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Algebra 1 Lesson 1-3 Common Core Real Numbers and the Number Line

Real Numbers Rational Numbers

Integers

Whole Numbers

Irrational Numbers

Natural Numbers

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Natural Numbers • what you see in nature • counting numbers • positive numbers – any number greater than zero – always to the right of zero

0

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Whole Numbers • includes all natural numbers • zero

1 2 3 4 5 6

Integers • Includes all whole numbers (0, 1, 2, 3, …) • Negative numbers (opposites of whole numbers) – any number less than zero – always left of zero

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

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Terminating Decimals • A number written as a decimal where the number has a limited number of digits. (It has an ending) • What are examples?

Repeating Decimals • A number written as a decimal where there is a repeating pattern of digits that NEVER ends. • What are examples?

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Rational Numbers • All integers • Decimals that have an ending (terminate) OR repeat indefinitely 0.5

0.25

-0.333…

1.272727…

• Fractions, a/b where a and b are integers and b ≠ 0. Examples:

1 2

-2 3

17 5

3 1

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Irrational Numbers • Cannot be written as a fraction (ratio of 2 intergers) • Decimal goes on forever without repeating (never ends) Example: Pi 3.1415926535897932384626433832795…

You can classify numbers using sets… • Set: a well-defined collect of terms – Example: The set of all integers {…-3, -2, -1, 0, 1, 2, 3…}

• Element of the Set: members of a set – Example: What are 3 elements of the above set? -1, 1, 100

• Subset: elements from a given set – Example: What are the natural numbers with the above set of integers? {1, 2, 3…}

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Real Numbers Rational Numbers – fractions & terminating or repeating decimals

Integers (… -3, -2, -1, 0, 1, 2, 3…) Whole Numbers (0, 1, 2, 3…)

Irrational Numbersdecimals that go on forever that DO NOT repeat

Natural Numbers (1, 2, 3…)

I’d Rather Be… Rational verses Irrational If you could be a number, would you rather be rational or irrational? Write your response on a sticky note, explain why and then post it on the whiteboard.

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Classify each number:

Classify each number

✔

✔ ✔

✔ ✔ ✔

✔

✔

✔

✔ ✔ ✔ ✔ ✔ ✔

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Square Roots • An operation that yields a number which, when multiplied by itself produces the given number • Example:

16 Radical sign

Parts of a Radical

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Inverse Operations (Opposite Operations) What is the opposite of… Subtraction Addition Division

Multiplication Squaring a Number

Square Root

Square Roots • If a ³0 and a2 = b, then a is the square root of b. Squares Square Roots 0= 0 02 = 0 12 = 1 1= 1 22 = 4 4= 2 2 3 =9 9= 3 42 =16 What is the 16 = 4 relationship between Squares and Square Roots?

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Inverse Operations (Opposite Operations) What is the opposite of… Subtraction Addition Multiplication Squaring a Number

Division Square Root

Square Roots • If a ³0 and a2 = b, then a is the square root of b. Squares Square Roots 0= 0 02 = 0 12 = 1 1= 1 22 = 4 4= 2 Perfect Squares2 3square = 9 roots that are 9= 3 whole 16 numbers 42 = (There are 11 of them 16 = 4 from 0 to 100. Can you name them?)

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Perfect Square Roots: square If a perfect square root gives a roots that have ayou whole number whole answer…What as anifanswer happens it is NOT 0= 0

a perfect square?

25 = 5

It is an irrational number

1=

1

4=

2

49 = 7

9=

3

64 = 8

16 = 4

81 = 9

100 = 10

36 = 6

BrainPop Video & Do It Review Square Roots & Perfect Square Roots

• http://www.brainpop.com/math/numbersando perations/squareroots/

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Try This… • Use a calculator to approximate the 3 • Write the number on a piece of paper • Enter it into your calculator, square it, then subtract 3. • Do you get 0? • EXPLAIN

Watch this video… …you never know when you may lose out on money because you do not know about square roots!

• http://www.youtube.com/watch?v=BbX44Y SsQ2I

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Non-Perfect Square Roots Use estimation to figure out the square root of 14…

14 is between what 2 perfect squares ? 9 = 3

9 &

16

16 = 4

Therefore the square root of 14 is a irrational number between 3 & 4. Check your calculator to see the exact irrational number.

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√25 & √36 √81 & √100 5&6 9 & 10 ≅6

≅9

-√4 & -√9 -2 & -3 ≅ -3

Another way to approximate square roots… Watch this video …. https://docs.google.com/file/d/0BV_py0ic4ugNlplcXRtdjJtc1k/edit?pli=1

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Try these in your calculator:

25 = 5 5

5

144 = 12 12

12

23,104 = 152 152

152

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Look at these examples again…

25 = 5 5

25 = 5

OR

5

-5

-5

What is a negative number multiplied by another negative number?

A positive

So…how many answers are there to every square root problem? Why?

Find the Square Root and then determine if it is rational or irrational 196 14 121 50 4 16 2 3

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Non-Real Numbers −4 −16 What is difference about these examples?

−25 −36 −100

Look at these examples again…

25 = 5 5

5

25 = 5

OR -5

-5

What is a negative number multiplied by another negative number?

A positive

Therefore, you will always have a positive number under your radical sign.

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3 Kinds of Square Roots When you first learned about square roots. You Principal Square Root only learned about principle -all #s are positive square roots. Now that you know your Integer Rules Negative Square Root and more about square -negative signroots…you appears know that ALL outside the radical signroots have 2 square

64 = 8

− 64= −8

answers! Both Square Roots -both the negative & positive signs are used

± 64 = ±8

3 Kinds of Square Roots Principal Square Root: The number that is multiplied by itself is a positive number Example:

64 = 8

You typically do not add the positive sign in front of the problem and answer. The positive is implied!

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3 Kinds of Square Roots Negative Square Root: The number that is multiplied by itself is a negative number (Remember: a negative times a negative is a positive) Example:

− 64= −8

The negative sign appears outside the radical sign. So your answer must have a negative sign too!

3 Kinds of Square Roots Both Square Roots: The number that is multiplied by itself could be BOTH a negative & positive

number Example:

± 64 = ±8

The negative sign and positive signs appear outside the radical sign. So your answer must have BOTH signs!

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Find each square root. If necessary, round to the nearest hundredth. 1.

49

2.

− 78

7 -8.83

€ 3. ± 513 € €

4.

±22.65

1 2

25 100

€

Cube Roots • An operation that yields a number which, when multiplied by itself three times produces the given number Example:

3

8= 2

What multiplied by itself three times equals 8?

Radical sign 3

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Perfect Cube Roots: cube roots that have a whole number as an answer 3

3

0 = 0 1= 1

3

3

125 = 5

1000 = 10

216 = 6

8= 2

3

343 = 7

3

27 = 3

3

512 = 8

3

64 = 4

3

729 = 9

3

3

E C A D F B

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What are the symbols of inequality?

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Write >, 5 1 < 7 = 0.2

€

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Write each set of numbers in order from least to greatest. 1.

0.42, 0.63, 0.63,

€

0.42,

4 3 4 3 €

2.

8, − 4.83, 0.4

−4.83, 0.4,

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The Code Name Organizer Students will need: • Name Decoder Chart • Poster Board • Markers

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