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