REAL NUMBERS

(as opposed to fake numbers?)

Objective • TSW identify the parts of the Real Number System • TSW define rational and irrational numbers • TSW classify numbers as rational or irrational

Real Numbers • Real Numbers are every number. • Therefore, any number that you can find on the number line. • Real Numbers have two categories.

What does it Mean? • The number line goes on forever. • Every point on the line is a REAL number. • There are no gaps on the number line. • Between the whole numbers and the fractions there are numbers that are decimals but they don’t terminate and are not recurring decimals. They go on forever.

Real Numbers REAL NUMBERS

154,769,852,35 4

1.333 -8

5,632.1010101256849765 … 61

π 549.2378 9

49 %

Two Kinds of Real Numbers • Rational Numbers • Irrational Numbers

Rational Numbers â€˘ A rational number is a real number that can be written as a fraction. â€˘ A rational number written in decimal form is terminating or repeating.

Examples of Rational Numbers

•16 •1/2 •3.56

•-8 •1.3333 … •- 3/4

Integers One of the subsets of rational numbers

What are integers? â€˘ Integers are the whole numbers and their opposites. â€˘ Examples of integers are 6 -12 0 186 -934

â€˘ Integers are rational numbers because they can be written as fraction with 1 as the denominator.

Types of Integers â€˘ Natural Numbers(N): Natural Numbers are counting numbers from 1,2,3,4,5,................ N = {1,2,3,4,5,................} â€˘ Whole Numbers (W): Whole numbers are natural numbers including zero. They are 0,1,2,3,4,5,............... W = {0,1,2,3,4,5,..............} W=0+N

REAL NUMBERS NATURAL Numbers IRRATIONAL Numbers

WHOLE Number s

INTEGERS RATIONAL Numbers

Irrational Numbers â€˘ An irrational number is a number that cannot be written as a fraction of two integers. â€˘ Irrational numbers written as decimals are nonterminating and nonrepeating.

Irrational numbers can be written only as decimals that do not terminate or repeat. They cannot be written as the quotient of two integers. If a whole number is not a perfect square, then its square root is an irrational number. Caution! A repeating decimal may not appear to repeat on a calculator, because calculators show a finite number of digits.

Examples of Irrational Numbers â€˘ Pi

Try this! a) b) c) d) e)

2

• a) Irrational

12

• b) Irrational

25 5

• c) Rational • d) Rational

11

66

• e) Irrational

Additional Example 1: Classifying Real Numbers Write all classifications that apply to each number.

A.

5 is a whole number that is not a perfect square. irrational, real 5

B. â€“12.75 â€“12.75 is a terminating decimal. rational, real 16 16 4 = =2 C. 2 2 2 whole, integer, rational, real

A fraction with a denominator of 0 is undefined because you cannot divide by zero. So it is not a number at all.

Additional Example 2: Determining the Classification of All Numbers

State if each number is rational, irrational, or not a real number. A.

21 irrational

B.

0 3

0 =0 3

rational

Additional Example 2: Determining the Classification o All Numbers

State if each number is rational, irrational, or not a real number. C.

4 0 not a real number

Objective â€˘ TSW compare rational and irrational numbers â€˘ TSW order rational and irrational numbers on a number line

Comparing Rational and Irrational Numbers • When comparing different forms of rational and irrational numbers, convert the numbers to the same form. 3 7

Compare -3

and -3.571

(convert -337 to -3.428571…

-3.428571… > -3.571

Practice

Ordering Rational and Irrational Numbers â€˘ To order rational and irrational numbers, convert all of the numbers to the same form. â€˘ You can also find the approximate locations of rational and irrational numbers on a number line.

Example • Order these numbers from least to greatest. ¹/₄, 75%, .04, 10%, ⁹/₇ ¹/₄ becomes 0.25 75% becomes 0.75 0.04 stays 0.04 10% becomes 0.10

⁹/₇ becomes 1.2857142… Answer:

0.04, 10%, ¹/₄, 75%, ⁹/₇

Practice Order these from least to greatest:

Objectives â€˘ TSW identify the rules associated computing with integers. â€˘ TSW compute with integers

Examples: Use the number line if necessary. -5

0

1) (-4) + 8 = 4 2) (-1) + (-3) = -4 3) 5 + (-7) = -2

5

Addition Rule

1) When the signs are the same, ADD and keep the sign. (-2) + (-4) = -6 2) When the signs are different, SUBTRACT and use the sign of the larger number. (-2) + 4 = 2 2 + (-4) = -2

Karaoke Time!

Addition Rule: Sung to the tune of â€œRow, row, row, your boatâ€? Same signs add and keep, different signs subtract, keep the sign of the higher number, then it will be exact! Can your class do different

-1 + 3 = ? 1. -4 2. -2 3. 2 4. 4 Answer Now

-6 + (-3) = ? 1. -9 2. -3 3. 3 4. 9 Answer Now

The additive inverses (or opposites) of two numbers add to equal zero. Example: The additive inverse of 3 is -3 Proof: 3 + (-3) = 0 We will use the additive inverses for subtraction problems.

What’s the difference between 7 - 3 and 7 + (-3) ? 7 - 3 = 4 and 7 + (-3) = 4 The only difference is that 7 - 3 is a subtraction problem and 7 + (-3) is an addition problem. “SUBTRACTING IS THE SAME AS ADDING THE OPPOSITE.” (Keep-change-change)

When subtracting, change the subtraction to adding the opposite (keep-change-change) and then follow your addition rule. Example #1:

- 4 - (-7) - 4 + (+7) Diff. Signs --> Subtract and use larger sign. 3 Example #2: -3-7 - 3 + (-7) Same Signs --> Add and keep the sign. -10

Which is equivalent to -12 â€“ (-3)? 1. 12 + 3 2. -12 + 3 3. -12 - 3 4. 12 - 3 Answer Now

7 – (-2) = ? 1. -9 2. -5 3. 5 4. 9 Answer Now

Review 1) If the problem is addition, follow your addition rule. 2) If the problem is subtraction, change subtraction to adding the opposite (keep-change-change) and then follow the addition rule.

State the rule for multiplying and dividing integersâ€Ś. If the signs are the same, the answer will be positive.

If the signs are different,

+

the answer will be negative.

1. -8 * 3 -24

Different What’s Signs The Negative Rule? Answer

2. -2 * -61 122

Tw Jus o tt at ak a e tim e

3. (-3)(6)(1) (-18)(1) -18

4. 6 ÷ (-3) -2 Start inside ( ) first

5. - (20/-5) - (-4) 4 Same Signs Positive Answer

6.

−408 −6 68

7. At midnight the temperature is 8°C. If the temperature rises 4°C per hour, what is the temperature at 6 am? How long Is it from Midnight to 6 am?

6 hours

How much does the temperature rise each hour?

+4 degrees

(6 hours)(4 degrees per hour) = 24 degrees

Add this to the original temp.

8° + 24° = 32°C

8. A deep-sea diver must move up or down in the water in short steps in order to avoid getting a physical condition called the bends. Suppose a diver moves up to the surface in five steps of 11 feet. Represent her total movements as a product of integers, and find the product.

Wha doe t s This mea n?

Multiply

(5 steps) (11 feet) (55 feet) 5 * 11 = 55

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