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Transcript
Chapter 1 Set of Real Numbers 1.1 Sets of Numbers and the Real Number Line A set is a collection of objects. We are interested in different sets of numbers. Notation: the set containing 1, 3, and 5 is written {1, 3, 5} Natural numbers {1, 2, 3, . . .} Whole numbers {0, 1, 2, 3, . . .} Note that to get the set of whole numbers, we took the set of natural numbers and included 0. Thus every natural number is also in the set of whole numbers. We say that the set of natural numbers is contained in or is a subset of the whole numbers. Integers {. . . –3, –2, –1, 0, 1, 2, 3, . . .} Note that to get the set of integers, we took the set of whole numbers and included the negatives of each of them. The set of whole numbers is a subset of the set of integers. Rational numbers: Any number that can be put in the form a where a and b b are integers with b ≠ 0. Are the following rational numbers? 2 3 5 -3 0 Irrational numbers: Any real number that is not rational. That is, it cannot be a b put in the form . Examples 2 π When rational and irrational numbers are written as decimal numbers, we see this difference: 1) When a rational number is written as a decimal number, it will either terminate or have a repeating block of digits. 2) When an irrational number is written as a decimal number, it will continue forever without a repeating block of digits. Examples Rational: 1 0.25 4 1 0.33333 3 3 1 0.142857142857 7 Irrational: π = 3.14159265359. . . 5 2.2360679775 Real numbers: The set of rational numbers combined with the set of irrational numbers. Example Classify: 0.569 Yes Rational Real ___ No____________ Natural Whole Integer Irrational Classify: 2 Yes ___ Irrational Real No____________ Natural Whole Integer Rational All the numbers we will be using will be real numbers. An example of a nonreal number would be 2 . How do the sets of numbers fit together? Real numbers: Rational numbers Integers Whole numbers Natural numbers Irrational numbers The set of real numbers can also be defined as all numbers corresponding to points on the number line. Thus, the real numbers are “ordered”. A number a is less than the number b if a is to the left of b on the number line. Order symbols < less than: 4 < 20 ≤ less than or equal to: 4 ≤ 20 5 ≤ 5 is true but 5 < 5 is not true > greater than: 16 > 5 ≥ greater than or equal to: 16 ≥ 5 16 ≥ 16 is true but 16 > 16 is not ≠ not equal to 5 ≠7 Put the appropriate sign(s) between the two numbers: 5 87 Signs that work: ≠ < 12 12 Signs that work: = ≤ ≥ -8 - 12 Signs that work: > ≥ ≠ ≤ Note that the inequality signs always “point to” the smallest number. The absolute value of a number is its distance from 0 on the number line. Note that this definition says nothing about the direction, only the distance. Notation: The absolute value of a: | a | | | | | | -4 -3 -2 -1 0 | 1 | 2 | | 3 4 | - 3| = | 3 | = 3 since 3 and – 3 are both 3 steps from 0 Note that since absolute value is a distance, | a | ≥ 0 for any number a. Two numbers are opposites if they have the same absolute value but lie in opposite directions from 0 on the number line. −3 and 3 are opposites. The sum of opposites is 0. Example Find the opposite of 2. The opposite of 2 is -2. Find the opposite of -3. The opposite of -3 is 3 -(-3) = 3 * KEY concept that will be used later!!!* | 32| = 32 | - 45| = 45 |0|=0 −|−6|=6 1.2 Order of Operations A variable is a letter that is used to hold the place of a number. It is called a variable because it can take on different values. Example Suppose your car gets 20 miles per gallon of gas. We can express how far you can drive your car as 20n where n represents how much gas you have. The 20 is called a constant since its value does not change. 20n is called an algebraic expression. When we put in some number for n and determine the distance you can drive, we are evaluating the expression. An algebraic expression consists of variables and/or numbers combined with operation symbols and/or grouping symbols. Example x + 25 3(n – 4) 2x2 – 6x 28 ÷ 3y An equation is a mathematical statement that sets two expressions as being equal. Example 3(n – 4) = 32 Note the difference between expressions and equations. Example evaluate 2(x – 3y) for x = 10 and y = 2 2(10 − 3·2) = 2(10 – 6) = 2(4) = 8 Exponents Exponent 23 Base The exponent tells how many times the base is used as a factor. Thus, 23 = 2 · 2 · 2 Example 24 2 3 3 32 (− 3)2 81 − 32 The square root of a number is the number we would have to square to get that number. The radical sign is used to find the principal square root (the positive root). The square root is the inverse operation to squaring a number. 25 = 5 since 52 = 25 Example 16 100 64 0 4 Order of Operations 1. Grouping Symbols: Do all operations within grouping symbols. Grouping symbols include: parenthesis, brackets, braces, radicals, absolute value bars and fraction bars. 2. Exponents, Radicals and Absolute Values: Evaluate all powers, radicals and absolute values. 3. Multiplications and Divisions: Perform multiplication and division from left to right in the order they appear in the problem. 4. Additions and Subtractions: Perform addition and subtraction from left to right in the order they appear in the problem. Example Simplify: – 2 + [(8 – 11) – (– 2 – 9)] – 2 + [– 3 – (– 11)] – 2 + [– 3 + 11] –2+8=6 Simplify: – 3 + [(– 2 – 5) – 2] – 3 + [– 7 – 2] – 3 + (– 9) = – 12 Simplify: 2[5 + 2(8 – 3)] 2[5 + 2(5)] 2[5 + 10] 2[15] = 30 Simplify: 3(10)2 8 22 = 3(100) – 8 ÷ 4 = 300 – 8 ÷ 4 = 300 – 2 = 298 Simplify: 6 2 8 32 18 3 6 6 32 18 3 669 18 3 21 15 7 5 In order to apply math to real life situations, we need to be able to translate English phrases and sentences into algebraic expressions and equations. Words for addition: add, the sum of, plus, more than, increased by Words for subtraction: subtract, minus, the difference of, less than, decreased by Words for multiplication: multiply, the product of, times, twice, of (with a fraction, like ½ of 8 means ½ times 8) Word for division: divide, the quotient of, ratio of, per Words for variables: For a phrase like “some number”, “a number”, “what number”, we use a variable. Words for equal: is, equals, results in Example The sum of 2 and 3: 2+3 The product of 6 and a number: 6n The quotient of 24 and 4: 24 ÷ 4 2 times the sum of a number and 5: 2(x + 5) The sum of 2 times a number and 5: 2x + 5 8 subtracted from twice a number: 2n – 8 The difference of 8 and twice a number: 8 – 2n The quotient of 18 and a number is 9: 18 ÷ n = 9 5 subtracted from 3 times a number is equal to 21: 3x – 5 = 21 Section 1.3 Addition of Real Numbers Adding real numbers can be visualized on the number line. To add a positive number, move right on the number line and to add a negative number, move left on the number line. To add numbers with the same sign: 1. Add their absolute values 2. Give the answer the common sign. To add numbers with different signs: 1. Subtract the smaller absolute value from the larger 2. Give the answer the sign of the larger. Note: We can think of – as meaning “negative”, “minus” or “the opposite of” Section 1.4 Subtraction of Real Numbers The opposite or additive inverse of a number a is written − a. Subtracting a number is the same as adding its opposite. 1. Change the subtraction symbol to the additional symbol 2. Change the sign of the number being subtracted 3. Add using the rules of addition So 5 – 7 is the same as 5 + (-7) 6 – (-2) is the same as 6 + 2 Section 1.5 Multiplication and Division of Real Numbers Multiplying and Dividing Real Numbers 1. Multiply, divide and absolute values. 2. If the two numbers have the same sign, the answer is positive. If the two numbers have different signs, the answer is negative. Notation for multiplication: 3·2 3x2 (3)2 3(2) (3)(2) Notation for division: 25 5 25 5 5 25 Zero property a·0=0 0 ÷ a = 0 for any real number a. Division by 0 is undefined Example − 9 · (− 2) −3 · (− 5) · (− 2) · (−1) (− 1)(− 1)(− 1)(− 1)(− 1)(− 1)(− 1) 28 28 28 = = 7 7 7 Property: a a a b b b 50 = 5 50 5 Property: a a b b Section 1.6 Properties of Real Numbers and Simplifying Expressions Expressions that represent the same number are called equivalent expressions. The expressions 3 + 8 and 8 + 3 are equivalent expressions. To solve equations, we use mathematical laws and properties to write equivalent expressions. For the following properties, we let a, b, c represent any real number. The Commutative Properties For addition: a + b = b + a For multiplication: ab = ba When we add or multiply numbers, the order of the numbers does not matter. The Associative Properties For addition: a + (b + c) = (a + b) + c For multiplication: a(bc) = (ab)c The associative property allows us to change the grouping of numbers when we add or multiply. The Distributive Property a(b + c) = ab + ac a(b – c) = ab – ac Identities The additive identity is 0. For any real number a: a 0 a The multiplicative identity is 1. any real number a: a 1 a For Inverse Properties Inverse property of addition: a (a) 0 Inverse property of multiplication: 1 b 1 b Combining like terms A term is a number or the product of numbers and variables. Examples 3 5x 7xy 2x3 When two terms have the same variable parts, the terms are called similar terms or like terms. The distributive property helps us combine similar terms. 2x + 3x = (2 + 3)x like 2 marbles + 3 marbles give us 5 marbles 7ab – 4ab = 3ab What about 4x + 7x2? They are not like terms, so they cannot be combined. Examples 3(2x − 5) + 3x 5 + 7y(2 − 3y) − 2y(4y − 3) 5 – (2y – 8) 4a + 2a2 - 6a + 5 + 3a2 - 2 9k – 6 – 3(2 – 5k)