Algebra 2 Topic 1 Real Numbers Properties

Sets of Numbers Naturals - Natural counting numbers. { 1, 2, 3… } Wholes - Natural counting numbers and zero. { 0, 1, 2, 3… } Integers - Positive or negative natural numbers or zero. { … -3, -2, -1, 0, 1, 2, 3… } Rationals - Any number which can be written as a fraction. Irrationals - Any decimal number which can’t be written as a fraction. A non-terminating and non-repeating decimal. Reals - Rationals and irrationals.

Sets of Numbers Reals RationalsIrrationals - any number that can be written as a fraction., 7, Fractions/Decimals Integers, , … -3, -2, -1, 0, 1, 2, 3... Negative IntegersWholes … -3, -2, -1 0, 1, 2, 3... Zero 0 Naturals 1, 2, non-terminating and non-repeating decimals

This is a Venn Diagram that displays the following sets of numbers: Naturals, Wholes, Integers, Rationals, Irrationals, and Reals. Naturals 1, 2, 3... Wholes 0 Integers Rationals Irrationals Reals Sets of Numbers

Identify each number below as natural, whole, integer, rational, irrational, or real. More than one may apply. Rational, Real Whole, Integer, Rational, Real Integer, Rational, Real Irrational, Real,Whole, Integer, Rational, Real Natural

Identify each number below as natural, whole, integer, rational, irrational, or real. More than one may apply. Rational, Real Integer, Rational, Real,Whole, Integer, Rational, Real Natural Integer, Rational, Real Irrational, Real

The word commute means to go back and forth. Many people commute to work or to school. If you travel from home to work and follow the same route from work to home, you travel the same distance each time. Commutative Properties Addition Multiplication Slide Use the commutative properties.

Solution: Use a commutative property to complete each statement. Slide EXAMPLE 1 Using the Commutative Properties

When we associate one object with another, we think of those objects as being grouped together. We can group the first two together or the last two together and get the same answer. Associative Properties Addition Multiplication Slide Use the associative properties.

Use an associative property to complete each statement. Solution: Slide EXAMPLE 2 Using the Associative Properties

Solution: Find the sum. Slide EXAMPLE 4 Using the Commutative and Associative Properties

If a child wears a costume on Halloween, the child’s appearance is changed, but his or her identity is unchanged. Likewise, the identity of a real number is left unchanged when identity properties are applied. The number 0 leaves the identity, or value, of any real number unchanged by addition. So 0 is called the identity element for addition, or the additive identity. Since multiplication by 1 leaves any real number unchanged, 1 is the identity element for multiplication, or the multiplicative identity. Identity Properties Addition Multiplication Slide Use the identity properties.

Solution: Use an identity property to complete each statement. Slide EXAMPLE 5 Using the Identity Properties

Solution: Simplify. Slide EXAMPLE 6 Using the Identity Property to Simplify Expressions

Each day before you go to work or school, you probably put on your shoes before you leave. Before you go to sleep at night, you probably take them off, and this leads to the same situation that existed before you put them on. These operations from everyday life are examples of inverse operations. Inverse Properties Addition Multiplication Slide Use the inverse properties.

Solution: Use an inverse property to complete each statement. Slide EXAMPLE 7 Using the Inverse Properties

Use the distributive property to rewrite each expression. Solution: Slide EXAMPLE 9 Using the Distributive Property

The everyday meaning of the word distribute is “to give out from one to several.” Look at the value of the following expressions:, which equals or 26, which equals, or 26. Since both expressions equal 26, Slide Use the distributive properties.

Solution: Write the expression without parentheses. Slide EXAMPLE 10 Using the Distributive Property to Remove (Clear) Parentheses

Now Let’s Practice

3 + 7 = Commutative Property of Addition 2.

8 + 0 = 8 Identity Property of Addition 3.

6 4 = 4 6 Commutative Property of Multiplication 5.

17 + (-17) = 0 Inverse Property of Addition 6.

2(5) = 5(2) Commutative Property of Multiplication 7.

(2 + 1) + 4 = 2 + (1 + 4) Associative Property of Addition 1.

even + even = even Closure Property 8.

3(2 + 5) = Distributive Property 9.

6(78) = (67)8 Associative Property of Multiplication 10.

5 1 = 5 Identity Property of Multiplication 11.

(6 – 3)4 = 64 – 34 Distributive Property 13.

1(-9) = -9 Identity Property of Multiplication 14.

3 + (-3) = 0 Inverse Property of Addition 15.

1 + [-9 + 3] = [1 + (-9)] + 3 Associative Property of Addition 16.

-3(6) = 6(-3) Commutative Property of Multiplication 17.

= -8 Identity Property of Addition 18.

37 – 34 = 3(7 – 4) Distributive Property 19.

6 + [(3 + (-2)] = (6 + 3) + (- 2) Associative Property of Addition 20.

7 + (-5) = Commutative Property of Addition 21.

(5 + 4)9 = Distributive Property 22.

-3(5 4) = (-3 5)4 Associative Property of Multiplication 23.

-8(4) = 4(-8) Commutative Property of Multiplication 24.

5 1 / = 5 1 / 7 Identity Property of Addition 25.

3 / 4 – 6 / 7 = – 6 / / 4 Commutative Property of Addition 26.

1 2 / 5 1 = 1 2 / 5 Identity Property of Multiplication 27.

(fraction)(fraction) = fraction Closure Property 28.

-8 2 / = -8 2 / 5 Identity Property of Addition 29.

[(- 2 / 3 )(5)]9 = - 2 / 3 [(5)(9)] Associative Property of Multiplication 30.

6(3 – 2n) = 18 – 12n Distributive Property 31.

2x + 3 = 3 + 2x Commutative Property of Addition 32.

ab = ba Commutative Property of Multiplication 33.

a + 0 = a Identity Property of Addition 34.

a(bc) = (ab)c Associative Property of Multiplication 35.

a1 = a Identity Property of Multiplication 36.

a +b = b + a Commutative Property of Addition 37.

a(b + c) = ab + ac Distributive Property 38.

a + (b + c) = (a +b) + c Associative Property of Addition 39.

a + (-a) = 0 Inverse Property of Addition 40.