Transcript Slide 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 Good Examples of a Tag KC9JPZ W3HUK The World of Numbers is full of surprises When? • Positive/Negative Integers (whole) • Fractions, Decimals • Numbers that can be expressed as the ratio of two whole numbers • Numbers that cannot • An example of an irrational number -1, 2, 3, -4 etc. ½, 1/6, 90/100 • Another irrational number, C/D • Complex Numbers • Matrices rational irrational Pre-history 2000BC 500BC 2 2 1 ~ 1850AD Systems of handling numbers Roman Numerals British System of Money* MCMLV = 1955 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 2x1=2 10 2x2x1=4 100 2x2x2x1=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 1 1 1 1 1 ... 1! 2 ! 3 ! 4 ! Etc. 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. Verify (but not by Wilson’s theorem) that 1973 is a Prime Number. Explain how you would attempt this. 1For a proof of Wilson’s Theorem, see Elementary Number Theory by J.V. Upensky and M.A. Heaslet, McGraw-Hill, 1939, p.153 Microsoft Excel is a Powerful Scientific Tool Where did this come from? Which results in Using Mathematica 4.0 My TI 85 calculator gave 638.103976901! Who gets the 4 mills per month? Done on a TI-85 calculator And finally using GW Basic Real Life Example My 2008 Honda Accord Principal % Rate / period Three Years Monthly Payment