1-8A Number Systems

Algebra 1 Glencoe McGraw-Hill Linda Stamper

Add closure property?

What are real numbers?

Pretend you are in the first grade.

Your teacher asks you to count.

What would you say?

Rational Numbers: Any number that can be written in the form of . As a decimal they repeat or terminate.

ex: Repeats ex: Terminates

Irrational Numbers:

ex: and These must

be represented by a symbol (ex: ), or as a rounded number, or in radical

form because the

decimal doesn’t

repeat or terminate

(stop).

Integers: Whole numbers and their opposites (this means positive and negative

whole numbers). ex: … 3, ־ 2, ־ 1, 0, 1, 2, 3, 4 ־

4, ־ …Whole Numbers: Natural Numbers and zero. ex: 0,1,2,3…

Natural or Counting Numbers

ex: 1,2,3,4,…

13

0 3333= . ... 14

025= . 2

ab

REAL NUMBERS

When you divide by zero and get no solution ( ), and = i (imaginary numbers).

So what isn’t a real number? - 1

A rational number is any number you can write as a

quotient of two integers, where b is not zero. b

a

Two numbers that are the same distance from 0 on a number line but on opposite sides of 0 are oppositesopposites.

The numbers –2 and 2 are opposites because each is 2 units from zero.

IntegersIntegers are the whole numbers, including zero, and their opposites.

-2 -1 0 1 2• •

Zero is neither positive nor negative, and zero has no opposite.

Name the set of numbers to which each real number belongs.

Example 1

Example 2

Example 3

Example 4

116

2 81 15

integer

rational

irrational

naturalwhole numberintegerrational

rational

Square Roots

You will be allowed to use a calculator for tomorrow’s lesson but NOT on the CHAPTER 1 test! NO GRAPHING CALCULATORS!

You know how to find the square of a number. For instance, the square of 3 (written as 32) is 9.

The square of –3 is also equal to 9 because (–3)2 = 9.

3

3

The inverse of a square number is the square root. Square roots are written with a radical symbol . The number or expression inside a radical symbol is called the radicand.

9radican

dradical symbo

l

932

All positive real numbers have two square roots:

positive square root (principal square root)

9

read as the positive square root of 9 is 3

negative square root 9

read as the negative square root of 9 is –3

This may be written together:

What two identical factors

= 9?

23

23What two

identical factors = 9?

3

3

read as plus or minus the square root of 9 is plus or minus 3.

9 233

All negative real numbers do NOT have square roots because two negative numbers multiplied produce a positive number.

9

What two identical factors = – 9?When two negatives are

multiplied the result is positive.

The square root of a negative radicand is undefined!

undefined

=

Zero has only one square root and that is zero!0 0

The square of an integer is called a perfect square.

3

332

3 is an integer

Therefore 9 is a perfect square.

3.52

3.5 is not an integer (integers

are whole numbers)

5.3

5.325.12 2

The figure is a square but it is not

composed of square sections.

339 2

3 is an integer

What two identical factors

= 12?

The square of an integer is called a perfect square.

3

332

3 is an intege

r

therefore 9 is a perfect square.

(3.4641016…)2

not an integer (irrational number)

12

12 is not a perfect square.

339 2

(3.4641016…)

(3.4641016…)

Determine whether the number is a perfect square.

49

yes

Example 5

Example 6 36

no

Example 7

Example 87

no

144

yes

What two identical factors = the given

number?Is your answer an

integer?

Evaluate the expression.

Example 9

Example 10

Example 11

Example 1281 81 81 81

9 9 9 undefined

To graph a set of numbers means to draw, or plot, the points named by those numbers on a number line. The number that corresponds to a point on a number line is called the coordinate of that point.

Graph –1, 2 and – 3 on a number line. Order the numbers from least to greatest.

-4 -3 -2 -1 0 1 2 3• ••

Draw a number line.Label the number line.Plot the points on the number

line.List the integers from least to greatest.

–3, –1, 2

Example 13 Graph – 4, , – 6 and 0 on a number line. Order the numbers from least to greatest.

-6 -5 -4 -3 -2 -1 0 1 2 • • •

Draw a number line.

Label the number line.

Plot the points on the number line.List the integers from least to greatest.

– 6, – 4, 0,

•

4

4

Graphing Inequalities

For this part of the lesson, you will need a ruler and a colored pencil.

O•

What is the name for the

geometric figure that

represents the solution?

The graph of an inequality in one variable is the set of points on a number line that represent all solutions of the inequality.

ray 4x 4

If the endpoint on the graph is a solution, draw a solid dot.

If the endpoint on the graph is not a solution, draw an open dot.

Then draw an arrowhead to show that the graph continues to infinity.

endpoint

Reading and Graphing an Inequality in One Variable x > 22

•a < 00

O

– 5 > y– 5

•Rewrite as y < – 5 All real numbers less

than or equal to – 5.

All real numbers greater than or equal to 2.

All real numbers less than 0.

When the variable is before the inequality symbol, what do you notice about the direction of the ray and the direction of the inequality symbol?

Graphing an Inequality in One Variable

1. Write inequality.

7

•x < 7

2. Draw a line (use arrowheads).

3. Draw open or solid dot and label the endpoint.

4. Draw the ray in the direction of the inequality symbol.

7 > x Rewrite with variable first.

You do NOT need to draw in

the tick marks.

Example 14 Graph the solutions of each inequality on a number line.

a) x > – 4

–4

•b) y < 1515

O

c) –3 > x–3•

Rewrite as x < – 3

1-A12 Pages