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CHAPTER 1: Preliminary Information

CHAPTER 1: Preliminary Information. Section 1-1: Sets of Numbers. Objectives:. Given the name of a set of numbers, provide an example. Given a number, name the set to which it belongs. Two Major Sets of Numbers. Real numbers:

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CHAPTER 1: Preliminary Information

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  1. CHAPTER 1: Preliminary Information Section 1-1: Sets of Numbers

  2. Objectives: • Given the name of a set of numbers, provide an example. • Given a number, name the set to which it belongs.

  3. Two Major Sets of Numbers • Real numbers: • Numbers used to represent “real” things such as measurement and counting. • Imaginary numbers: • Square roots of negative numbers. • Used in: • Electromagnetism • Fluid dynamics • Vibration analysis

  4. The Real Numbers • Rational Numbers: numbers that can be expressed as a ratio of two integers. • Integers: whole numbers and their opposites. • Natural Numbers: positive integers (aka- counting numbers) • Digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 • Irrational Numbers: numbers that cannot be expressed exactly as a ratio of two numbers: • Radicals:__________________ • Transcendental numbers:__________________

  5. CHAPTER 1: Preliminary Information Section 1-2: The Field Axioms

  6. Objective: • Given the name of an axiom that applies to addition or multiplication, give an example that shows you understand the meaning of the axiom.

  7. Closure • {real numbers} is closed under addition and multiplication. • If x and y are real numbers, then: • x + y is a unique, real number. • xy is a unique, real number.

  8. Commutativity • Addition and multiplication of real numbers are commutative operations. • That is, if x and y are real numbers, then: • x + y = y + x • xy = yx • Many operations are not commutative: • Subtraction • Division • Exponentiation (raising to powers)

  9. Associativity • Addition and multiplication of real numbers are associative operations. • That is, if x, y, and z are real numbers, then: • (x + y) + z = x + (y + z) • (xy) z = x(yz) • Subtraction is not associative.

  10. Distributivity • Multiplication distributes over addition. • That is, if x, y, and z are real numbers, then: • x (y + z) = xy + xz • Multiplication does not distribute over multiplication.

  11. Identity Elements • {real numbers} contains: • A unique identity element for addition: • x + _______ = x • A unique identity element for multiplication: • x (_____) = x

  12. Inverses • {real numbers} contains: • A unique additive inverse for every real number x: • x + _______ = 0 • A unique multiplicative inverse for every real number x: • x (_____) = 1

  13. Notes: • Any set that obeys all eleven of these axioms is called a field. • The number –x is called: • The opposite of x. • The additive inverse of x. • Negative x. • The number 1/x is called: • The multiplicative inverse of x. • The reciprocal of x.

  14. HOMEWORK Page 2 #1-9 Page 7 #1-10

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