Download The Beauty of Numbers - Oldham Sixth Form College

Special Numbers
PHI
Perfect
Numbers
Harmonic
Numbers
Phi – The Phinest number around
This is the “Golden Ratio”.
It can be derived from:
Since n2-n1-n0=0
n2-n-1=0
n2=n+1
The root of which is 0.5(51/2+1)
Which can be approximated to:
Phi to the first 1000 decimal places
• 1.618033988749894848204586834365638117720309179805762862
13544862270526046281890244970720720418939113748475408807
53868917521266338622235369317931800607667265443338908659
59395829056383226613199282902678806752087668925017116962
07032221043216269548626296313614438149758701220340805887
95445474924618569536486444924104432077134494704956584678
85098743394422125448770664780915884607499887124007652170
57517978834166256249407589069704000281210427621771117778
05315317141011704666599146697987317613560067087480710131
79523689427521948435305678300228785699782977834784587822
89110976250030269615617002504643382437764861028383126833
03724292675263116533924731671112115881863851331620384005
22216579128667529465490681131715993432359734949850904094
76213222981017261070596116456299098162905552085247903524
06020172799747175342777592778625619432082750513121815628
55122248093947123414517022373580577278616008688382952304
59264787801788992199027077690389532196819861514378031499
7411069260886742962267575605231727775203536
Phee Phi Pho Phum
I smell the blood of a Mathematician
BUT WHY DOES
THAT MATTER?!?!
Well…
Check this out!
You can make a ruler based on this ratio
looking like this:
And you can see that this ratio appears
everywhere!
IN YOUR FACE!!!
In nature
So what does it all mean?
Some people take this to be a proof that
god exists as all of these things could not
be based on this same ratio purely by
chance. This suggests a creator or
designer…
?
Perfect Numbers
A perfect number is a positive integer which is the sum of all
it’s positive divisors (e.g. 6 being the sum of 1, 2 and 3)
The first 4 perfect numbers are 6, 28, 496 and 8128
1+2+3=6
1+2+4+7+14=28
1+2+4+8+16+31+62+124+248=496
1+2+4+8+16+32+64+127+254+508+1016+2032+4064=8128
(The first records of these came from Euclid around 300BC)
This starts going up very quickly
As you can see:
6, 28, 496, 8128,33550336, 8589869056,
137438691328, 2305843008139952128,
2658455991569831744654692615953842176,
19156194260823610729479337808430363813
0997321548169216,
13164036458569648337239753460458722910
223472318386943117783728128
You can take my word for it or if you want you can work them out. =P
How to find a perfect number:
According to Euclid, if you start with 1 and keep
adding the double of the number preceding it
until the sum is a prime number
e.g. 1+2+4=7
Then take the last number (4) and the sum (7)
then you should get a perfect number 4x7=28
Also from 1+2+3+4…+2k-1=2k-1
We can rearrange to 2k-1(2k-1) should be a perfect
number (so long as 2k-1 is prime).
Nicomachus (c. 60 –c. 120)
Nicomachus added some extra rules for perfect
numbers:
1.)The nth perfect number has n digits.
2.) All perfect numbers are even.
3.) All perfect numbers end in 6 and 8 alternately.
4.) Euclid's algorithm to generate perfect numbers will give all perfect
numbers i.e. every perfect number is of the form 2k-1(2k - 1), for some
k > 1, where 2k - 1 is prime.
5.) There are an infinite amount of perfect numbers.
At this time however, only the first 4 perfect numbers had been found,
do these rules apply to the rest of them?
Check
The 4th rule
Take the example of when k=11:
210(211-1)=1024x2047=2096128
Check again
Therefore the 4th rule is also incorrect
5th rule
Can’t dispute it.
To date there are 39 known perfect numbers
The last of which is: 213466916(213466917 - 1).
Perfect Harmony
Perfect numbers are all thought to be
Harmonic numbers integer whose divisors
have a harmonic mean that is an integer.
e.g. 6 which has the divisors 1, 2, 3 and 6
And 140:
=5
This sequence goes a little bit like
this:
1, 6, 28, 140, 270, 496, 672, 1638, 2970,
6200, 8128, 8190 …
Including the perfect numbers: 6, 28, 496,
8128
However: This could also be as wrong as
Nicomachus so beware!