 # Properties of Real Numbers The properties of real numbers allow us to manipulate expressions and equations and find the values of a variable.

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Properties of Real Numbers The properties of real numbers allow us to manipulate expressions and equations and find the values of a variable.

Number Classification Natural numbers are the counting numbers. Whole numbers are natural numbers and zero. Integers are whole numbers and their opposites. Rational numbers can be written as a fraction. Irrational numbers cannot be written as a fraction. All of these numbers are real numbers.

Number Classifications Subsets of the Real Numbers I - Irrational Integers Whole Natural Q - Rational

There are also numbers that are NOT real. These are called imaginary or complex numbers. These numbers complete the categories of all numbers. Number Classifications

Irrational Integers Whole Natural Rational Imaginary

Classify each number -1 real, rational, integer real, rational, integer, whole, natural real, irrational real, rational real, rational, integer, whole real, rational 6 - 2.222 0

Properties of Real Numbers Commutative Property Think… commuting to work. Deals with ORDER. It doesn’t matter what order you ADD or MULTIPLY. a+b = b+a 4 6 = 6 4

Properties of Real Numbers Associative Property Think…the people you associate with, your group. Deals with grouping when you Add or Multiply. Order does not change.

Properties of Real Numbers Associative Property (a + b) + c = a + ( b + c) (nm)p = n(mp)

Properties of Real Numbers Additive Identity Property s + 0 = s Multiplicative Identity Property 1(b) = b

Distributive Property a(b + c) = ab + ac (r + s)9 = 9r + 9s Properties of Real Numbers

5 = 5 + 0 5(2x + 7) =10x + 35 8 7 = 7 8 24(2) = 2(24) (7 + 8) + 2 = 2 + (7 + 8) Additive Identity Distributive Commutative Name the Property

7 + (8 + 2) = (7 + 8) + 2 1 v + -4 = v + -4 (6 - 3a)b = 6b - 3ab 4(a + b) = 4a + 4b Associative Mult. Identity Distributive

Properties of Real Numbers Reflexive Property a + b = a + b The same expression is written on both sides of the equal sign.

Properties of Real Numbers If a = b then b = a If 4 + 5 = 9 then 9 = 4 + 5 Symmetric Property

Properties of Real Numbers Transitive Property If a = b and b = c then a = c If 3(3) = 9 and 9 = 4 +5, then 3(3) = 4 + 5

Properties of Real Numbers Substitution Property If a = b, then a can be replaced by b. a(3 + 2) = a(5)

Name the property 5(4 + 6) = 20 + 30 5(4 + 6) = 5(10) 5(4 + 6) = 5(4 + 6) If 5(4 + 6) = 5(10) then 5(10) = 5(4 + 6) 5(4 + 6) = 5(6 + 4) If 5(10) = 5(4 + 6) and 5(4 + 6) = 20 + 30 then 5(10) = 20 + 30 Distributive Substitution Reflexive Symmetric Commutative Transitive

Solving Equations To solve an equation, find replacements for the variables to make the equation true. Each of these replacements is called a solution of the equation. Equations may have {0, 1, 2 … solutions.

Solving Equations 3(2a + 25) - 2(a - 1) = 78 6a + 75 – 2a + 2 = 78 (6a – 2a) + (75+2) = 78 4a + 77 = 78 4a = 1 a = 1/4

Solving Equations 4(x - 7) = 2x + 12 + 2x 4x – 28 = 4x +12 4x = 4x + 40 0 = 40 This is NOT true!!! There is NO solution to this.

Solving Literal Equations Solve: V = πr 2 h, for h The object is to get h by itself. We do this by reversing everything that is being done to h, in REVERSE of the Order of Operations rules. Right now, h is being multiplied by πr 2. We reverse this by dividing both sides by πr 2. So, h = V/ πr 2

Solving Literal Equations Solve: de - 4f = 5g, for e Again, we need to isolate e. We do this by following the Order of Operations in REVERSE!! We will add the 4f first, then divide by d. de = 5g + 4f e = (5g + 4f) / d

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