Algebra 2 Properties of Real Numbers Lesson 1-2

Goals Goal To graph and order real numbers. To Identity properties of real numbers. Rubric Level 1 – Know the goals. Level 2 – Fully understand the goals. Level 3 – Use the goals to solve simple problems. Level 4 – Use the goals to solve more advanced problems. Level 5 – Adapts and applies the goals to different and more complex problems.

Essential Question Big Idea: Properties How can you use the properties of real numbers to simplify algebraic expressions? –Students will understand the set of real numbers has several subsets related in particular ways. –Students will understand that algebra involves operations on and relations among numbers, including real and imaginary numbers. –Students will understand how rational and irrational numbers form the set of real numbers.

Vocabulary Opposite Additive Inverse Reciprocal Multiplicative Inverse

A set is a collection of items called elements. The rules of 8-ball divide the set of billiard balls into three subsets: solids (1 through 7), stripes (9 through 15), and the 8 ball. A subset is a set whose elements belong to another set. The empty set, denoted , is a set containing no elements. Sets of Numbers

Other useful set notation –U Union (or statement) –∩ Intersection (and statement) –\ excludes whatever follows –∈ element of a set

There are many ways to represent sets. For instance, you can use words to describe a set. You can also use roster notation, in which the elements in a set are listed between braces, { }. WordsRoster Notation The set of billiard balls is numbered 1 through 15. {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15} Sets of Numbers

A Venn Diagram can also be used to display sets of numbers and their relationships. The following sets of numbers are displayed: Reals, Rationals, Irrationals, Integers, Wholes, and Naturals. Naturals 1, 2, 3... Wholes 0 Integers Rationals Irrationals Reals Sets of Numbers

Natural Numbers - ℕ The numbers we use to count things are the set of natural numbers, N = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, …} – The three dots, ellipses, show that the same pattern continues in the same pattern indefinitely. – Each natural number other than 1 is either a prime number or a composite number. – A prime number is a number greater than one that is evenly divisible only by itself or 1. – A natural number other than 1 that is not a prime number is a composite number.

Whole Numbers - The set of whole numbers include 0 and the natural numbers, W = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, …}

Integers - ℤ The set of numbers including positive and negative counting numbers and zero are called the set of integers:  The negative integers are {…,–5,–4,–3,–2,–1}.  The positive integers are {1, 2, 3, 4, 5, …}. Z = {…, –5, –4, –3, –2, –1, 0, 1, 2, 3, 4, 5, …}

Rational - ℚ A rational number can be expressed as the quotient of two integers (a fraction), with the denominator not 0.  Rational numbers can also be written in decimal form, either as terminating; ¾=.75, -⅛ = -.125, or 1 ¼= 2.75  or as repeating decimals; ⅓ =.33333… or ⅙ =.1666….

Irrational - ℚ Decimal numbers that neither terminate nor repeat are not rational, and thus are called irrational numbers.  The square root of any number that is not a perfect square is an irrational number;  as well as numbers represented by familiar symbols;  or e. A repeating decimal may not appear to repeat on a calculator, because calculators show a finite number of digits. Caution!

Write all classifications that apply to each number. 5 is a whole number that is not a perfect square. 5 irrational, real –12.75 is a terminating decimal. –12.75 rational, real 16 2 whole, integer, rational, real = = A. B. C. Example: Classifying Real Numbers

Write all classifications that apply to each number. 9 whole, integer, rational, real –35.9 is a terminating decimal. –35.9 rational, real 81 3 whole, integer, rational, real = = A. B. C. 9 = 3 Your Turn:

NumbersRealRationalIntegerWholeNaturalIrrational 2.3  Consider the numbers Classify each number by the subsets of the real numbers to which it belongs. Example: Classifying Real Numbers

Classify each number by the subsets of the real numbers to which it belongs. Consider the numbers – 2, , – 0.321, and. Your Turn: NumbersRealRationalIntegerWholeNaturalIrrational –2  –0.321

Example: Classifying a Variable Your school is sponsoring a charity race. Which set of numbers best describes the number of people p who participate in the race? The number of people p is a natural number. The correct answer is A.

Your Turn: If each participant in a charity race makes a donation d of $15.50 to a local charity, which subset of real numbers best describes the amount of money they will raise? A. Natural numbers B. Integers C. Rational numbers D. Irrational numbers

Sets of Numbers A set can be finite like the set of billiard ball numbers or infinite like the natural numbers {1, 2, 3, 4 …}. A finite set has a definite, or finite, number of elements. An infinite set has an unlimited, or infinite number of elements. The Density Property states that between any two numbers there is another real number. So any interval that includes more than one point contains infinitely many points.

Many infinite sets, such as the real numbers, cannot be represented in roster notation. Therefore, we use a number line. A number line is a line on which each point is associated with a real number. 2– – 1– 3– 4– 5 The Number Line The number line represents the set of all real numbers. The number line can be used to graph individual real numbers, sets of real numbers both finite and infinite, and order real numbers.

To construct a number line, 1.Choose any point on a horizontal line and label it 0. 2.Choose a point to the right of 0 and label it 1. 3.The distance from 0 to 1 establishes a scale that can be used to locate more points, with positive numbers to the right of 0 and negative numbers to the left of 0.  The number 0 is neither positive nor negative. 2– – 1– 3– 4– 5 Negative numbers Positive numbers – The Number Line

You can use a number line to compare and order real numbers. A number on a number line is greater than all of the numbers to it’s left. 1.Approximate Irrational Numbers & fractions as a decimal (Remember: a fraction is just division). 2.Graph the decimal numbers on the number line. Determine if the number is positive or negative and plot it by the whole number it is closest to. 3.To compare the numbers, determine the larger of two numbers and place the appropriate inequality between them.

Example: Graphing Numbers on the Number Line What is the graph of the numbers Change the numbers to decimals.  Graph the decimal numbers on the number line. – 4 – 3 – 2 – ||||||||||||||||||

Your Turn: What is the graph of the numbers

Order the numbers from least to greatest. Write each number as a decimal to make it easier to compare them. Use a decimal approximation for .  ≈ 3.14 Use < to compare the numbers. Rewrite in decimal form. Use a decimal approximation for. Consider the numbers –5.5 < 2.23 < 2.3 < < 3.14 The numbers in order from least to great are Example: Ordering Real Numbers

Order the numbers from least to greatest. Consider the numbers – 2, , – 0.321, and. Your Turn: Write each number as a decimal to make it easier to compare them. = 1.5 ≈ –1.313  ≈ 3.14 Use < to compare the numbers. Use a decimal approximation for . –2 < –1.313 < –0.321 < 1.50 < 3.14 The numbers in order from least to great are –2,, –0.321,, and . Rewrite in decimal form. Use a decimal approximation for.

Inequality Symbols An inequality symbol is used to compare numbers: Symbols include: greater than: greater than or equal to: less than: less than or equal to: not equal to: Examples:.

The Number Line The set of real numbers between 3 and 5, which is also an infinite set, can be represented on a number line or by an inequality < x < 53 < x < 5

Graphing a Set on the Number Line If the end point of a set is included, then a “closed circle” is used. –x ≥ 3 If the end point of a set is excluded, then an “open circle” is used. –x > 3

Example: Write the following intervals on the number line as an inequality. 1)  x > -2  If the variable is on the left side of the inequality, then the inequality sign is the same direction as the arrow. – 4 – 3 – 2 – ||||||||||||||||||

Your Turn: 2)  x ≤ 1 – 4 – 3 – 2 – ||||||||||||||||||

Your Turn: 3)  Conjunction “and statement”  x ≥ -3 and x < 2  -3 ≤ x < 2 (compound inequality) – 4 – 3 – 2 – ||||||||||||||||||

Your Turn: 4)  Disjunction “or statement”  x ≤ -3 or x > 0  Disjunction must be written with the word or (or symbol ∪ ).  x ≤ -3 ∪ x > o (compound inequality) – 4 – 3 – 2 – ||||||||||||||||||

Your Turn: 5)  x equals all real numbers except 0  x 0  x ℝ \0 – 4 – 3 – 2 – ||||||||||||||||||

Properties of Real Numbers The Rules of the Game

Properties of Real Numbers Why are the properties of real numbers important? –Properties are the rules for the game involving real numbers. To get the correct answer, you must follow the rules. –If the rules are changed a different kind of math is played. –You use the properties of real numbers for all math problems that involve real numbers.

Definitions Opposite – the opposite or additive inverse of any number a is –a. The sum of opposites is zero, the additive identity. –Example: 3 + (-3) = (-5.2) = 0 Additive Inverse – the additive inverse or opposite of any number a is –a. The sum of additive inverses is zero, the additive identity. –Example: (-6.8) = 0

Definitions Reciprocal – the reciprocal or multiplicative inverse of any nonzero number a is 1/a. The product of reciprocals is 1, the multiplicative identity. –Example: 5 1/5 = 1 Multiplicative Inverse – the multiplicative inverse or reciprocal of any nonzero number a is 1/a. The product of multiplicative inverses is 1, the multiplicative identity. –Example: 1/3 3 = 1

Find the additive and multiplicative inverse of each number. 12 The opposite of 12 is –12. additive inverse: –12 Check – = 0 The Additive Inverse Property holds. The reciprocal of 12 is. multiplicative inverse: The Multiplicative Inverse Property holds. Check Example: Finding Inverses

Find the additive and multiplicative inverse of each number. additive inverse: multiplicative inverse: The opposite of is. The reciprocal of is Example: Finding Inverses

Find the additive and multiplicative inverse of each number. 500 The opposite of 500 is –500. The Additive Inverse Property holds. Check (–500) = 0 additive inverse: –500 multiplicative inverse: The reciprocal of 500 is. The Multiplicative Inverse Property holds. Check Your Turn:

Find the additive and multiplicative inverse of each number. –0.01 The opposite of –0.01 is additive inverse: 0.01 The reciprocal of –0.01 is –100. multiplicative inverse: –100 Your Turn:

Properties of Real Numbers The four basic math operations are addition, subtraction, multiplication, and division. Because subtraction is addition of the opposite and division is multiplication by the reciprocal, the properties of real numbers focus on addition and multiplication.

Properties of Real Numbers Identity Properties of Addition & Multiplication Identity = Same (no change) What can you add to a number & get the same number back? What can you multiply a number by and get the number back? 0 (zero) 1 (one)

For all real numbers n, WORDS Additive Identity Property The sum of a number and 0, the additive identity, is the original number. NUMBERS = 0 ALGEBRA n + 0 = 0 + n = n Properties Real Numbers Additive Identity Properties of Real Numbers

For all real numbers n, WORDS Multiplicative Identity Property The product of a number and 1, the multiplicative identity, is the original number. NUMBERS ALGEBRA n  1 = 1  n = n Properties Real Numbers Multiplicative Identity Properties of Real Numbers

Inverse Properties of Addition & Multiplication Inverse = Opposite What is the opposite (inverse) of addition? What is the opposite of multiplication? Subtraction (add the negative) Division (multiply by reciprocal)

For all real numbers n, WORDS Additive Inverse Property The sum of a number and its opposite, or additive inverse, is 0. NUMBERS 5 + (–5) = 0 ALGEBRA n + (–n) = 0 Properties Real Numbers Additive Inverse Properties of Real Numbers

For all real numbers n, WORDS Multiplicative Inverse Property The product of a nonzero number and its reciprocal, or multiplicative inverse, is 1. NUMBERS ALGEBRA Properties Real Numbers Multiplicative Inverse Properties of Real Numbers

Closure Properties of Addition & Multiplication over the Set of Real Numbers A number set has closure under an operation if performance of that operation on members of the set always produces a member of the same set.

For all real numbers a and b, WORDS Closure Property The sum or product of any two real numbers is a real number NUMBERS = 5 2(3) = 6 ALGEBRA a + b   ab   Properties Real Numbers Addition and Multiplication Properties of Real Numbers

Commutative Properties of Addition & Multiplication Commute = Travel (move) It doesn’t matter how you move addition or multiplication around…the answer will be the same!

For all real numbers a and b, WORDS Commutative Property You can add or multiply real numbers in any order without changing the result. NUMBERS = (11) = 11(7) ALGEBRA a + b = b + a ab = ba Properties Real Numbers Addition and Multiplication Properties of Real Numbers

Associative Properties of Addition & Multiplication Associate = Group It doesn’t matter how you group (associate) addition or multiplication…the answer will be the same!

For all real numbers a and b, WORDS Associative Property The sum or product of three or more real numbers is the same regardless of the way the numbers are grouped. NUMBERS (5 + 3) + 7 = 5 + (3 + 7) (5  3)7 = 5(3  7) ALGEBRA a + (b + c) = a + (b + c) (ab)c = a(bc) Properties Real Numbers Addition and Multiplication Properties of Real Numbers

Stop and think! Does the Associative Property hold true for Subtraction and Division? Does the Commutative Property hold true for Subtraction and Division? Is 5-2 = 2-5? Is 6/3 the same as 3/6? Is (5-2)-3 = 5-(2-3)? Is (6/3)-2 the same as 6/(3-2)? Properties of real numbers are only for Addition and Multiplication

Properties of Real Numbers Distributive Property of Multiplication over Addition/Subtraction If something is sitting just outside a set of parenthesis, you can distribute it through the parenthesis with multiplication and remove the parenthesis.

For all real numbers a and b, WORDS Distributive Property When you multiply a sum by a number, the result is the same whether you add and then multiply or whether you multiply each term by the number and add the products. NUMBERS 5(2 + 8) = 5(2) + 5(8) (2 + 8)5 = (2)5 + (8)5 ALGEBRA a(b + c) = ab + ac (b + c)a = ba + ca Properties Real Numbers Addition and Multiplication Properties of Real Numbers

Let’s play “Name that property!”

State the property or properties that justify the following = Commutative Property of Addition

State the property or properties that justify the following. 10(1/10) = 1 Multiplicative Inverse Property

State the property or properties that justify the following. 3(x – 10) = 3x – 30 Distributive Property

State the property or properties that justify the following. 3 + (4 + 5) = (3 + 4) + 5 Associative Property of Addition

State the property or properties that justify the following. (5 + 2) + 9 = (2 + 5) + 9 Commutative Property of Addition

3 + 7 = Commutative Property of Addition State the property or properties that justify the following.

8 + 0 = 8 Additive Identity Property State the property or properties that justify the following.

6 4 = 4 6 Commutative Property of Multiplication State the property or properties that justify the following.

17 + (-17) = 0 Additive Inverse Property State the property or properties that justify the following.

2(5) = 5(2) Commutative Property of Multiplication State the property or properties that justify the following.

(2 + 1) + 4 = 2 + (1 + 4) Associative Property of Addition State the property or properties that justify the following.

2 + 2 = 4 Closure Property State the property or properties that justify the following.

3(2 + 5) = Distributive Property State the property or properties that justify the following.

6(78) = (67)8 Associative Property of Multiplication State the property or properties that justify the following.

5 1 = 5 Multiplicative Identity Property State the property or properties that justify the following.

(6 – 3)4 = 64 – 34 Distributive Property State the property or properties that justify the following.

1(-9) = -9 Multiplicative Identity Property State the property or properties that justify the following.

3 + (-3) = 0 Additive Inverse Property State the property or properties that justify the following.

1 + [-9 + 3] = [1 + (-9)] + 3 Associative Property of Addition State the property or properties that justify the following.

-3(6) = 6(-3) Commutative Property of Multiplication State the property or properties that justify the following.

= -8 Additive Identity Property State the property or properties that justify the following.

37 – 34 = 3(7 – 4) Distributive Property State the property or properties that justify the following.

6 + [(3 + (-2)] = (6 + 3) + (- 2) Associative Property of Addition State the property or properties that justify the following.

7 + (-5) = Commutative Property of Addition State the property or properties that justify the following.

(5 + 4)9 = Distributive Property State the property or properties that justify the following.

-3(5 4) = (-3 5)4 Associative Property of Multiplication State the property or properties that justify the following.

-8(4) = 4(-8) Commutative Property of Multiplication State the property or properties that justify the following.

5 1 / = 5 1 / 7 Additive Identity Property State the property or properties that justify the following.

3 / 4 – 6 / 7 = – 6 / / 4 Commutative Property of Addition State the property or properties that justify the following.

1 2 / 5 1 = 1 2 / 5 Multiplicative Identity Property State the property or properties that justify the following.

(1/3)(1/2) = 1/6 Closure Property State the property or properties that justify the following.

-8 2 / = -8 2 / 5 Additive Identity Property State the property or properties that justify the following.

[(- 2 / 3 )(5)]9 = - 2 / 3 [(5)(9)] Associative Property of Multiplication State the property or properties that justify the following.

6(3 – 2n) = 18 – 12n Distributive Property State the property or properties that justify the following.

2x + 3 = 3 + 2x Commutative Property of Addition State the property or properties that justify the following.

ab = ba Commutative Property of Multiplication State the property or properties that justify the following.

a + 0 = a Additive Identity Property State the property or properties that justify the following.

a(bc) = (ab)c Associative Property of Multiplication State the property or properties that justify the following.

a1 = a Multiplicative Identity Property State the property or properties that justify the following.

a +b = b + a Commutative Property of Addition State the property or properties that justify the following.

a(b + c) = ab + ac Distributive Property State the property or properties that justify the following.

a + (b + c) = (a +b) + c Associative Property of Addition State the property or properties that justify the following.

a + (-a) = 0 Additive Inverse Property State the property or properties that justify the following.

Assignment Section 1-2, Pg 15 – 17; #1 – 9 all, 10 – 38 even, 42 – 50 even, 54 – 58 even