Chapter 1 Section 1 Properties of Real Numbers

1-1 ALGEBRA 2 LESSON 1-1 Simplify. 1.–(–7.2)2.1 – (–3) 3.–9 + (–4.5) 4.(–3.4)(–2) Properties of Real Numbers 5.–15 ÷ 3 6.– + ( – )

Solutions 1.–(–7.2) = –9 + (–4.5) = – –15 ÷ 3 = –5 6.– + ( – ) = = – = – –2 + (–3) – (–3) = = 4 4.(–3.4)(–2) =

Classifications of Numbers Imaginary Numbers will be introduced later.

Real Numbers The largest classification we will deal with Include any number that you can tell me  Ex: Split into Rational and Irrational Numbers

Real Numbers Irrational Numbers Numbers that cannot be written as ratios Decimals that never terminate and never repeat Square roots of positive non-perfect squares Ex: √2, -√7, √(8/11), …

Real Numbers Rational Numbers All the numbers that can be written as a ratio (fraction) This includes terminating and repeating decimals. Ex: 8, 10013, -54, 7/5, -3/25, 0, 0/6, -1.2,.09,.3333….

Real Numbers Rational Numbers Integers “Complete” numbers (no parts – fractions or decimals) Negative, Zero, and Positive Each negative is the additive inverse (or opposite) of the positive Ex: -543, 76, 9, 0, -34

Real Numbers Rational Numbers Integers Whole Numbers Zero and positive integers Ex: 0, 1, 2, 3, 4, …

Real Numbers Rational Numbers Integers Whole Numbers Natural Numbers Also known as Counting Numbers Think of young children Ex: 1, 2, 3, 4, 5, 6, …

Which set of numbers best describes the variable? a) The cost C in dollars for admission for n people b) The cost C of admission is a rational number and the number n of people is a whole number c) The maximum speed s in meters per second on a roller coaster of height h in meters d) Since the speed s is calculated using a formula with a square root, s is real (either rational or irrational). The height h is measured in rational numbers. e) The park’s profit (or loss) P in dollars for each week w of the year f) The profit P is a rational number and the week number w is natural.

Try This Problem p. 6 Check Understanding The number r is the ratio of the number of adult tickets sold to the number of children’s tickets sold. Which set of numbers best describes the values of r? Which set of numbers best describes the average cost c per family for tickets? r is a ratio, so it is a rational number; c is the cost which means a terminating decimal, so it is a rational number.

Graphing Numbers on a Number Line Make sure your number lines have zero Make them fairly accurate Label Important points

Ordering Real Numbers Less Than<  Or equal to≤ Greater Than >  Or equal to ≥

Properties of Real Numbers Opposite or Additive Inverse – of any number a is –a  The sum of opposites is 0. Reciprocal or Multiplicative Inverse – of any number a is  Think of it as a flip Remember you may have to make a decimal into a fraction before flipping it. Use the place (hundredths) to write a fraction.  The product of reciprocals is 1.

Find the opposite and reciprocal of each number. a). b) -3.2

Try These Problems p. 7 Check Understanding State the opposite and reciprocal of each number. a)400 b)4 1/5 c)-.002 d)-4/9 OppositeReciprocal / /55/ /9-9/4

Properties of Real Numbers PropertyAdditionMultiplication Closurea+b is a real number ab is a real number Commutative commute = to move a + b = b + aAb = ba Associative associate = regroup (a+b)+c = a+(b+c)(ab)c = a(bc) Identitya+0=a,0+a=aa*1=a, 1*a=a Inversea+(-a)=0a*(1/a)=1,a≠0 Distributivea(b+c) = ab + ac

Identify the property Example 5 Which Property is illustrated? a) 6 + (-6) = 0 a) Inverse Property of Addition b) (-4 ∙ 1) – 2 = -4 – 2 a) Identity Property of Multiplication You MUST state Addition or Multiplication. Appropriate abbreviations: Prop. Of Add. Or Prop. Of Mult. Comm. Assoc. Ident. Inv. Dist

Try these Problems p. 7 Check Understanding Which Property is illustrated? a) (3 + 0) – 5 = 3 – 5 a) Identity Property of Addition b) -5 + [2 + (-3)] = (-5 + 2) + (-3) a) Associative Property of Addition

Absolute Value The absolute value of a number is the distance from zero on the number line. Its always positive. Be careful to watch for negatives outside the absolute value bars (then the answer is negative). Absolute Value Symbols │a│

Example 6 a) │-4 │ = b) │0 │= c) │-1 ∙ (-2) │= d) │-10 │= e) │1.5 │= f) │0 - 3 │= a) 4 b) 0 c) 2 d) 10 e) 1.5 f) 3