Chapter 9: Real Numbers

Presentation by Heath Booth

Real Numbers are possible outcomes of measurement.Excludes imaginary or complex numbers.Includes

Whole numbersNatural numbersIntegersRational numbersIrrationals numbers

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.

2Source:http://en.wikipedia.org/wiki/Square_root_of_2

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?

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.

History:

Approximations of

Babylonians – 2000 BCTablets of approximations of square and cube

rootsTablet YBC 7289Approximately 1.41421297

2

History:Possible Babylonian method:Find the range Midpoint

10 3

101

3

1010 10

310

10 33

Second approximation:

103

3 3.1672

1010

a

aa

10

3.162282

aa

History:Approximations of

China - 12th century BC = 3

Egypt – 1650 BC =

India – 628 AD = 3 , and (as of 499) Depending upon desired accuracy.

284( )

9

10177

31250

Archimedes (287-212 BC)

First recorded theoretical derivation

Resulted in or 3.1408450 < < 3.1428571

223 22

71 7

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

2

Developmental:Upper level elementary students often

complete simplified versions of the early attempts methods to approximate Pi. Ratio – C/D – direct measurementArchimedes – trap method

There is little discussion about how to treat these approximations in basic mathematical operations.

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.

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 - .351.475 – 1.584 grams

Arithmetic with the Reals:.2 .3

1

2

41 2

.2 .2222 .00002 .2222

.3 .3333 .00003 .3333

0 , 10

Now try in your groups..3 .142857

1 2.2 .3 (.2222 ) (.3333 ) .5555

Subtraction:Similar to addition

1 2

1 2

1 2

1 2

.2 .3 (.2222 ) (.3333 )

.2 .3 .2222 .3333

.2 .3 .2222 .3333

.2 .3 .1111

Multiplication:

4

3*.3 3*(.3333 )

3*.3 .9999 3

0 3 3*10

BUT:

13*.3 *3 1

3

Consider:

1 .999999999999 .0000000000001

1 .999999999999

So 1=.9

Multiplication:

Try in your groups9*.1