Download presentation

Presentation is loading. Please wait.

Published byAlessandro Reams Modified over 3 years ago

1
Special Numbers PHI Perfect Numbers Harmonic Numbers

2
Phi – The Phinest number around This is the Golden Ratio. It can be derived from: Since n 2 -n 1 -n 0 =0 n 2 -n-1=0 n 2 =n+1 The root of which is 0.5(5 1/2 +1) Which can be approximated to:

3
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 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

4
BUT WHY DOES THAT MATTER?!?! Well…

5
Check this out! You can make a ruler based on this ratio looking like this: And you can see that this ratio appears everywhere!

6
IN YOUR FACE!!!

7
In nature

8
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… ?

9
Perfect Numbers A perfect number is a positive integer which is the sum of all its 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)

10
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

11
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…+2 k-1 =2 k -1 We can rearrange to 2 k-1 (2 k -1) should be a perfect number (so long as 2 k -1 is prime).

12
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 2 k-1 (2k - 1), for some k > 1, where 2 k - 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

13
The 4 th rule Take the example of when k=11: 2 10 (2 11 -1)=1024x2047=2096128 Therefore the 4 th rule is also incorrect Check again

14
5 th rule Cant dispute it. To date there are 39 known perfect numbers The last of which is: 2 13466916 (2 13466917 - 1).

15
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

16
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!

Similar presentations

OK

Chapter 5 Number Theory © 2008 Pearson Addison-Wesley. All rights reserved.

Chapter 5 Number Theory © 2008 Pearson Addison-Wesley. All rights reserved.

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Mammalian anatomy and physiology ppt on cells Free ppt on motions of the earth Pdf to ppt online nitro Ppt on japanese culture Ppt on adventure tourism in india Download ppt on indus valley civilization culture Controller area network seminar ppt on 4g Ppt on attendance management system using rfid Ppt on earth day 2014 Ppt on nestle india ltd company