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Types of Numbers 10 Real Rational Irrational Integer Whole Counting For each of the following numbers, circle all sets to which it belongs: 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. √8 -2 51.3 0 83 0.5555… π 0.35 -9 /2 -0.341 real real real real real real real real real real irrational irrational irrational irrational irrational irrational irrational irrational irrational irrational rational rational rational rational rational rational rational rational rational rational integer integer integer integer integer integer integer integer integer integer whole whole whole whole whole whole whole whole whole whole counting counting counting counting counting counting counting counting counting counting PROPERTIES Commutative Property The order in which you add or multiply two numbers does not change the sum or product. a+b=b+a a•b = b•a 3 + (-2) = -2 + 3 4•(-5) = -5•4 Associative Property The way you group three numbers in a sum or product does not change the sum or product. (a + b) + c = a + (b + c) (-3 + 2) + 1 = -3 + (2 + 1) (a•b)•c = a•(b•c) (-2•7) •4 = -2•(7•4) Identity Property The sum of zero and a number is that number. The product of a number and 1 is that number. a+0=a a•1 = 1•a = a -5 + 0 = -5 (-5)•1 = -5 Inverse Property The sum of a number and its opposite is 0. 3 1 1 3 a + (-a) = -a + a = 0 -6 + 6 = 0 Property of Zero The product of a number and 0 is 0. a•0 = 0•a = 0 -3•0=0 Property of -1 The product of a number and -1 is the opposite of the number. a•(-1) = -1•a = -a -2•(-1) = 2 Distributive Property a(b + c) = ab + ac 3(4 + 2) = 3(4) + 3(2) Reflexive Property of Equality For any real number a, a = a. –7y = –7y This is obvious: anything equals itself. They used the reflexive property. Symmetric Property of Equality For any real numbers a and b, if a = b then b = a. If 10 = y, then y = 10. When solving an equation, I might rearrange things so I end up with the variable on the left. But I only switched sides; I didn't actually change anything: the symmetric property. Transitive Property of Equality For any real numbers a, b and c, if a = b and b = c, then a = c. If 3x + 2 = y and y = 8, then 3x + 2 = 8. You might be torn here between the transitive property and the substitution property. What they did here was "cut out the middleman" by removing the "y" in the middle, so they used the transitive property. Combining Like Terms! Suppose you have 4 apples + 2 oranges + 6 apples + 3 oranges This expression can be simplified by combining the like terms to the following expression: 10 apples + 5 oranges Like terms in algebra are terms that have exactly the same variables and the variable is raised to the exact same power. Examples of Like Terms 2x2 and 6x2 6xy and -4xy xyz3 and 4xyz3 Examples of NON-Like Terms 2x2 and 6x 6xy and -4xy2 5 and 3x Example of Combining Like Terms: 1. 4x + 3 y + 3 y2 + 2 x2 + 5y + 7x + 2y2 2. 9y +2x – 3y + 4x + 10z – 12z WORKING WITH THE DISTRIBUTIVE PROPERTY Example: 3(2x – 5) + 5(3x +6) = Since in the order of operations, multiplication comes before addition and subtraction, we must get rid of the multiplication before you can combine like terms. We do this by using the distributive property: 3(2x – 5) + 5(3x +6) Original Problem 3(2x) – 3(5) + 5(3x) + 5(6) Distributive Property 6x – 15 + 15x + 30 Simplify 6x + 15x – 15 + 30 Gather Like Terms 21x + 15 Combine Like Terms You Try: 5(8x + 3) – 7(4x + 11) Use the distributive property to rewrite the expression without parentheses. 3 x 4 1. 2. 2 2 y 1 3. 1 10 15r 5 Simplify the expression by combining like terms. 12m 5m 4. 5. 6a 2a 2 4a a 2 6. 10r 6t 2 2t 4r Simplify the expression completely. 10 b 1 4b 7. 8. 4 4 p 3 4 p 1 9. 1 2 6 3r Understanding Check: 1. 2. Which statement is always true? A B C D 3. Which is equivalent to (5x2 + 4x + 1) + (-7x + 2)? A B C D -2x2 + 6x + 1 5x2 – 3x – 1 5x2 – 3x + 3 5x2 + 11x + 3 4. 4+a=4a a + (-4 + 4) = a + 0 a÷4=4÷a 4–a=a–4 The number -35 is best represented by which one set of numbers? A. Real B. Rational C. Integer D. Whole Algebra 1: Lesson 2.5-2.7 Homework Types of Real Numbers and the Properties of the Real Number System Identify the Property Being Illustrated. 1. 8 8 0 2. (3 4) 5 3 (4 5) 3. (5)(0) 0 4. (12.5)(1) 12.5 5. (1.25)(6) (6)(1.25) 1 6. 3 1 3 7. 4(3 a) 12 4a 8. (4 5) 2 (5 4) 2 9. If y = 4 and 4 = x, then y = x 10. If 3 = y, then y = 3 Simplify the Expression. 11. 6 10x 3 12. 9 2( x 4) 13. 7 x 3(4 2 x) 14. 8 x 3(2 x 1) 15. 7( x 2) 5 16. x(6 2 x) 3x 17. 2(3x 1) 4 2 x 18. 2( x 4) 3 19. ( x 4) 3 20. 5 x(2 x 1) 4 x 2 11 Between What Two Integers Does the Square Root Belong? 21. 24 22. 60 23. 7 24. 98 Put a Check in Each box that Names that Number. Number 144 49 5.36 1.35987… 1 3 Real Rational Put the Numbers in Order from Least to Greatest. 30. 64 , 5 , 9 , 2 31. 3 , 5.5 , 18 , 0 32. 1 1 , 2, , 3 3 2 Irrational Integer Whole Chapter 1+2 Quiz Practice 12 Evaluate the Expression. 1. 2 x 3 when x = 6 Perform Order of Operations. 2. 3(5 1) 2 3. 16 (3 1) 2 2 Translate the verbal phrase into an expression, equation or inequality. 4. Three times a number is at least 4. 5. 5 less than twice a number. Is the given number a solution to the equation or inequality (Yes, No)? 7. r 8 21 , if r = 13 6. 4 m 20 , if m = 4 2 Complete the Chart by Putting a Check in the Box for each that applies to the number. Number 8. -16 9. Real Rational 13 Put the Numbers in order from least to greatest. 10. 2, 2,1.2, 1 2 Identify the Property being illustrated. 11. -3 + 3 = 0 12. If 3 = x, then x = 3 Irrational Integer Whole Find the Sum or Difference. 13. –4 + (-1) 14. 8 – ( -2) Find the Product or Quotient. 15. (3)(4)(-2) 16. 28 (7) Tell Whether the Statement is True or False. 17. If a number is an integer, then the number is also a rational number. Simplify the Expression. 18. 3(2x + 3) – 5x 19. -6(5x – 3) + 2(2 – 8) What Two Whole Numbers is the Square Root Between? 20. 68