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Transcript
1.1 Real Numbers A. Sets A set is a list of numbers: We separate the entries with commas, and close off the left and right with and . The empty set is the set containing nothing: . It is given the symbol . B. Special Sets 1. Natural Numbers: 2. Whole Numbers: 3. Integers: (these are the counting numbers) (same as , but throw in zero) (same as the whole numbers, but allow negative whole numbers) 4. Rational Numbers: This contains: a. all integers, like , , or b. all fractions positive and negative, like and c. all decimals that terminate (stop), like d. all decimals that have a repeating block, like It is a fact that all rational numbers can be turned into fractions: stay tuned 1 5. Irrational Numbers: This contains only infinite decimals that don’t have a repeating block. Examples: It is a fact that all roots that don’t come out “cleanly” are irrational. Thus , , are all irrational. 6. Real Numbers: This consists of anything that is either rational or irrational. Hence all numbers (in this course) are real. 2 C. Picture of Special Sets real numbers rational numbers integers whole #’s naturals Notice that irrational numbers are in a separate group. 3 D. Practice Problems on Special Sets Try to decide which sets each number belongs to. A number may belong to only one set, or several; list all possible. 1. 2. 3. 4. 5. 6. 7. 8. 4 E. Properties of Real Numbers 1. Commutative: we can add or multiply in any order commutative property of addition commutative property of multiplication (note: means multiply) 2. Associative: In repeated adding or multiplying, we can move parentheses associative property of addition associative property of multiplication Parenthesis mean do this first 3. Identity: Adding zero or multiplying by 1 does nothing Note: identity property of addition identity property of multiplication is called the additive identity, and is called the multiplicative identity. 4. Inverse: Two numbers add to zero or multiply to 1 Note: inverse property of addition inverse property of multiplication is the additive inverse; is the multiplicative inverse (reciprocal) 5. Distributive: Multiplying a sum distributive property of multiplication over addition 5 F. Practice Problems on Properties For each mathematical statement below, correct identify the name of the property shown. 1. 2. 3. 4. 5. 6. 8. 9. 6