How do we use numbers?

Can you give examples of each of the three types?

A good example of a “Tag” is my Amateur Radio License KC9JPZ

Good Examples of a Tag KC9JPZ W3HUK

The World of Numbers is full of surprises

• Positive/Negative Integers (whole) -1, 2, 3, -4 etc. Pre-history • Fractions, Decimals ½, 1/6, 90/100 2000BC

• Numbers that can be expressed as theratio of two whole numbers rational

• Numbers that cannot irrational 500BC

• An example of an irrational number

• Another irrational number, C/D • Complex Numbers

• Matrices ~ 1850AD

2

12

When?

Systems of handling numbers

Roman Numerals MCMLV = 1955British System of Money*pounds, shilling, pence

20 shillings to the pound, 5 pence for the shilling

*In 1971 the British System was changed to 100 pence to the pound

Decimal System common use throughout the west for about 850 years. Special symbols are digits (0,1, 2, 3, 4 … 9) and all other numbers are written in terms of fundamental digits

Example: 8, 629, 798, 478, 111 = eight trillion, six hundred twenty nine billion, seven hundred ninety-eight million, four hundred seventy-eight thousand, one hundred and eleven

The Binary System: Perfect for Computers

To Begin with, we make up a table of successive products of 2

1 = 1 1 a bit

2 x 1 = 2 10

2 x 2 x 1 = 4 100

2 x 2 x 2 x 1 = 8 1000

2 x 2 x 2 x 2 x 1 = 16 10000 a nibble

2 x 2 x 2 x 2 x 2 x 1 = 32 etc. 100000

28 = 256 100000000 a byte

Compute 111012 in decimal and in hexi-decimal

Ans: 29 and 1D

From: The Lore of Large Numbers by Philip J. Davis, Random House Printing 1961

Problem Set # 1

How many different one digit numbers are there?

How many different two digit numbers are there?

Which is large LXXXVIII or C?

trillion = 1012 (3 x 3) + 3 = 12

quadrillion = 1015 (3 x 4) + 3 = 15

vigintillion = 1063 (3 x 20) + 3 = 63

Invent your own for any remaining

e.g. Kasher and Newman in their book “Mathematics and the Imagination”

10100 = one googol

10googolplex = pme googolplex

Standard prefixes for large multiples

Adopted by an international committee on weights and measures in Paris in the fall of 1958

Algebra and Calculator order of operations

Example: Order of Operations

Use the correct order of operations

1st law of exponents 2 n + 2 m = 2 (m + n)

2nd law of exponents [a m] n = a m n

Significant Figures, Accuracy and Order of Magnitude

The order of magnitude of a number is the number which results when the given number has been rounded to one signficant figure.

A hardware store displays a gallon jug full of pea beans. The prize is for a bicycle. The person guessing the closest to the actual number wins. Put in an estimate.

Small numbers

And their names

OOPS! What is wrong with this.

Hint! Sometimes Division by Zero is camouflaged!

Let us investigate this converging series

1st term 2nd 3rd 4th Etc.

...!4

1

!3

1

!2

1

!1

11 Do this for up to the

5th term in the series and see how close you get to e. Is the series converging? How would you prove it?

More on the Lore of Numbers!

Wilson’s Theorem1 states that a number N is a prime if and only if it divides the number:

1 x 2 x 3 x 4 x 5 x … x (N – 1) + 1

Try this for the number 11 which is known to be a prime! Try this for a number known not to be a prime.

1For a proof of Wilson’s Theorem, see Elementary Number Theory by J.V. Upensky and M.A. Heaslet, McGraw-Hill, 1939, p.153

Verify (but not by Wilson’s theorem) that 1973 is a Prime Number. Explain how you would attempt this.

Microsoft Excel is a Powerful Scientific Tool

Where did this come from?

Which results in Using Mathematica 4.0

Who gets the 4 mills per month?

My TI 85 calculator gave 638.103976901!

Done on a TI-85 calculator

And finally using GW Basic

Real Life Example

My 2008 Honda Accord

Three Years

% Rate / period

Principal

Monthly Payment