Fibonacci
Number Trick

Here is a number trick that is not too well known and so you can use it to impress your friends. In fact this doesn't require a computer and can even be done with paper and pencil. It involves a series known as the Fibonacci number sequence, named after the Italian mathematician Leonardo Fibonacci (1170-1250).
A Fibonacci number sequence is formed by starting with any two numbers, adding those to get a third number, adding the second and third to produce a fourth number and so on.
This is much easier to see with a short example:

      2
      3
      5
      8
    13
    21
    34
    55
    89
  144
As can be seen, the sequence is formed by adding the previous two numbers.
2 plus 3 = 5, 3 plus 5 = 8, 5 plus 8 = 13, 8 plus 13 = 21 and so on.
Using the calculator below, if you input 2 and 3 into the first two boxes, when you click "Calculate", you will see all 10 boxes filled in with the same numbers in the list above.











Now, for the "trick" with the Fibonacci number sequence.
Ask your friend for two numbers.
You could then enter the numbers in this computer page but it is much more impressive if this trick is done on paper.
Adding the two numbers, create a Fibonacci sequence that is exactly ten steps long.
When you reach the tenth number, tell your friend you can total all ten numbers in your head!

And what's the secret?

Whenever you have a Fibonacci sequence of 10 numbers, the total will always be the seventh number times 11.

For practice, input two and three in the first two boxes and then click "CALCULATE".
Yes, you could get the total by clicking "Calc Total" or by using the trick.

If you entered two and three for the first two numbers, the seventh number will be 34 and multiplying this by 11 gives a result of 374.
It isn't that difficult to multiply by 11 in your head.
For example, to multiply 34 by 11, think of summing 34 and 34 but shift one number one decimal place.

               34
               34  
               374

Obviously, when you ask a friend for two numbers make sure you mention they should be kept relatively small.

Doing this trick on paper is much more impressive than on a computer so why not use this computer page just for practice?

Remember, before you show this to anybody, the best advice is to practice.

* * * * * * * * * * * * * * * *
And, if you are curious, here are the first 100 Fibonacci numbers:

  1   1
  2   1
  3   2
  4   3
  5   5
  6   8
  7   13
  8   21
  9   34
  10   55
  11   89
  12   144
  13   233
  14   377
  15   610
  16   987
  17   1,597
  18   2,584
  19   4,181
  20   6,765
  21   10,946
  22   17,711
  23   28,657
  24   46,368
  25   75,025
  26   121,393
  27   196,418
  28   317,811
  29   514,229
  30   832,040
  31   1,346,269
  32   2,178,309
  33   3,524,578
  34   5,702,887
  35   9,227,465
  36   14,930,352
  37   24,157,817
  38   39,088,169
  39   63,245,986
  40   102,334,155
  41   165,580,141
  42   267,914,296
  43   433,494,437
  44   701,408,733
  45   1,134,903,170
  46   1,836,311,903
  47   2,971,215,073
  48   4,807,526,976
  49   7,778,742,049
  50   12,586,269,025
  51   20,365,011,074
  52   32,951,280,099
  53   53,316,291,173
  54   86,267,571,272
  55   139,583,862,445
  56   225,851,433,717
  57   365,435,296,162
  58   591,286,729,879
  59   956,722,026,041
  60   1,548,008,755,920
  61   2,504,730,781,961
  62   4,052,739,537,881
  63   6,557,470,319,842
  64   10,610,209,857,723
  65   17,167,680,177,565
  66   27,777,890,035,288
  67   44,945,570,212,853
  68   72,723,460,248,141
  69   117,669,030,460,994
  70   190,392,490,709,135
  71   308,061,521,170,129
  72   498,454,011,879,264
  73   806,515,533,049,393
  74   1,304,969,544,928,657
  75   2,111,485,077,978,050
  76   3,416,454,622,906,707
  77   5,527,939,700,884,757
  78   8,944,394,323,791,464
  79   14,472,334,024,676,221
  80   23,416,728,348,467,685
  81   37,889,062,373,143,906
  82   61,305,790,721,611,591
  83   99,194,853,094,755,497
  84   160,500,643,816,367,088
  85   259,695,496,911,122,585
  86   420,196,140,727,489,673
  87   679,891,637,638,612,258
  88   1,100,087,778,366,101,931
  89   1,779,979,416,004,714,189
  90   2,880,067,194,370,816,120
  91   4,660,046,610,375,530,309
  92   7,540,113,804,746,346,429
  93   12,200,160,415,121,876,738
  94   19,740,274,219,868,223,167
  95   31,940,434,634,990,099,905
  96   51,680,708,854,858,323,072
  97   83,621,143,489,848,422,977
  98   135,301,852,344,706,746,049
  99   218,922,995,834,555,169,026
  100   354,224,848,179,261,915,075


An interesting aspect of the Fibonacci Number Sequence is that if you divide one Fibonacci number by the previous Fibonacci number, this produces a quotient called the phi ratio φ which is also known as the golden ratio.

For example, Fibonacci #20 divided by Fibonacci #19 =

6,765 ÷ 4,181 = 1.618033963166...


Fibonacci #50 divided by Fibonacci #49 =

12,586,269,025 ÷ 7,778,742,049 = 1.6180339887499...


As we go further down the Fibonacci sequence, this number approaches a limit of {1 + Square Root (5) } ÷ 2
= 1.6180339887498948482...

Another way to express the phi ratio is:
φ = 0.5 + 5.5 • 0.5 = 1.6180339887498948482...

* * * * * * * * * * * * * * * *
The Fibonacci Sequence appears in many places.

Here is a website that explains the arrangement of seeds in a sunflower being based on the Fibonacci Sequence.

* * * * * * * * * * * * * * * *
Leonardo Fibonacci (1170 - 1250) is thought to be the western world's most skillful mathematician of the Middle Ages.
In his 1202 book Libre Abaci he strongly advocated for the use of Arabic Numerals as opposed to Roman Numerals.
He stated that Arabic Numerals were much easier to read and calculations could be done more quickly and accurately with them.

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