can be written as a multiplication.  For example, .  With variables: .  Exponents serve a similar purpose, but for multiplication of several of the same number.  For example, .  With variables, . MAKE SURE YOU KNOW THE DIFFERENCE BETWEEN THESE FORMS OF NOTATION!  Exponents can only be affected by multiplication.

  In exponential notation, the b in the previous example is called the base and the 5 is the power to which the base is raised.  If no exponent is given, the base is understood to have the exponent 1.  That is, .  We will later see that  for all real numbers other than zero.

Rules for Exponents

For any real number x and for natural numbers m and n:


For any real number y:


 

Examples:

Simplify.


 

Order of Operations and Evaluating Expressions

When an expression includes several different operations, the operations must be performed in the following order:

1.      Parentheses (or other grouping symbols)—evaluate all expressions within parentheses

2.      Exponents—evaluate all exponent operations

3.      Multiplication/Division—perform multiplication and division operations as they occur from left to right.

4.      Addition/Subtraction—perform addition and subtraction operations as they occur from left to right.

Example: Evaluate

An the variables in an expression are assigned values, the values can be substituted in the expression.  The expression can then be evaluated.  Example:

Evaluate the given expression for x = -2, y = 3, and z = 4

Scientific Notation

Format: , where  and n is an integer.

Convert from decimal to scientific notation:

1.      Go to the decimal and move it either left or right until there is a single digit to the left of the decimal point.

2.      Write a  after the last digit and the exponent is the same as the number of places you moved the decimal.  If the movement was to the left, the exponent is positive; if to the right, the exponent is negative.

3.      Examples:




Convert from scientific notation to decimal:

1.      Go to the decimal point and move it either:

a.       The same number of spaces to the right as the exponent on the 10 if the exponent is positive

b.      The same number of spaces to the left as the exponent on the 10 if the exponent is negative

2.      Write your answer without the

3.      Examples