1. 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 - 12 th 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