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How do we use numbers? Can you give examples of each of the three types? A good example of a...
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![Page 1: 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.](https://reader036.fdocuments.us/reader036/viewer/2022070412/56649d485503460f94a24824/html5/thumbnails/1.jpg)
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
![Page 2: 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.](https://reader036.fdocuments.us/reader036/viewer/2022070412/56649d485503460f94a24824/html5/thumbnails/2.jpg)
Good Examples of a Tag KC9JPZ W3HUK
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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?
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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
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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
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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?
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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
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Standard prefixes for large multiples
Adopted by an international committee on weights and measures in Paris in the fall of 1958
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Algebra and Calculator order of operations
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Example: Order of Operations
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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
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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.
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Small numbers
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And their names
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OOPS! What is wrong with this.
Hint! Sometimes Division by Zero is camouflaged!
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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?
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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.
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Microsoft Excel is a Powerful Scientific Tool
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Where did this come from?
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Which results in Using Mathematica 4.0
Who gets the 4 mills per month?
My TI 85 calculator gave 638.103976901!
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Done on a TI-85 calculator
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And finally using GW Basic
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Real Life Example
My 2008 Honda Accord
Three Years
% Rate / period
Principal
Monthly Payment