Number Fun December 3,

Transcription

1 Number Fun December 3, 2008 John L. Lehet

2 Numbers Fibonacci Numbers Digital Roots Vedic Math Original Puzzles MathMagic Tricks

3 Predict the Sum? (PredictTheSum.xls)

4 Overview of Numbers

5 Numbers

6 Numbers

7 Numbers Natural Numbers 1, 2, 3, 4, 5, 6, 7, 8, 9... Natural Numbers (N) 1, 2, 3...

8 Numbers Whole Numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, 9... Natural Numbers (N) 1, 2, 3... Whole Numbers 0, 1, 2, 3...

9 Numbers Integers... 3, 2, -1, 0, 1, 2, 3... Natural Numbers 1, 2, 3... Whole Numbers 0, 1, 2, 3... Integers (Z) -3, -2, -1, 0, 1, 2, 3...

10 Numbers Rationals 3, 1/2, 22/9, 8 2/5, 17 Natural Numbers 1, 2, 3... Whole Numbers 0, 1, 2, 3... Integers (Z) -3, -2, -1, 0, 1, 2, 3... Rationals (Q) -3, ½, 22/9, 8 2/5, 17

11 Numbers Algebraic Numbers -4, 7, -3 5, 17/3 Natural Numbers 1, 2, 3... Whole Numbers 0, 1, 2, 3... Integers (Z) -3, -2, -1, 0, 1, 2, 3... Rationals (Q) -3, ½, 22/9, 8 2/5, 17 Algebraic Numbers -4, 7, -3 5, 17/3

12 Numbers Real Numbers -4,, e, 7, log 2, sin 17 Natural Numbers 1, 2, 3... Whole Numbers 0, 1, 2, 3... Integers (Z) -3, -2, -1, 0, 1, 2, 3... Rationals (Q) -3, ½, 22/9, 8 2/5, 17 Algebraic Numbers -4, 7, -3 5, 17/3 Real Numbers ( R ) -4,, e, 7, log 2, sin 17

13 Numbers Complex Numbers 3, i, 3i + 2, i, 17 Natural Numbers 1, 2, 3... Whole Numbers 0, 1, 2, 3... Complex Numbers ( C ) -4, i, 3i+3, i, 17 Integers (Z) -3, -2, -1, 0, 1, 2, 3... Rationals (Q) -3, ½, 22/9, 8 2/5, 17 Algebraic Numbers -4, 7, -3 5, 17/3 Real Numbers ( R ) -4,, e, 7, log 2, sin 17

14 Four 4 s Using Four 4 s Can you combine them with +, -, x, to make the numbers 1 through 10? example 0 = = ( ) 4

15 Four 4 s - Solution 0 = = = (4 4) + (4 4) 3 = ( ) 4 4 = 4 + (4-4) 4 5 = ((4 x 4) + 4) 4 6 = (4 + 4) = = = (4 4) 10 = (44-4) 4

16 Digital Roots

17 Digital Roots 1. Pick a number from between 2 and 9: 2. Multiply the number from step #1 by 9: 3. Find the sum of the digits of the number from step #2: 4. Subtract 5 from the number from step #3: 5. Map the number from step #4 to the alphabet: (1-a; 2-b; 3-c; 4-d; 5-e; etc) 6. Pick a country in Europe starting with the letter in step #5: 7. Pick an animal with a long tail starting with the last letter of the country in step #6:

18 Digital Roots

19 Digital Roots A Digital Root of a number is the sum of the digits (until a single digit value is found) example Digital Root of 241 is = 7 Digital Root of 2487 is = = 3 Digital Root of is?

20 Digital Roots The Number 9 When is a number divisible by 9? Let s make a 6 digit number?????? Is this number divisible by 9?

21 Digital Roots The Number 9 A Number is Divisible by 9, When the Digital Root is 9! so, 243 is divisible by 9 since its digital root is 9 and, is divisible by 9 since its digital root is 9 but, 3452 is NOT divisible by 9 since its digital root is 5

22 Multiples 2 2,4,6,8,10,12,16,18, Ending Digit 2,4,6,8,0, Digital Root 2,4,6,8,1,3,5,7,9, 3 3,6,9,12,15,18,21,24,27,30,33, 3,6,9,2,5,8,1,4,7,0, 3,6,9, 4 4,8,12,16,20,24,28, 4,8,2,6,0, 4,8,3,7,2,6,1,5,9, 5 5,10,15,20, 5,0, 5,1,6,2,7,3,8,4,9, 6 6,12,18,24,30,36,42, 6,2,8,4,0, 6,3,9, 7 7,14,21,28,35,42,49,56,63,70, 7,4,1,8,5,2,9,6,3,0, 7,5,3,1,8,6,4,2,9, 8 8,16,24,32,40,48, 8,6,4,2,0, 8,7,6,5,4,3,2,1,9, 9 9,18,27,36,45,54,63,72,81,90 9,8,7,6,5,4,3,2,1,0, 9,

23 Numbers Divisibility Tests 2 all even numbers 3 digital root is 3, 6 or 9 4 last two digits are divisible by 4 5 last digit is 0 or 5 6 digital root is 3, 6, or 9 AND an even number 7 if twice the last digit subtracted from the remaining digits is divisible by 7 (may as well divide!) 8 last three digits are divisible by 8 9 digital root is 9

24 Generate a 9 Generate a 9 Method #1 #1. Select any three digit number in which none of the digits are the same example 517 #2. Rewrite the number in step #1, reversing the digits #3. Subtract the smaller number from the larger number #4. The number in the tens digit is always 9 and the sum of the hundreds and ones digits is

25 Generate a 9 #1. Select any number #2. Sum the digits Generate a 9 Method #2 #3. Subtract the number from step #2 from the number from step #1 #4. Find the digital root of the number from step #3 example = =18 1+8=9 9

26 Generate a 9 #1. Select any 2-digit number #2. Sum the digits Generate a 9 Method #3 example =5 #3. Multiply the result from step #2 by 8 #4. Add the original number and the number from step #3 #5. Find the digital root of the number from step #4 5 x 8= =9

27 Generate a 9 Generate a 9 Method #4 #1. Select any single digit number #2. Multiply this number by 5 example 8 8x5=40 #3. Reverse the digits in the result from step #2 #4. Add the results from steps #2 and # =44 #5. Subtract the original number from the result in step #4 44-8=36 #6. Sum the digits of the result in step #5 3+6=9

28 Generate a 9 Generate a 9 Method #5 #1. Select any 2-digit number #2. Multiply this number by 2 #3. Reverse the digits in the result from step #2 and multiply by 2 #4. Rearrange the digits in the result from step #3 and subtract the original number #5. Add the results from steps 2,3 and 4 example #6. Find the digital root of the result in step #5 3+6=9

29 Fibonacci Numbers

30 Fibonacci Numbers Starting with the numbers 0 and 1 Construct the next Fibonacci Number as the sum of the previous two... F 0 = 0 F 1 = 1 F 2 = 1 (0+1) F 3 = 2 (1+1) F 4 = 3 (1+2) F 5 = 5 (2+3) F 6 = 8 (3+5) F 7 = 13 (5+8) F 8 = 21 (8+13) F 9 = 34 (13+21) F 10 = 55 (21+34)

31 Fibonacci Numbers - Ratio F 0 = 0 F 1 = 1 F 2 = 1 F 3 = 2 F 4 = 3 F 5 = 5 F 6 = 8 F 7 = 13 F 8 = 21 F 9 = 34 F 10 = 55 F 11 = 89 = 144 F 12 The ratio of two consecutive Fibonacci Numbers F i F i+1

32 Fibonacci Numbers - Ratio F 0 = 0 F 1 = 1 F 2 = 1 F 3 = 2 F 4 = 3 F 5 = 5 F 6 = 8 F 7 = 13 F 8 = 21 F 9 = 34 F 10 = 55 F 11 = 89 = 144 F 12 The ratio of two consecutive Fibonacci Numbers Converges to a number F i F i+1

33 Fibonacci Numbers - Ratio F 0 = 0 F 1 = 1 F 2 = 1 F 3 = 2 F 4 = 3 F 5 = 5 F 6 = 8 F 7 = 13 F 8 = 21 F 9 = 34 F 10 = 55 F 11 = 89 = 144 F 12 The ratio of two consecutive Fibonacci Numbers Undefined Converges to a number F i+1 F i

34 Fibonacci Numbers - Ratio φ = The Golden Ratio a b a + b φ = = OR 5 = (2φ 1) 2

35 Fibonacci Numbers - Ratio Pentagram 5-Sided Star red green = green blue blue = purple = φ Each Acute Isosceles Triangle is a Golden Triangle

36 Fibonacci Numbers - Ratio φ = φ 1 = = φ-1 φ 2 = = φ+1 φ 3 = φ+1 φ-1

37 Fibonacci Numbers In Nature The Human Hand Four Bones per Finger Lengths are 8, 5, 3 and 2 units

38 Fibonacci Numbers In Nature The center of the flower is comprised of spirals both clockwise and counter-clockwise

39 Fibonacci Numbers In Nature

40 Fibonacci Numbers Rectangles There are 9 squares, the smallest is 1x1 and the largest is 34x34 Within each square, there is a quarter-circle creating a Fibonacci Spiral which takes on the shape of a Nautilus in Nature

41 Fibonacci Numbers Rabbits The original problem that Fibonacci investigated, in the year 1202, was about how fast rabbits could breed in ideal circumstances. "A pair of rabbits, one month old, is too young to reproduce. Suppose that in their second month, and every month thereafter, they produce a new pair. If each new pair of rabbits does the same, and none of the rabbits dies, how many pairs of rabbits will there be at the beginning of each month?"

42 At the end of Month 1 At the end of Month pair of new born rabbits 2. One pair mates At the end of Month pair of new born rabbits 2. 2 pair mate Fibonacci Numbers Rabbits 1. NO new borns (the rabbits are too young) 2. They mate At the end of Month pair of new born rabbits 2. 3 pair mate At the end of Month pair of new born rabbits 2. 5 pair mate At the end of Month pair of new born rabbits 2. 8 pair mate R1 R2 R1 R2 r3 r4 R1 R2 R3 R4 r5 r6 1 Pair 2 Pair R1 R2 R3 R4 R5 R6 r7 r8 r9 r10 3 Pair 5 Pair R1 R2 R3 R4 R5 R6 R7 R8 R9 R10 r11 r12 r13 r14 r15 r16 R1 R2 R3 R4 R5 R6 R7 R8 R9 R10 R11 R12 R13 R14 R15 R16 r17 r18 r19 r20 r21 r22 r23 r24 r25 r26 8 Pair 13 Pair

43 Fibonacci Numbers MathMagic Trick 1. Pick any two integers greater than 0 and less than 5 2. Using these two values, generate a Fibonacci Sequence of 10 elements 3. Sum the 10 elements (and keep to yourself) Example: Select 2 and 4 as the two integers The Fibonacci Sequence is 2,4,6,10,16,26,42,68,110,178 The sum is 462 The Trick: Given a some or all of the numbers in the generated sequence, I will immediately tell you the sum!

44 Fibonacci Numbers Sum Any number that is not a Fibonacci Number can be written as the sum of non-adjacent Fibonacci Numbers F 0 = 0 F 1 = 1 F 2 = 1 F 3 = 2 4 = 1+3 F 4 = 3 6 = 1+5 F 5 = 5 7 = 2+5 F 6 = 8 F 7 = 13 F 8 = 21 F 9 = 34 F 10 = 55 F 11 = 89 F12 = = = = = : 27 =

45 Fibonacci Numbers Squares F 0 = 0 F 1 = 1 F 2 = 1 F 3 = 2 = 3 Square each Fibonacci Number Add consecutive pairs Do you see an interesting pattern? F 4 F 5 = 5 F 6 = 8 F 7 = 13 F 8 = 21 F 9 = 34 F 10 = 55 F 11 = 89 F12 = 144

46 Magic Puzzles

47 HoneyComb and OcTangle Puzzles

48 Special Numbers

49 The Number 142, x 2 = x 3 = x 4 = x 5 = x 6 = x 7 = x 8 = x 9 = x 10 = x 11 = x 12 = = = = = = 5

50 The Number Perfect Number A Number is a Perfect Number if its divisors sum to the number example 6 is a Perfect Number The divisors of 6 are 1,2 and = 6 Is 100 a Perfect Number? Is 496 a Perfect Number?

51 The Number Perfect Number 496 is a Perfect Number The divisors of 496 are 1, 2, 4, 8, 16, 31, 62, 124 and = 496 There is one other Perfect Number less then 100 Can you find it?

52 The Number Friendly Numbers The divisors of 220 are 1,2,4,5,10,11,20,22,44,55, = 284 The divisors of 284 are 1,2,4,71, = and 284 are referred to as friendly numbers

53 The Number Perfect Cubes Both 8 and 27 are perfect cubes 8 = 2 3 and 27 = = 512 whose digital root is 8! 27 3 = whose digital root is 27!

54 Vedic Math

55 Traditional Method 98 x Vedic Math Multiplying Numbers Close to 100 FOIL Method 98 = = x 88 = (100 2)(100 12) = 100(100) 2(100) 12(100) + 24 = = = = 8624

56 Traditional Method 98 x Vedic Math Multiplying Numbers Close to 100 Vedic Method = or = 2x12 98 is 2 below is 12 below 100 So, what s 97 x 85? 3(15) = 8245

57 Traditional Method 102 x Vedic Math Multiplying Numbers Close to 100 FOIL Method 102 = = x 112 = ( )( ) = 100(100) + 2(100) + 12(100) + 24 = = = = 11424

58 Traditional Method 102 x Vedic Math Multiplying Numbers Close to Vedic Method = or = 2x is 2 above is 12 above 100 So, what s 103 x 108? 3(8) = 11124

59 Traditional Method 112 x Vedic Math Multiplying Numbers Close to 100 FOIL Method 112 = = x 98 = ( )(100-2) = 100(100) + 12(100) - 2(100) - 24 = = = ( ) + (100-24) = = 10976

60 Traditional Method 112 x Vedic Math Multiplying Numbers Close to Vedic Method = or = -2x is 12 above is 2 below100 So, what s 103 x 96? = 9888

61 Traditional Method 75 x Vedic Math Squaring Numbers Ending in 5 FOIL Method = = = (75)(75) = (80-5)(70 + 5) = 80(70) + 5(80) - 5(70) = (80-70) - 25 = = = = 5625

62 Traditional Method 75 x Vedic Math Squaring Numbers Ending in 5 Vedic Method the answer has two parts top part = 7(8) = 56 bottom part = 25 So 75 2 = 5625 So, what s 35 2? 3(4)=12 is the top part 25 is the bottom part 1215

63 Traditional Method 24 x 43 Vedic Math Multiplying 2-Digit Numbers Intermediate Method 24 x 43 Vedic Method =3x =3x =40x =40x = 2x4 upper half So, what s 46 x 52? = = 4x4 6 = 2x3 12 = 4x3 lower half = 1032

64 Traditional Method 45 x Vedic Math Multiplying by 11 So, what s 57 x 11? carry the 1 and add to 5 57 sum Vedic Method 45 sum

65 Vedic Math Multiplying by 11 Traditional Method 243 x Vedic Method So, what s 561 x 11? carry the 1 and add to

66 MathMagic Tricks

67 Find the Number #1 #1. Using the numbers 1-9, make a 4-digit number using a number only once #2. Using the remaining five numbers, make a 3-digit number using a number only once #3. Using the remaining two numbers, make a 2-digit number using a number only once #4. Sum the numbers from steps #1, #2 and #3 #5. Write down any number from 1-9 and circle #6. Sum the numbers from steps #4 and #5 example

68 Find the Number #2 #1. Using the numbers 1-9, make a 3-digit number using a number only once #2. Using the remaining six numbers, make a 3-digit number using a number only once example #3. Using the remaining three numbers, make a digit number using a number only once #4. Sum the numbers from steps #1, #2 and #3 #5. Write down any number from 1-9 and circle #6. Subtract the numbers from step #5 from the number from step #

69 Find the Number #3 #1. Using the numbers 1-9, select a single digit number and circle #2. Using the remaining eight numbers, make a 3-digit number using a number only once #3. Using the remaining five numbers, make a 3-digit number using a number only once #4. Using the remaining two numbers, make a 2-digit number using a number only once example #5. Sum the number from steps #2, #3 and #4 1057

70 Find the Number #4 #1. Using the numbers 7-9, select a single digit number and circle example 7 24 #2. Using the numbers 1-6, make three 2-digit 51 numbers using each digit only once 36 #3. Sum the number from step #1 and the three numbers from step #2 118

71 Questions and Comments