number fun - mathmaverick.com

Number FunDecember 3, 2008

John L. LehetJohn L. [email protected]

NumbersFibonacci Numbers

Digital RootsDigital RootsVedic Math

Original PuzzlesMathMagic Tricks

Predict the Sum?(PredictTheSum.xls)(PredictTheSum.xls)

Overview of Numbers

Numbers

Numbers

Natural Numbers1, 2, 3, 4, 5, 6, 7, 8, 9 . . .

Natural Numbers (N)1, 2, 3 . . .

Numbers

Whole Numbers0, 1, 2, 3, 4, 5, 6, 7, 8, 9 . . .

Natural Numbers (N)1, 2, 3 . . .

Whole Numbers0, 1, 2, 3 . . .

Numbers

Integers. . . –3, –2, -1, 0, 1, 2, 3 . . .

Natural Numbers1, 2, 3 . . .


Integers (Z)-3, -2, -1, 0, 1, 2, 3 . . .

Numbers

Rationals–3, 1/2, 22/9, 8 2/5, 17



Integers (Z)-3, -2, -1, 0, 1, 2, 3 . . .

Rationals (Q)-3, ½, 22/9, 8 2/5, 17

Numbers

Algebraic Numbers-4, √7, -3 √5, 17/3



Integers (Z)-3, -2, -1, 0, 1, 2, 3 . . .

Rationals (Q)-3, ½, 22/9, 8 2/5, 17


Numbers

Real Numbers-4, ∏, e, √7, log 2, sin 17



Integers (Z)-3, -2, -1, 0, 1, 2, 3 . . .

Rationals (Q)-3, ½, 22/9, 8 2/5, 17

Real Numbers ( R )-4, ∏, e, √7, log 2, sin 17


Numbers

Complex Numbers–3, i, 3i + 2, ∏i, 17



Complex Numbers ( C )-4, i, 3i+3, ∏i, 17

0, 1, 2, 3 . . .

Integers (Z)-3, -2, -1, 0, 1, 2, 3 . . .

Rationals (Q)-3, ½, 22/9, 8 2/5, 17


Real Numbers ( R )-4, ∏, e, √7, log 2, sin 17

Four 4’s

Using Four 4’s

Can you combine them with+, -, x, ÷

to make the numbers 1 through 10 ?

example

0 = 44 - 44

3 = (4 + 4 + 4) ÷ 4

Four 4’s - Solution

44 - 440 =

1 =

2 =

3 =

4 =

5 =

44 ÷ 44

(4 ÷ 4) + (4 ÷ 4)

(4 + 4 + 4) ÷ 4

4 + (4 - 4) ÷ 4

((4 x 4) + 4) ÷ 45 =

6 =

7 =

8 =

9 =

((4 x 4) + 4) ÷ 4

(4 + 4) ÷ 4 + 4

44 ÷ 4 - 4

4 + 4 + 4 - 4

4 + 4 + (4 ÷ 4)

10 = (44 - 4) ÷ 4

Digital Roots

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 #3:4. 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 letterof the country in step #6:

Digital Roots

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 72 + 4 + 1 = 72 + 4 + 1 = 7

Digital Root of 2487 is 32 + 4 + 8 + 7 = 21

2 + 1 = 3

Digital Root of 694832 is ___?

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?

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,157248 is divisible by 9since its digital root is 9

but,3452 is NOT divisible by 9

since its digital root is 5

Multiples

2 2,4,6,8,10,12,16,18,

3

4

5

3,6,9,12,15,18,21,24,27,30,33,

4,8,12,16,20,24,28,

5,10,15,20,

2,4,6,8,0,…

3,6,9,2,5,8,1,4,7,0,…

4,8,2,6,0,…

5,0,…

2,4,6,8,1,3,5,7,9,…

3,6,9,…

4,8,3,7,2,6,1,5,9,…

5,1,6,2,7,3,8,4,9,…

Ending Digit Digital Root

5

6

7

89

6,12,18,24,30,36,42,

7,14,21,28,35,42,49,56,63,70,

8,16,24,32,40,48,

9,18,27,36,45,54,63,72,81,90

6,2,8,4,0,…

8,6,4,2,0,…

9,8,7,6,5,4,3,2,1,0,…

7,4,1,8,5,2,9,6,3,0,…

6,3,9,…

8,7,6,5,4,3,2,1,9,…

9,…

7,5,3,1,8,6,4,2,9,…

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 55 last digit is 0 or 5

6 digital root is 3, 6, or 9 AND an even number

7if 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

Generate a 9

#1. Select any three digit number in which noneof the digits are the same

#2. Rewrite the number in step #1, reversing the digits

517

example

715

Generate a 9 – Method #1

the digits

#3. Subtract the smaller number from the largernumber

#4. The number in the tens digit is always 9 andthe sum of the hundreds and ones digits is 9

715

715-517198

198

Generate a 9

#1. Select any number

#2. Sum the digits

3864

example

3+8+6+4=

21


#3. Subtract the number from step #2 from thenumber from step #1

#4. Find the digital root of the number from step #3

3864-21

38433+8+4+ 3=18

1+8=9

9

Generate a 9

#1. Select any 2-digit number

#2. Sum the digits

41

example

4+1=5


40#3. Multiply the result from step #2 by 8

#4. Add the original number and the number from step #3

41+ 40

81

8+1=9

5 x 8=40

#5. Find the digital root of the number from step #4

Generate a 9

#1. Select any single digit number

#2. Multiply this number by 5

#3. Reverse the digits in the result from step #2

8

example

8x5=40


4#3. Reverse the digits in the result from step #2

#4. Add the results from steps #2 and #3

44-8=36

4

#5. Subtract the original number from the result instep #4

40+4=44

#6. Sum the digits of the result in step #5 3+6=9

Generate a 9

#1. Select any 2-digit number

#2. Multiply this number by 2

#3. Reverse the digits in the result from step #2

49

example

98


178#3. Reverse the digits in the result from step #2and multiply by 2

#4. Rearrange the digits in the result from step #3and subtract the original number

945

178

#5. Add the results from steps 2,3 and 4

669

#6. Find the digital root of the result in step #5 3+6=9

Fibonacci Numbers

Fibonacci Numbers

Starting with the numbers 0 and 1Construct the next Fibonacci Number as

the sum of the previous two . . .

= 0F0

= 1F1

= 1 (0+1)F2

= 2 (1+1)F = 2 (1+1)F3

= 3 (1+2)F4

= 5 (2+3)F5

= 8 (3+5)F6

= 13 (5+8)F7

= 21 (8+13)F8

= 34 (13+21)F9

= 55 (21+34)F10

Fibonacci Numbers - Ratio

The ratio of two consecutive Fibonacci Numbers

= 0F0

= 1F1

= 1 F2

= 2 F3

= 3 F4

= 5 F

01 0.50.66 0.6

Fi

Fi+1

= 5 F5

= 8 F6

= 13F7

= 21F8

= 34F9

= 55F10

= 89F11

0.6 0.6250.6153846…

0.6176470…0.6181818…0.6179775…

0.6190475…

= 144F12 0.6180555…



= 0F0

= 1F1

= 1 F2

= 2 F3

= 3 F4

= 5 F

01 0.50.66 0.6

Fi

Fi+1

Converges to a number …

0.61803398878…

= 5 F5

= 8 F6

= 13F7

= 21F8

= 34F9

= 55F10

= 89F11

0.6 0.6250.6153846…

0.6176470…0.6181818…0.6179775…

0.6190475…

= 144F12 0.6180555…



= 0F0

= 1F1

= 1 F2

= 2 F3

= 3 F4

= 5 F

Undefined1 1.51.66 1.6

Fi

Fi+1

Converges to a number …

1.61803398878…

= 5 F5

= 8 F6

= 13F7

= 21F8

= 34F9

= 55F10

= 89F11

1.6 1.6251.6153846…

1.6176470…1.6181818…1.6179775…

1.6190475…

= 144F12 1.6180555…

φ =1.61803398878…


The Golden Ratio

a b

a + ba + b

φ = 1.61803398878 = 1 + √5

2 OR

5 = (2φ – 1)2


Pentagram – 5-Sided Star

red green

greenblue

bluepurple

= = = φ

Each Acute Isosceles Triangle is a Golden Triangle


φ =1.61803398878…

φ2 =2.61803398878…= φ+1

=0.61803398878…= φ-1φ1

φ2 =2.61803398878…= φ+1

φ3 =φ+1φ-1

Fibonacci Numbers – In Nature

The Human Hand

Four Bones per FingerLengths are 8, 5, 3 and 2 units


The center of the floweris comprised of spirals

both clockwise andboth clockwise andcounter-clockwise

Fibonacci Numbers – Rectangles

There are 9 squares, the smallest is 1x1 and the largest is 34x34Within each square, there is a quarter-circle

creating a Fibonacci Spiral which takes on the shape of a Nautilus in Nature

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 and none of the rabbits dies, how many pairs of rabbits

will there be at the beginning of each month?"

At the end of Month 31. 1 pair of new born rabbits 2. 2 pair mate

R1 R2 R3 R4r5 r6

3 Pair

Fibonacci Numbers – Rabbits

At the end of Month 21. 1 pair of new born rabbits 2. One pair mates

R1 R2r3 r4

2 Pair

At the end of Month 11. NO new borns (the rabbits are too young)2. They mate

R1 R2 1 Pair

2. 2 pair mate


R1 R2 R3 R4 R5 R6r7 r8 r9 r10

5 Pair


R1 R2 R3 R4 R5 R6 R7 R8 R9 R10r11 r12 r13 r14 r15 r16

8 Pair


R1 R2 R3 R4 R5 R6 R7 R8 R9 R10R11 R12 R13 R14 R15 R16

r17 r18 r19 r20 r21 r22 r23 r24 r25 r26

13 Pair

Fibonacci Numbers – MathMagic Trick

1. Pick any two integers greater than 0 and less than 5

2. Using these two values, generate a Fibonacci Sequenceof 10 elements

3. Sum the 10 elements (and keep to yourself)

Example:Example:Select 2 and 4 as the two integersThe Fibonacci Sequence is 2,4,6,10,16,26,42,68,110,178The sum is 462

The Trick:Given a some or all of the numbers in the generatedsequence, I will immediately tell you the sum!

Fibonacci Numbers – Sum

Any number that is not a Fibonacci Number can be written as the sum of non-adjacent Fibonacci Numbers

4 = 1+36 = 1+57 = 2+5

= 0F0

= 1F1

= 1 F2

= 2 F3

= 3 F4

= 5 F 7 = 2+59 = 8+110 = 8+211 = 8+312 = 8+3+1:27 = 21+5+2

= 5 F5

= 8 F6

= 13F7

= 21F8

= 34F9

= 55F10

= 89F11

= 144F12

Fibonacci Numbers – Squares

Square each Fibonacci NumberAdd consecutive pairs

Do you see an interesting pattern?

= 0F0

= 1F1

= 1 F2

= 2 F3

= 3 F4

= 5 F = 5 F5

= 8 F6

= 13F7

= 21F8

= 34F9

= 55F10

= 89F11

= 144F12

Magic Puzzles

HoneyComb andOcTangle Puzzles

Special Numbers

The Number 142,857

142857 x 2 = 285714

142857 x 3 = 428571

142857 x 4 = 571428

142857 x 5 = 714285

142857 x 6 = 857142

1+6 = 7

142857 x 7 = 999999

142857 x 8 = 1142856

142857 x 9 = 1285713

142857 x 10 = 1428570

142857 x 11 = 1571427

142857 x 12 = 1714284

1+3 = 4

1+0 = 1

1+7 = 8

1+4 = 5

The Number – Perfect Number

A Number is a Perfect Number if its divisorssum to the number

example

6 is a Perfect NumberThe divisors of 6 are 1,2 and 3

1+2+3 = 6

Is 100 a Perfect Number?

Is 496 a Perfect Number?

The Number – Perfect Number

496 is a Perfect NumberThe divisors of 496 are

1, 2, 4, 8, 16, 31, 62, 124 and 248

1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248 = 496

There is one other Perfect Number less then 100Can you find it?

The Number – Friendly Numbers

The divisors of 220 are1,2,4,5,10,11,20,22,44,55,110

1 + 2 + 4 + 5 + 10 + 11 + 20 + 22 + 44 + 55 + 110 = 284

The divisors of 284 are1,2,4,71,142

220 and 284 are referred to as “friendly” numbers

1,2,4,71,142

1 + 2 + 4 + 71 + 142 = 220

The Number – Perfect Cubes

Both 8 and 27 are perfect cubes

8 = 23 and 27 = 33

83 = 512 whose digital root is 8!

273 = 19683 whose digital root is 27!

Vedic Math

Vedic Math

Multiplying Numbers Close to 100

98x 88

784784

TraditionalMethod

“FOIL”Method

98 = 100 - 288 = 100 - 12

98 x 88 = (100 – 2)(100 – 12)

784

8624= (100 – 2)(100 – 12)= 100(100) – 2(100) – 12(100) + 24= 10000 – 200 – 1200 + 24= 10000 – 1400 + 24= 8600 + 24= 8624

Vedic Math


98x 88

784784

86

TraditionalMethod

VedicMethod

98

88

2

12

98 is 2 below 100

88 is 12 below 100

24784

86 248624

86 = 98–12 or 88-2 24 = 2x12

So, what’s 97 x 85? 3(15) + 8200 = 8245

Vedic Math


102x 112

204102

TraditionalMethod

“FOIL”Method

102 = 100 + 2112 = 100 + 12

102 x 112 = (100 + 2)(100 + 12)102

102

11424

= (100 + 2)(100 + 12)= 100(100) + 2(100) + 12(100) + 24= 10000 + 200 + 1200 + 24= 10000 + 1400 + 24= 11400 + 24= 11424

102

Vedic Math


114

VedicMethod

102

112

2

12

102 is 2 above 100

112 is 12 above 100

24

102x 112

204102

TraditionalMethod

102 114 24

114 = 112+2 or 102+12 24 = 2x12

So, what’s 103 x 108? 3(8) + 11100 = 11124

102

11424

102

Vedic Math


112x 98

8961008

TraditionalMethod

“FOIL”Method

112 = 100 + 1298 = 100 - 2

112 x 98 = (100 + 12)(100 - 2)

1008

10976= (100 + 12)(100 - 2)= 100(100) + 12(100) - 2(100) - 24= 10000 + 1200 - 200 - 24= 10000 + 1000 – 24= (11000 -100) + (100-24)= 10900 + 76= 10976

Vedic Math


110

VedicMethod

112

98

+12

-2

112 is 12 above 100

98 is 2 below100

-24

TraditionalMethod

112x 98

8961008

110 -24

110 = 112-2 or 98+12 -24 = -2x12

So, what’s 103 x 96? 9900 – 12 = 9888

1008

10976

Vedic Math

Squaring Numbers Ending in 5

75x 75

375

TraditionalMethod

“FOIL”Method

75 = 80 - 5

752

= (75)(75)

752

75 = 70 + 5

375525

= (75)(75)= (80 - 5)(70 + 5)= 80(70) + 5(80) - 5(70) - 52

= 5600 + 5(80-70) - 25= 5600 + 50 – 25= 5600 + 25= 5600 + 25= 5625

5625

Vedic Math

Squaring Numbers Ending in 5

75x 75

375

TraditionalMethod

VedicMethod

the answer has two partstop part = 7(8) = 56

bottom part = 25

752 752

375525

bottom part = 25

So 752 = 56255625

So, what’s 352? 3(4)=12 is the top part25 is the bottom part

1215

Vedic Math

Multiplying 2-Digit Numbers

8

VedicMethod

2

4

4

3

12

TraditionalMethod

24x 43

7296 16

Intermediate Method

24x 43

1260

160

=3x4=3x20

8 12

8 = 2x4upperhalf

12 = 4x3lowerhalf

So, what’s 46 x 52? 2012 + 380 = 2392

96

1032166

16 = 4x46 = 2x3

60

1032

160800

=3x20=40x4=40x20

812 + 220 = 1032

Vedic Math

Multiplying by 11

VedicMethod

sum

TraditionalMethod

45x 11

4545

45

4 59

495

So, what’s 57 x 11?

45

495

495

sum57

5 712

627carry the 1and add to 5

Vedic Math

Multiplying by 11

VedicMethod

TraditionalMethod

243x 11

243243

243

2 372+4

64+3

So, what’s 561 x 11?

243

2673 2673

5 17

6171

5+6

116+1

561

carry the 1and add to 5

MathMagic Tricks

Find the Number #1

#1. Using the numbers 1-9, make a 4-digit numberusing a number only once

#2. Using the remaining five numbers, make a 3-digitnumber using a number only once

2591example

783

#3. Using the remaining two numbers, make a 2-digit 46#4. Sum the numbers from steps #1, #2 and #3

#5. Write down any number from 1-9 and circle

3427

#3. Using the remaining two numbers, make a 2-digitnumber using a number only once

#6. Sum the numbers from steps #4 and #5

46

3420

7

Find the Number #2

#1. Using the numbers 1-9, make a 3-digit numberusing a number only once

#2. Using the remaining six numbers, make a 3-digitnumber using a number only once

732example

148

#3. Using the remaining three numbers, make a 569

#4. Sum the numbers from steps #1, #2 and #3

#5. Write down any number from 1-9 and circle

1443

#3. Using the remaining three numbers, make a3-digit number using a number only once

#6. Subtract the numbers from step #5 from thenumber from step #4

1449

6

569

Find the Number #3

#1. Using the numbers 1-9, select a single digitnumber and circle

#2. Using the remaining eight numbers, make a 3-digit number using a number only once

5example

186#3. Using the remaining five numbers, make a 472#4. Using the remaining two numbers, make a

2-digit number using a number only once

#5. Sum the number from steps #2, #3 and #4

#3. Using the remaining five numbers, make a3-digit number using a number only once

39

1057

472

Find the Number #4

#1. Using the numbers 7-9, select a single digitnumber and circle

#2. Using the numbers 1-6, make three 2-digit

7example

2451#2. Using the numbers 1-6, make three 2-digit

numbers using each digit only once 5136

#3. Sum the number from step #1 and the threenumbers from step #2 118

Questions and Comments

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