Chapter 1 – REAL NUMBERS

Natural Numbers – the counting numbers

N =

Whole Numbers – the natural numbers with zero included

W =

Integers – positive naturals, zero, and negative naturals

Z =

Each integer is made up of two parts:

  1. ______- the distance between the number and zero
  1. ______- a designation of which side of zero the number falls on

Place the integers 0, -3, 2, 1, and –5 on the number line.

DEF: ______- a letter used to represent a number.

INEQUALITIES

If a and b are two numbers and a is to the left of b on the number line, we say a is ______b (in symbols this is written )

If a and b are two numbers and a is to the right of b on the number line, we say that a is ______b (in symbols this is written )

To remember the orientation of the inequality symbols:

Examples of inequalities:

What symbol goes between5- 4812

-3-70-3

 means “less than or equal to”  means “greater than or equal to

Exercise:

Let x  {1, 3, 5, 7, 9}

List all elements of the set such that x  5Answer: ______

Let y  {-5, -3, 0, 1, 6, 19}

List all elements of the set such that y  0Answer: ______

DEF: Two numbers which have the same magnitude but lie on opposite sides of zero on the number line are called ______or, more commonly, ______.

What is the opposite of :

Number

/ Opposite
15
0
-2.8

The negative symbol can be thought of as an opposite symbol:

-(9) means the opposite of 9 and equals ______

- (-3) means the opposite of –3 and equals ______

DEF: The ______of a number refers to the distance between the number and zero on the number line. The notation is two vertical bars.

Absolute value is always a ______number.

Examples:

1.2 ADDITION AND SUBTRACTION OF INTEGERS

To add two integers:

  • If the integers have the same sign:

______& ______.

Examples:

  • If the integers have opposite signs:

______& ______.

Examples:

To subtract two integers:

  • Change subtraction to ______!!!!
  • Follow rules for addition.

Examples:

1.3 MULTIPLICATION AND DIVISION OF INTEGERS

To Multiply or Divide Integers

  • If the integers have the same sign, multiply or divide the whole number parts & the sign will be ______
  • If the integers have different signs, multiply or divide the whole number parts & the sign will be ______

Examples:

NOTE: These all are equivalent.

Properties of Zero in Division:

  • for any a  0
  • ______

WHY?

Examples:

1.4 OPERATIONS WITH RATIONAL NUMBERS

Rational Numbers – any number that can be expressed as the quotient or “ratio” of two integers is called a rational number

Rational Numbers are more commonly referred to as ______.

Q =

Every integer is a rational number, where the denominator is equal to ____

A fraction is in simplestform when the numerator (top) and denominator (bottom) share no common factors.

The Greatest Common Factor of two numbers is found by rewriting each number as a product of its prime factors, then setting the GCF equal to the product of those factors found on each factor list.

Examples:

What is the GCF of 12 and 30?

What is the GCF of 54 and 36?

To Simplify a Fraction: Divide both the numerator and denominator by their greatest common factor.

Examples:

Simplify

Simplify

To Rewrite a Fraction as a Decimal:

  • Change to a long division problem (numerator goes “inside” the box)
  • Divide until the decimal ______or ______

Examples:

Rewrite as a decimal

Rewrite as a decimal

Rewrite as a decimal

To Rewrite a Fraction as a Percent:

  • Multiply the fraction by 100% and rewrite the resulting improper fraction as a mixed number in lowest terms.

Examples:

Rewrite as a percent

Rewrite as a percent

To Multiply Fractions: Multiply numerators, multiply denominators

Examples:

To Divide Fractions: Flip the second, and switch to multiplication

Examples:

To Rewrite Percents as Fractions: Remove the percent sign and multiply by

Examples:

To Rewrite Percents as Decimals: Remove the percent sign and multiply by 0.01.

Examples:

To add or subtract fractions:

  • Get a common denominator
  • Add or subtract the numerators and keep the common denominator

NEVER ADD FRACTIONS BY ADDING TOPS AND ADDING BOTTOMS!!!

Common Denominators can be found by generating the Least Common Multiple of the two original denominators.

To Create the Least Common Multiple of two numbers, rewrite each number as a product of its prime factors. The LCM will be the product of each individual factor the largest number of times it appears on any one list.

Examples:

Find the LCM of 20 and 30

Find the LCM of 12 and 48

Find the LCM of 6 and 35

Examples of Addition or Subtraction of Fractions:

1.5 ORDER OF OPERATIONS

Exponents are used to represent a repeated self-multiplication

 Here 3 is the ______and 4 is the ______

A positive base will always simplify as a positive result regardless of the exponent.

A negative base will simplify as a:

  • ______result if the exponent is ______
  • ______result if the exponent is ______

Examples:

ORDER OF OPERATIONS:

To help remember, use the saying Please Excuse My Dear Aunt Sally

P = parenthesis - this includes anything that looks like parenthesis (such as absolute value, and the tops and bottoms of fractions)

E = exponent- this includes any square roots

M = multiply- do whichever of these comes first as you read left to right

D = divide(they carry equal precedence)

A = add- do whichever of these comes first as you read left to right

S = subtract(they carry equal precedence)

Examples of Order of Operation Problems: