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P.1 Real Numbers and Algebraic Expressions

P.1 Real Numbers and Algebraic Expressions. Negative numbers. Positive numbers. -4 -3 -2 -1 0 1 2 3 4. Units to the left of the origin are negative. Units to the right of the origin are positive. the Origin. The Real Number Line.

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P.1 Real Numbers and Algebraic Expressions

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  1. P.1 Real Numbers and Algebraic Expressions

  2. Negative numbers Positive numbers -4 -3 -2 -1 0 1 2 3 4 Units to the left of the origin are negative. Units to the right of the origin are positive. the Origin The Real Number Line The real number line is a graph used to represent the set of real numbers. An arbitrary point, called the origin, is labeled 0;

  3. Ordering the Real Numbers On the real number line, the real numbers increase from left to right. The lesser of two real numbers is the one farther to the left on a number line. The greater of two real numbers is the one farther to the right on a number line. -2 -1 0 1 2 3 4 5 6 Since 2 is to the left of 5 on the number line, 2 is less than 5. 2 < 5 Since 5 is to the right of 2 on the number line, 5 is greater than 2. 5 > 2

  4. Symbols Meaning Example Explanation a<b a is less than or equal to b. 3 < 7 Because 3 < 7 7 < 7 Because 7 =7 b>a b is greater than or equal to a. 7 > 3 Because 7 > 3 -5 > -5 Because -5 = -5 Inequality Symbols Be especially careful when the numbers are negatively signed. The relationship will be the opposite of what it would be if positive. Ex: 5<7, yet –5>-7. Always “grab” the larger (further to the right) number. Ex: -  ______ -3.14 (ans: < since 3.14 ends & neg.)

  5. For example, the real number line below shows that |-3| = 3 and |5| = 5. |–3| = 3 |5| = 5 -3 -2 -1 0 1 2 3 4 5 The absolute value of –3 is 3 because –3 is 3 units from 0 on the number line. The absolute value of 5 is 5 because 5 is 5 units from 0 on the number line. Absolute Value Absolute value describes the distance from 0 on a real number line. If a represents a real number, the symbol |a| represents its absolute value, read “the absolute value of a.”

  6. { x if x> 0 |x| = -x if x < 0 Definition of Absolute Value The absolute value of x is given as follows: Note: the absolute value of any real number will always be positive or zero. If it is originally negatively signed (less than zero), we must take the opposite sign. (|-3| = - (-3)) If it is originally positively signed (greater than zero) we want to keep it positively signed. Ex: Find | \/7 - | (ans: pi-root7 since neg) Ex: Evaluate the given expression for x = -5 and y = 7 |x-y|/(x-y) (ans: -1) Q: If |x| = 5, what possible values could x equal? (ans5 or –5)

  7. Distance Between Two Points on the Real Number Line You live 80 miles east off I-4, your partner lives 20 miles east off I-4. What is the distance between you and your partner? If your partner lived 20 miles WEST off I-4 what would the distance be? If a and b are any two points on a real number line, then the distance between a and b is given by |a – b| or |b – a| Show how we could write the above using integers and absolute value symbols. (ans: |80-20| and if WEST: |80-(-20)|)

  8. ? -5 -4 -3 -2 -1 0 1 2 3 Ex: Solution Because the distance between a and b is given by |a – b|, the distance between –5 and 3 is | -5-3 | = | -8 | = 8. Find the distance between –5 and 3 on the real number line. We obtain the same distance if we reverse the order of subtraction: | 3-(-5) | = | 8 | = 8.

  9. Algebraic Expressions A combination of variables and numbers using the operations of addition, subtraction, multiplication, or division, as well as powers or roots, is called an algebraic expression. Note: there is NO EQUAL SIGN, and we can only SIMPLIFY, not SOLVE! Here are some examples of algebraic expressions: x + 6, x – 6, 6x, x/6, 3x + 5. Note: In none of these cases can you SOLVE for the variable, the best you can do is SIMPLIFY. In what case could you “solve”? What would that be called (instead of an expression?)

  10. The Order of Operations Agreement • Perform operations within the innermost parentheses (or any other grouping symbol such as a radical sign, absolute value, or division bar) and work outward. If the algebraic expression involves division, treat the numerator and the denominator as if they were each enclosed in parentheses. • Evaluate all exponential expressions. • Perform multiplication or division as they occur, working from left to right. • Perform addition or subtraction as they occur, working from left to right. • That is: remove parenthesis and use order of operations to combine “like” terms. Ex: Simplify 123•2 (ans: 8, NOT 2!)

  11. Text Example The algebraic expression 2.35x + 179.5 describes the population of the United States, in millions, x years after 1980. Evaluate the expression when x = 20. Describe what the answer means in practical terms. Solution We begin by substituting 20 for x. Because x = 20, we will be finding the U.S. population in 1980 + 20 or the year 2000. 2.35x + 179.5 Replace x with 20. = 2.35(20) + 179.5 = 47 + 179.5 Perform the multiplication. = 226.5 Perform the addition. Thus, in the year 2000 the population of the United States was 226.5 million. Note the FORM of the simplification. Your test papers should look like this.

  12. Example Solution 6(2x – 4y) + 10(4x + 3y) Simplify: 6(2x – 4y) + 10(4x + 3y). = Use the distributive property. = Multiply. = Combine like terms. Note: when we add “like” terms, we add the coefficient parts, but leave the variable part (the like part) the same. (ans: 53x+6y) Think: 2(ab) vs 2(a+b): In which case do we use the distributive law and why? (2nd only, 1st is all mult.)

  13. A set is a collection of objects whose contents can be clearly determined. From this point I have provided these sheets as review only. Please review and answer the questions. Answers will be discussed next class.The Basics About Sets The set {1, 3, 5, 7, 9} has five elements. • The objects in a set are called the elements of the set. • We use braces to indicate a set and commas to separate the elements of that set. For example, The set of counting numbers can be represented by {1, 2, 3, … }. The set of even counting numbers are {2, 4, 6, …}. The set of even counting numbers is a subset of the set of counting numbers, since each element of the subset is also contained in the set.

  14. Name Description Examples Natural Numbers N {1, 2, 3, …} These are the counting numbers 4, 7, 15 Whole Numbers W {0, 1, 2, 3, … } Add 0 to the natural numbers 0, 4, 7, 15 Integers Z {…, -2, -1, 0, 1, 2, 3, …} Add the negative natural numbers to the whole numbers -15, -7, -4, 0, 4, 7 Important Subsets of the Real Numbers

  15. Name Description Examples Rational Numbers Q These numbers can be expressed as an integer divided by a nonzero integer: Rational numbers can be expressed as terminating or repeating decimals. Irrational Numbers I This is the set of numbers whose decimal representations are neither terminating nor repeating. Irrational numbers cannot be expressed as a quotient of integers. Important Subsets of the Real Numbers

  16. The set of real numbers is formed by combining the rational numbers and the irrational numbers. The Real Numbers Rational numbers Irrational numbers Integers Whole numbers Natural numbers

  17. -4 -3 -2 -1 0 1 2 3 4 Graphing on the Number Line Real numbers are graphed on the number line by placing a dot at the location for each number. –3, 0, and 4 are graphed below. Q: How could we graph –13/5 on the real number line? (ans: -13/5 = -2and 3/5, so it would be a little over half way from –2 to –3.)

  18. |a| a b |b| Properties of Absolute Value For all real number a and b, 1. |a| > 0 2. |-a| = |a| 3. a < |a| 4. |ab| = |a||b| 5. = , b not equal to 0 6. |a + b| < |a| + |b| (the triangle inequality)

  19. Example • Find the following: |-3| and |3|. |-5•7| and |-5+7| (Ans: 3 and 3, 35 and 2)

  20. Name Meaning Examples Commutative Property of Addition Two real numbers can be added in any order. a + b = b + a • 13 + 7 = 7 + 13 • 13x + 7 = 7 + 13x Commutative Property of Multiplication Two real numbers can be multiplied in any order. ab = ba • x · 6 =6x Associative Property of Addition If 3 real numbers are added, it makes no difference which 2 are added first. (a + b) + c = a + (b + c) • 3 + ( 8 + x) • = (3 + 8) + x • = 11 + x Properties of the Real Numbers Think: How is 3(8+x) different and why can’t we use the associative property? (Ans: two operations: mult & addtn.)

  21. Name Meaning Examples Associative Property of Multiplication If 3 real numbers are multiplied, it makes no difference which 2 are multiplied first. (a · b) · c = a · (b · c) • -2(3x) = (-2·3)x = -6x Distributive Property of Multiplication over Addition Multiplication distributes over addition. a · (b + c) = a · b + a · c • 5 · (3x + 7) • = 5 · 3x + 5 · 7 • = 15x + 35 Identity Property of Addition Zero can be deleted from a sum. a + 0 = a 0 + a = a • 0 +6x = 6x Properties of the Real Numbers

  22. Name Meaning Examples Identity Property of Multiplication One can be deleted from a product. a · 1 = a and1 · a = a • 1 · 2x = 2x Inverse Property of Addition The sum of a real number and its additive inverse gives 0, the additive identity. a + (-a) = 0 and (-a) + a = 0 • (-6x) +6x= 0 Inverse Property of Multiplication The product of a nonzero real number and its multiplicative inverse gives 1, the multiplicative identity. a · 1/a =1and1/a · a =1 • 2 · 1/2 = 1 Properties of the Real Numbers

  23. Definitions of Subtraction and Division Subtraction: a – b = a + (-b) We call –b the additive inverse or opposite of b. Let a and b represent real numbers. Division: a ÷ b = a · 1/b, where b = 0 We call 1/b the multiplicative inverse or reciprocal of b. The quotient of a and b, a ÷ b, can be written in the form a/b, where a is the numerator and b the denominator of the fraction. Rewrite using multiplication: 10  2 10  (1/2) (ans: 10*(1/2) and 10 * (2/1) or 10*2)

  24. Properties of Negatives • Let a and b represent real numbers, variables, or algebraic expressions. • (-1)a = -a • -(-a) = a • (-a)(b) = -ab • a(-b) = -ab • -(a + b) = -a - b • -(a - b) = -a + b = b - a

  25. Algebra Translation Dictionary You should add to this list as you discover new translations. “Of” means use parenthesis (see p15 #111 ~118 to practice.) Translation: Example:

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