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Chapter 9: Real Numbers

Chapter 9: Real Numbers . Presentation by Heath Booth. Real Numbers are possible outcomes of measurement. Excludes imaginary or complex numbers. Includes Whole numbers Natural numbers Integers Rational numbers Irrationals numbers.

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Chapter 9: Real Numbers

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  1. Chapter 9: Real Numbers Presentation by Heath Booth

  2. Real Numbers are possible outcomes of measurement. • Excludes imaginary or complex numbers. • Includes • Whole numbers • Natural numbers • Integers • Rational numbers • Irrationals numbers

  3. Likely the first number determined to be irrational. Measurements on this simple triangle generate a number, the square root of two, which can only be represented by a non-repeating non-terminating decimal. Source:http://en.wikipedia.org/wiki/Square_root_of_2

  4. Calculations of the value of the square root of 2 1.4142135623746... • Has been calculated a trillion decimal places. • We still cannot predict the next digit • We have all seen the proof which rely on the simplest form property of the rationals to show a contradiction. • How many rational numbers can we name?

  5. How many irrational numbers can we name? • Believe it or not almost all of the real numbers are irrational! Source ( http://en.wikipedia.org/wiki/Irrational_number) • This makes sense when we consider the infinite nature of the real numbers combined with notion that the rational numbers are countable. • Write on the board a decimal which is irrational.

  6. History: • Approximations of • Babylonians – 2000 BC • Tablets of approximations of square and cube roots • Tablet YBC 7289 • Approximately 1.41421297

  7. History: Possible Babylonian method: Find the range Midpoint Second approximation:

  8. History: • Approximations of • China - 12th century BC = 3 • Egypt – 1650 BC = • India – 628 AD = 3 , and (as of 499) • Depending upon desired accuracy.

  9. Archimedes (287-212 BC) First recorded theoretical derivation Resulted in or 3.1408450 < < 3.1428571

  10. Developmental: • Doesn’t come up until square roots are introduced • We can not accurately measure the diagonal of the unit square. • Students are puzzled by this idea

  11. Developmental: • Upper level elementary students often complete simplified versions of the early attempts methods to approximate Pi. • Ratio – C/D – direct measurement • Archimedes – trap method • There is little discussion about how to treat these approximations in basic mathematical operations.

  12. Developmental: • Consider the gold problem again: • Amounts collected and accuracy of scale used • 1.14 grams - scale accurate to .01 gram • .089 grams – scale accurate to .001 gram • .3 grams – scale accurate to .1 gram How much gold do we have? Work in your groups.

  13. Developmental: • Added directly the total is 1.529 • Does this account for the type of scale used? • 1.14 accurate to .01 = 1.135 – 1.145 • .089 accurate to .001 = .0885 - .0895 • .3 accurate to .1 = .25 - .35 • 1.475 – 1.584 grams

  14. Arithmetic with the Reals: Now try in your groups.

  15. Subtraction: • Similar to addition

  16. Multiplication: BUT: Consider:

  17. Multiplication: Try in your groups

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