Unit 2Comparing and Ordering Rational Numbers

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Real Numbers

Are all numbers on the number line.

This includes (but is not limited to) positive and negative integers...

rational numbers, square roots, cube roots , π (pi), etc.

Venn Diagram

Venn diagrams are diagrams that graphically show possible relationships between a collection of sets (groups of things).

Now, let’s look at a Venn diagram of the set of Real Numbers

The Set of Real Numbers

Natural Numbers

Rational Numbers are a “sub-set” of the Real Numbers.

The set of Rational numbers can be divided into 4 more categories.

Our first group is the set of Natural numbers which are also called the counting numbers {1, 2, 3, ...}.

Natural Numbers

Here is what they look like on the number line.

1 2 3 4 5 6

Next Step... Whole Numbers

Whole Numbers

Whole numbers {0, 1, 2, 3, ...}. This set includes “0” Here is what they look like on the number line.

0 1 2 3 4 5

Now... Integers

Integers

Integers now include all the Whole Numbers and their negatives {... -3, -2, -1, 0, 1, 2, 3, ...}.

Here is what they look like on the number line.

-3 -2 -1 0 1 2

Rational Numbers

Rational Numbers

The set of Rational Numbers now includes the ratios of integers, also called fractions.

Hence the word Rational.

The only exception is that the denominator cannot equal “0”

Examples: 1/2 = 0.5 or 1/3 = 0.333...

Rational decimal end or repeat.

Rational Numbers

The term, Rational Number, refers to any number that can be written as a fraction where the denominator does not equal “0”.

This includes: fractions that are reduced,

fractions that can be reduced,

mixed numbers, improper fractions,

and even integers and whole numbers.

An integer, like 4, can be written as a fraction by putting the number 1 under it.

Rational Numbers

Here is what they look like on the number line.

-3 -2 -1 0 1 21/2-1 1/3-2 1/4 -1/3 1 3/4

Irrational Numbers If we now add to our list

the set of Irrational Numbers - we have now have all of the Real Numbers.

√2 = 1.41421 and Pi... are irrational.

Irrational decimal neither end nor repeat.

Real Numbers

Real numbers are all the numbers on the continuous number line with no gaps. Every decimal expansion is a real number. Real numbers may be rational or irrational0

Classwork

Now that you have an understanding of the Real Number system…

Complete the blank Venn diagram given to you.

Label each set of numbers and give a short example for each.

Classwork

Next let’s make a note of the Key Words using Cornell Notes.

Name Defintion

Key Words

Real numbers are all the numbers on the continuous number line with no gaps.

Natural Numbers are the counting numbers - not including zero.

Whole Numbers are the counting numbers including zero.

Key Words

Integers are positive and negative whole numbers and zero.

Rational Numbers are simply numbers that can be written as fractions or ratios (this is where the term rational comes from). Decimals repeat or stop.

Irrational Numbers have decimals that don’t repeat or stop. They cannot be written as a ratio of 2 integers

Unit 2- Review

How can you compare rational numbers that are in different formats... such as fraction or decimal format?

Put them all into the same format… you may have to convert some of them!

How do you convert fractions into decimals?

By simply dividing the numerator by the denominator.

More Key Words

Ascending Order - in numerical order from lowest to highest.

Example: 5, 6, 7, 8

Descending Order - in numerical order from highest to lowest.

Example: 9, 8, 7, 6

Show You Know

Put the following rational numbers in ascending order…

0.3 -0.6 -3/4 11/5 -1

0.75 -21/4 21/4 -1/2 0.35

1/5 3/8 5/6 2/3 2/7

Show You Know

Answers

-1 -3/4 -0.6 0.3 11/5

-21/4 -1/2 0.35 0.75 21/4

1/5 2/7 3/8 2/3 5/6

Show You Know

Put the following rational numbers in descending order…

-21/4 1.3 -7/8 -11/5 -1.3

.33 -23/5 -1/2 0.335 -0.56

-1/5 -3/8 -5/6 -2/3 -2/7

Show You Know

Answers

1.3 -7/8 -11/5 -1.3 -21/4

0.335 .33 -1/2 -0.56 -23/5

-1/5 -2/7 -3/8 -2/3 -5/6

Homework

Complete Practice Sheet