Chapter 1 Basic Concepts

Chapter Sections 1.1 – Study Skills for Success in Mathematics, and Use of a Calculator 1.2 – Sets and Other Basic Concepts 1.3 – Properties of and Operations with Real Numbers 1.4 – Order of Operations 1.5 – Exponents 1.6 – Scientific Notation Chapter 1 Outline

Properties of and Operations with Real Numbers § 1.3 Properties of and Operations with Real Numbers

Additive Inverse Two 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 Inverse For any real number a, its additive inverse is –a Double Negative Property For any real number, a –(-a) = a

Absolute Value The 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 -1 1 2 3 4 5 4 units

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 -1 1 2 3 4 5 The second arrow starts at 4 and is drawn 2 units to the left , since the second number is negative. -5 -4 -3 -2 -1 1 2 3 4 5 4 + (– 2) = 2

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) = 11 The sum of two positive numbers will always be positive and the sum of two negative numbers will always be negative.

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) = -4 The 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.

Least Common Denominator The least common denominator (LCD) of a set of denominators is the smallest number that each denominator divides into without remainder.

Least Common Denominator Example: The LCD is 27. Rewriting the first fraction with the LCD gives the following.

Subtraction of Real Numbers If a and b represent two real numbers, then a – 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) = -5 b.) – 6 – 4 = – 6 + (– 4) = – 10

Subtracting a Negative Number If a and b represent two real numbers, then a – (-b) = a + b Example: a.) -4 – (-11) = -4 +11 = 7

More Examples Example: a.) – 42 – 35 = -77 b.)

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) = –6.72 b.) (-18)(-1/2) = 9

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 = –6 Multiplication – 3(–5) = 15 2(–4) = –8

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) = –6 b.) –6.45  (–0.4) = 16.125

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.

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 .

Dividing with Zero If a represents any real number except 0, then Division by 0 is undefined.