Transcript calculator_tricks_2

Step 1
Number the first 25 lines on your
paper, (1,2,3…)
Step 2
Write any two whole numbers on
the first two lines
Step 3
Add the two numbers and write
the sum on the third line
Step 4
Add the last two numbers and
write the sum on the next line
Continue this process
(add the last two, write the sum)
until you have 25 numbers on your
list).
Select any number among
the last five
on your list and divide it by the
number above it
Remember I do not know your
original two numbers
 or any of the 25 numbers on your sheet
of paper
 So I can’t know which of the last 5
numbers you have chosen to divide by
the number above it
Now I need to concentrate on
the number presently shown
on your calculator.
If I close my eyes and think about your
number, I will be able to prove to you that
I know what your number will be.
 If you select any number between the
last five (#21 to #25) and divides it by the
number above it, the answer will always
be1.618033989…, which just happens to
be the Golden mean! (provided, of
course, you have done all the addition
correctly in steps 3-5 above)
 It’s an incredible bit of mathematical trivia.
 Begin with any two whole numbers, make
a Fibonacci-type addition list, take the ratio
of two consecutive entries, and the ratio
approaches the Golden Mean!
 The further out we go, the more accurate
it becomes.
That’s why we need 25
numbers to obtain sufficient
accuracy.
The proof requires familiarity with the
Fibonacci Sequence, pages of algebra,
and a knowledge of limits, all of which go
far beyond the scope of explanation.
 If you divide one of your last five
numbers by the next number (instead of
the previous number), the result is the
same decimal without the leading 1.
In Book VI of the Elements, Euclid defined the "extreme and
mean ratios" on a line segment. He wished to find the point
(P) on line segment AB such that, the small segment is to the
large segment as the large segment is to the whole segment.
In other words how far along the line is P such that:
AP PB

PB AB
Euclid of
Alexandria
(325 – 265 BC.)
We use a different approach to Euclid and use algebra to help us find this ratio,
however the method is essentially the same.
 is the Greek
letter Phi.
A
B
1
P

Let AP be of unit length and PB = . Then we require  such that
2
1



1
 1
  2    1  0    1  5  1.618033.....
Solving this quadratic and
taking the positive root.
2
We get the irrational number shown.
Add the numbers shown
along each of the
shallow diagonals to find
Leonardo
of Pisa
another
well known
- 1250
sequence1180
of numbers.
The Fibonacci Sequence
1
1
1
1
1
2
3
1
1
3
1
2
3
5
1
1
1
The sequence first appears as a
recreational maths problem
about the growth in population
of rabbits in book 3 of his
famous work, Liber – abaci (the
book of the calculator).
10 45 120 210 252 210 120 45 10
1
Fibonacci
1
66 220
495 792 924 792 495 220 66 12 1
Sequence
11 55 165 330 462 462 330 165 55 11
12
13
21 34 55 89 144 233 377
Fibonacci travelled
1 4 6 4 1
extensively throughout
1 5 10 10 5 1
the Middle East and
elsewhere. He strongly
1 6 15 20 15 6 1
recommended that
Europeans adopt the
1 7 21 35 35 21 7 1
Indo-Arabic system of
numerals including the 1 8 28 56 70 56 28 8 1
use of a symbol for
1 9 36 84 126 126 84 36 9 1
zero “zephirum”
1
8
1 13 78 286 715 1287 1716 1716 1287 715 286 78 13
1