REAL NUMBERS: Understanding Rational & Irrational Numbers CCSS 8.NS.A.1, 8.NS.A.2

REAL NUMBERS:Understanding Rational &

Irrational NumbersCCSS 8.NS.A.1, 8.NS.A.2

Classifying Numbers Foldable

CLASSIFYING NUMBERS

Natural Numbers

Whole Numbers

Integers

Rational Numbers

Real Numbers

Natural Numbers

1 15 8

2 7 23

79 6

Also called counting numbers.

N: {1, 2, 3, 4, 5, …}

1 2 3 4 5 6

Whole Numbers

WThe Natural Numbers

and ZERO

W: {0, 1, 2, 3, 4, …}

0 1 2 3 4 5

1 15 8

2 7 23 0

79 6

Integers

The Whole Numbers and their opposites

(negatives)

Z: {…-2, -1, 0, 1, …}

-3 -2 -1 0 1 2 3 4

- 1 -345 -44

1 15 8

-3 2 7 23 0 -17

79 6

-28 -91 -1475

Rational Numbers

Any number that can be written as a

fraction ,where a & b are both integers and

b ≠ 0(includes all the integers)

Division by zero is undefined

Q:Fractions: ½, ¼, ⅓, ⅝, ⅞, …

Terminating Decimals: 2.3, 5.85,(decimals that end) -17.6783

Repeating Decimals:

-1 - ⅔ - ⅓ 0 ⅓ ⅔ 1b

a

78.45 ,3.0

½ 2.71 ¼

⅝ ⅓

3.5

⅞

4.0

2.0

3712.0

73

2

- 1 -345 -44

1 15 8

-3 2 7 23 0 -17

79 6

-28 -91 -1475

9.0

Real Numbers

Irrational Numbers

INon-Terminating / Non-Repeating

Decimals(decimals that never end or repeat)

Square Roots of non perfect squares:

π

9.8754... ...,1234.0

... ,7 ,6 ,5 ,3 ,2IRRATIONALNUMBERS

RATIONAL NUMBERS

INTEGERS

WHOLE NUMBERS

NATURAL NUMBERS

The Rational Numbers and the

Irrational Numbers

REAL NUMBERS

REALNUMBERS

RATIONAL

IRRATIONAL

INTEGERS

FRACTIONS

REPEATING DECIMALS

TERMINATING DECIMALS

NON-REPEATING/NON-TERMINATING DECIMALS

SQUARE ROOTS OF NON-PERFECT SQUARES

π

WHOLE NUMBERS

OPPOSITES(NEGATIVES)

NATURAL NUMBERS

aka counting numbers

0

(a.k.a. counting numbers)

Square roots of non-perfect squares

RealNumbers

RationalNumbers

IrrationalNumbers

Integers

Fractions

Terminating Decimals

Repeating Decimals

Non-terminating Decimals

Non-repeating Decimals

π

WholeNumbers

NaturalNumbers

ZeroNegatives

ImaginaryNumbers

i

ComplexNumbers

A+Bi

Number LinesA number line can be any length you want.

The lines on a number line are calledTICK MARKS.

The space between the tick marks is called an INTERVAL.

All intervals on a number line must be equal(the same distance apart).

Number Lines

TICK MARK TICK MARKTICK MARK

INTERVAL INTERVAL INTERVAL

Showing Decimals on aNumber Line

2 32. 2. 2. 2. 2. 2. 2. 2. 2.

Taking it further

2 32. 2. 2. 2. 2. 2. 2. 2. 2.

2.3 2.31 2.32 2.33 2.34 2.35 2.36 2.37 2.38 2.39 2.4

Showing Fractions on aNumber Line

2 36

12

6

22

6

32

6

42

6

52

6

12

3

12

2

12

3

22

6

52

How many intervals are there?

6The denominator of the fraction

must match the number of intervals!

SIMPLIFY your fractions!

Inequality Symbols

There are two inequality symbols we will look at today.

They are:< less than

and> greater than

Using Inequality SymbolsWe can use the inequality symbols to order

numbers.i.e. 5 < 7

We can also use the inequality symbol to show that a number is between two numbers:

i.e. 5 < n < 7What are some values of n?

6, 5.5, 6.5, 6.75, 6.854

EQUIVALENCE

Two numbers are EQUIVALENT if they have the same position on the number line.

Another word for EQUIVALENT is EQUAL.

Equivalent Fractions

To write an equivalent fraction, just SIMPLIFY the fraction or EXPAND it.

Example:8

3

2

2

8

3 NOTICE:

Multiplying a number by 1 does not change its identity!

12

2

16

6

16

6and are equivalent.

8

3

Equivalent Fractions

To write an equivalent fraction, just SIMPLIFY the fraction or EXPAND it.

Example:15

12

315

312

5

4

5

4and are equivalent.

15

12

PLACE VALUE TABLES

Thousands Place1000s

Hundreds Place100

Tens Place10

Ones Place1

2 3 0

All numbers can be placed in a place value table.

Each digit has a place. Even ZERO must be entered into place value tables.

Ex: 230 (standard form)

EXPANDED FORM

EXPANDED FORM is when you write the number out indicating each place value.

Ex: 230 (standard form)

Thousands Place1000

Hundreds Place100

Tens Place10

Ones Place1

2 3 0

EXPANDED FORM: (2 ∙ 100) + (3 ∙ 10) + (0 ∙ 1)

DECIMALSDecimals can also be put on a place value table

and written in expanded form.Ex: 4.36

Tens Place

10Ones Place

1

Tenths Place

Hundredths Place

4 3 6

10

11.0

100

101.0

decimal point

●When writing or saying a DECIMAL POINT in words, the word

“and” is used.

DECIMALS

Tens Place

10Ones Place

1

Tenths Place

Hundredths Place

4 3 6

10

11.0

100

101.0

decimal point

●

EXPANDED FORM:(sum of decimals)

EXPANDED FORM:(sum of fractions)

100

16

10

1314

01.061.0314

POWERS OF TEN

The PLACE VALUES are also POWERS OF TEN.

3101000 210100 11010 0101

11010

11.0

210100

101.0

3101000

1001.0

In Words:

Standard Form:24.67

●

Place Value:

Expanded Form:(sum of fractions)

Expanded Form:(powers of ten)

Expanded Form:(sum of decimals)

HUNDREDS TENS ONES TENTHS HUNDREDTHS THOUSANDTHS