2 Chapter Sections1.1 – Study Skills for Success in Mathematics, and Use of a Calculator1.2 – Sets and Other Basic Concepts1.3 – Properties of and Operations with Real Numbers1.4 – Order of Operations1.5 – Exponents1.6 – Scientific NotationChapter 1 Outline
3 Properties of and Operations with Real Numbers § 1.3Properties of and Operations with Real Numbers
4 Additive InverseTwo numbers that are the same distance from 0 on the number line but in opposite directions are called additive inverses, or opposites, of each other.Additive InverseFor any real number a, its additive inverse is –aDouble Negative PropertyFor any real number, a –(-a) = a
5 Absolute ValueThe absolute value of a number is its distance from the number 0 on the real number line. The absolute value of every number will be either 0 or positive-5-4-3-2-1123454 units
6 Number Lines Add 4 + (– 2) using a number line 4 + (– 2) = 2 Always begin with 0.Since the first number is positive, the first arrow starts at 0 and is drawn 4 units to the right.-5-4-3-2-112345The second arrow starts at 4 and is drawn 2 units to the left , since the second number is negative.-5-4-3-2-1123454 + (– 2) = 2
7 Add Real Numbers To add real numbers with the same sign, add their absolute values. The sum has the same sign as the numbers being added.Example:–4 + (–7) = 11The sum of two positive numbers will always be positive and the sum of two negative numbers will always be negative.
8 Adding with Different Signs To add real numbers with the different signs, subtract the smaller absolute value from the larger absolute value. The sum has the sign of the number with the larger absolute value.Example:5 + (–9) = -4The sum of two numbers with different signs may be positive or negative. The sign of the sum will be the same as the sign of the number with the larger absolute value.
9 Least Common Denominator The least common denominator (LCD) of a set of denominators is the smallest number that each denominator divides into without remainder.
10 Least Common Denominator Example:The LCD is 27. Rewriting the first fraction with the LCD gives the following.
11 Subtraction of Real Numbers If a and b represent two real numbers, thena – b = a + (– b)In other words, to subtract b from a, add the additive inverse of b to a.Example:a.) 3 – (8) =3 + (– 8) = -5b.) – 6 – 4 = – 6 + (– 4) = – 10
12 Subtracting a Negative Number If a and b represent two real numbers, thena – (-b) = a + bExample:a.) -4 – (-11) = = 7
14 Multiply Two Real Numbers To multiply two numbers with like signs, multiply their absolute values. The product is positive.To multiply two numbers with unlike signs, one positive and the other negative, multiply their absolute values. The product is negative.Example:a.) (4.2)(–1.6) = –6.72b.) (-18)(-1/2) = 9
15 Caution!It is very easy to mix up subtraction and multiplication problems.– 3 – 5 is not the same as –3(–5).2 – 4 is not the same as 2(–4)Subtraction– 3 – 5 = –8– 2 – 4 = –6Multiplication– 3(–5) = 152(–4) = –8
16 Divide Real Numbers Example: a.) -24 (4) = –6 To divide two numbers with like signs, either both positive or both negative, divide their absolute values. The quotient is positive.To divide two numbers with unlike signs, one positive and the other negative, divide their absolute values. The quotient is negative.Example:a.) -24 (4) = –6b.) –6.45 (–0.4) =
17 Multiplication vs. Division For multiplication and division of two real numbers:(+)(+) = + (+) ÷ (+) = +(–)(–) = + (–) ÷ (–) = +(+)(–) = – (+) ÷ (–) = –(–)(+) = – (–) ÷ (+) = –Like signs give positive products and quotients.Unlike signs give negative products and quotients.
18 Signs of a Fraction If a and b represent any real numbers, b 0, then We generally do not write fractions with a negative sign in the denominator.The fraction would be written as or
19 Dividing with Zero If a represents any real number except 0, then Division by 0 is undefined.