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1. Classify numbers and graph them on number lines.

2. Tell which of two real numbers is less than the other.

3. Find the additive inverse of a real number.

4. Find the absolute value of a real number.

Objectives

1.3 Real Numbers and the Number Line

Section 1.3, Slide 1

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Section 4.1 Slide 2

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Study Mathematics Skills

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Review Powerpoint Presentation

Take notes and summarize steps

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Be on Time

Pay attention

Take notes in your notebook

Ask questions and participate.

3 Home Work

Complete All assigned homework.

Do the quiz until you score 100%

Take notes and Summaries the steps

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practice test until Score 100%.

Take notes and summarize steps.

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Mini Lesson - Warm Up

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A set is a collection of objects. In mathematics, these

objects are usually numbers. The objects that belong to

the set, called elements of the set, are written between

braces.

Classifying Numbers and Graphing Them on Number Lines

Natural Numbers

{1, 2, 3, 4, 5, . . . }

The set of numbers used for counting is called the

natural numbers.

Whole Numbers

{0, 1, 2, 3, 4, 5, . . . }

The set of natural numbers plus zero gives us the set of

whole numbers.

Section 1.3, Slide 6

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The indicated points

correspond to whole numbers.

Numbers can be represented on a number line.

Classify Numbers and Graph Them on Number Lines

0

1

2

3

4

5

6

The indicated points

correspond to natural numbers.

Each number to the left of 0 is the opposite, or negative,

of a natural number. The natural numbers, their opposites,

and 0 form a new set of numbers called the integers.

Section 1.3, Slide 7

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Positive numbers

Negative numbers

Integers

{. . ., – 3, – 2, – 1, 0, 1, 2, 3, . . . }

Classify Numbers and Graph Them on Number Lines

2

–1

0

1

3

–2

–3

Zero

(neither positive nor negative)

Opposites

The points correspond to integers.

Section 1.3, Slide 8

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Rational Numbers

{x|x is a quotient of two integers, with denominator not 0}

is the set of rational numbers.

Quotients of integers, like ½ or 3¾ are called rational

numbers.

Since any integer can be written as the quotient of itself

and 1 (i.e. –3 = –3/1), all integers are rational numbers.

Classify Numbers and Graph Them on Number Lines

Section 1.3, Slide 9

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Example 2 Graph each number on the number line.

Classify Numbers and Graph Them on Number Lines

To locate the improper fractions on the number line,

write them as mixed numbers or decimals.

Section 1.3, Slide 10

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Irrational Numbers

{x|x is a nonrational number represented by a point on

the number line} is the set of irrational numbers.

Some numbers, such as

Classify Numbers and Graph Them on Number Lines

can be found on the number

line but cannot be written as the quotient of two integers.

Such numbers are called irrational numbers.

Section 1.3, Slide 11

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Real Numbers

{x|x is a rational or irrational number} is the set of real

numbers.

Both rational and irrational numbers can be represented

by points on the number line and are called real numbers.

Classify Numbers and Graph Them on Number Lines

Section 1.3, Slide 12

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Classify Numbers and Graph Them on Number Lines

Noninteger Numbers

Rational Numbers

Real Numbers

Irrational Numbers

Integers

Whole Numbers

Negative Integers

Zero

Natural Numbers

Section 1.3, Slide 13

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Examples of Irrational Numbers (With Lists)

No list enumerates all the irrational numbers. There are more irrational numbers than rational numbers. It is crazy to even think about listing all of them! But we can use some of their properties to discover them. We can also take the help of prime numbers to do this. Few lists of irrational numbers:

These lists are not exclusive but do provide a way to create irrational numbers.

Irrational Number – Any real number that is not rational is irrational.

Endless digits after the decimal (however, if the digits in a number end, like 1.25, it is easily converted to a quotient = 125/100 )
The digits after decimal have no repeating pattern (however, any repeating pattern, even endless, like in 1.3252525…, can be converted to a quotient = 1312/99 )

https://mathnovice.com/examples-of-irrational-numbers-lists/

Section 1.3, Slide 14

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http://www.geom.uiuc.edu/~huberty/math5337/groupe/digits.html

Section 1.3, Slide 15

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Ordering Real Numbers

Ordering of the Real Numbers

For any two real numbers a and b, a is less than b if a is to the left of b on a number line.

a is to the left of b,

a < b.

a

b

Example 4

Is it true that –3 < –1?

To find out, locate –3 and –1 on a number line, as shown.

Because –3 is to the left of –1 on the number line, –3 is less than –1. The statement –3 < –1 is true.

2

–1

0

1

3

–2

–3

Section 1.3, Slide 16

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Additive Inverse

The additive inverse of a number x is the number that is the same distance from zero on the number line as x, but on the opposite side of 0. (The number 0 is its own additive inverse.)

Finding the Additive Inverse of a Real Number

Double Negative Rule

For any real number a,

–(–a) = a.

The chart shows several numbers and their additive inverses.

Number	Additive Inverse
–4	–(–4) or 4
0	0
5	–5

The additive inverse of a nonzero number is found by changing the sign of the number.

Section 1.3, Slide 17

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The absolute value of a number can never be negative.

The absolute value of a number is the undirected

distance between 0 and the number on the number line.

The symbol for the absolute value of the number a is |a|,

which is read “the absolute value of a.”

Find the Absolute Value of a Real Number

For example, the distance between 2 and 0 on the

number line is 2 units, so |2| = 2.

Also, the distance between –2 and 0 on the number line

is 2, so |–2| = 2.

Section 1.3, Slide 18

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Example 5

Simplify by finding the absolute value.

(a) |5| =

Find the Absolute Value of a Real Number

5

(c) – |–5| =

– (5) = –5

Replace |–5| with 5.

(b) |–5| =

5

(d) – |–13| =

– (13) = –13

(e) |8 – 5|

|8 – 5| = |3|

(f) –|8 – 5| =

–|3|

(g) –|12 –3| =

–|9|

Simplify within the absolute value bars first.

= 3

= –3

= –9

The absolute value of a number can never be negative.

Section 1.3, Slide 19

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Practice

Makes

Perfect

Section 1.2, Slide 20