Using only integers, we do not have a number to describe
dividing one whole pie into

four equal pieces. The rational numbers were created to solve this problem.

A RATIONAL NUMBER is a number that can be re presented as a
ratio of two integers

**a/b **for any integer **a** and any nonzero
integer **b **

Note that “a over b” and “a divided by b” mean the same
thing, solving the “division

problem” is built into the definition of a rational number!

Note also that every integer is also a rational number
since an integer z can also be

expressed as **z/1** “z divided by **1**”.

Now that we know what a rational number is, how do we add,
subtract, multiply and

divide them?

** ADDITION and SUBTRACTION **

Adding and Subtracting Fractions :

o Find a least common denominator using method for LCM

o Change the numerators of each fraction

o Add or subtract the numerators (keep denominator unchanged)

o Reduce

**Examples:**

**MULTIPLICATION and DIVISION**

Multiplying Fractions:

o Simplify the fractions if not in lowest terms.

o Multiply the numerators of the fractions to get the new numerator.

o Multiply the denominators of the fractions to get the new denominator.

**Examples:**

**Dividing Fractions:**

o Multiply the first fraction by the reciprocal of the second

**Examples:**

POWERS – SHORTHAND for multiplication and division

Define

Rules for multiplying and dividing with exponents:

**Rules for Exponents:**

**Examples:**

Different Ways of Expressing a Rational Number

Proper and Improper Fractions

PROPER FRACTION IMPROPER FRACTION

Mixed Numbers

Decimals – Fractions meet the base ten number system

How do you turn a fraction into a decimal? Divide with
long division

Repeating Decimals

Are RATIONAL NUMBERS!!

Notation:

An example of a decimal number that goes on forever but
does not repeat:

**1.242442444244442444442444444...**

Order and Rational Numbers

The expression x < y is equivalent to saying y – x > 0.

How to compare two rational numbers:

1. In Decimal Form

0.4443252 ____ 0.4433324

3.44 ____ 3.442

1003 _____ 1030

0.332 _____ 1.0004

2. In Fraction Form

a. Express both numbers as fractions – proper or improper

b. Find the common denominator

c. Express both fractions as equivalent fractions over the common

denominator.

d. Compare the numerators if and only if a <
b.

Practice:

Summary: We now have the rational numbers, and we can
perform all four operations

with them and get a rational number answer. So, we have all the numbers we
need!!

OR DO WE???

Problem 1: Given a square with sides of length one inch,
how long is the diagonal?

Problem 2: Find the ratio of the circumference of a circle
to its diameter.

Neither of there two problems have answers that are
rational numbers!!

We need more numbers to describe REAL PHYSICAL LENGTHS.

Irrational Numbers are real numbers (real points on the
number line ) that CANNOT be

expressed as a ratio of two integers (fraction).

Examples: pi, the square root of 2, ..

We will work with square roots in this class. Other
classes will introduce you to other

irrational numbers.

**Simplifying Radicals**

Note: When there isn’t a number for n, it means there’s really a 2 there.

**Examples:**