1. p
2. Prime Numbers
3. Perfect Numbers
4. Amicable Numbers
5. Perfectly Divisible Numbers
6. Magic Square
7. Time, Dates & Years
8. Fibonacci Number

p

3.1415926535897932384626433832795...
My favorite constant. I can recite up to 10 decimal places only. The most accurate version of pi (p) has been calculated to 2,260,321,336 decimal places by brothers Gregory Volfovich and David Volfovich Chudnovsky, on their homemade supercomputer in New York in Summer 1991. This constant is the most recognizable of all constants and it should applies to any 2 or 3 dimensional realm. This means any planet or moon in any solar system in any galaxy will have this constant as well. This is what I mean when I say “Mathematics is the language of the universe”. No matter where you are, mathematics is the language that all intelligent life forms can understand and can communicate in. 

For 1 million decimal places of p goto this site. Pi=3.

Prime Numbers

The most basic properties of numbers. A prime number is any positive integer (excluding unity 1) having no integral factors other than itself and unity, e.g. 2, 3, 5, 7, 11, 13 and so on. The lowest prime number is 2. According to the Guinness Book of Records 1995, the highest known prime number is 2⁸⁵⁹⁴³³ - 1. It has 258,716 digits and it was discovered by using a CRAY C90 super computer in Cray Research Inc in 1994. I am sure there is a higher number known by now.

Prime numbers were used in the movie Contact for aliens to communicate when they sent us a message.

A number is said to be perfect if it is equal to the sum of all divisors of the number (other than itself), e.g. 1 + 2 + 4 + 7 + 14 = 28. The lowest perfect number is 6; 1 + 2 + 3 = 6. Perfect numbers were first named in Ancient Greece by the Pythagoreans around 500BC. To date, only 33 perfect numbers were discovered, the largest being (2⁸⁵⁹⁴³³ - 1) x 2⁸⁵⁹⁴³³. That’s a 517,430 digit number. The first four perfect numbers were discovered before AD100 and these are
6 = 1 + 2 + 3
28 = 1 + 2 + 4 + 7 + 14
496 = 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248
8,128 = 1 + 2 + 4 + 8 + 16 + 32 + 64 + 127 + 254 + 508 + 1,016 + 2,032 + 4,064

The fifth perfect number however was not found until the 15th century. That number is
33,550,336 = 1 + 2 + 4 + 8 + 16 + 32 + 64 + 128 + 256 + 512 + 1,024 + 2,048 + 4,096 + 8,191 + 16,382 + 32,664 + 65,528 + 131,056 + 262,112 + 524,224 + 1,048,448 + 2,096,896 + 4,193,792 + 8,387,584 + 16,775,168

Amicable Numbers

Amicable numbers are pairs which are mutually equal to the sum of all their divisors (other than itself), e.g. 220 and 284. The sum of all the divisors (other than itself) of 220 are 1 + 2 + 4 + 5 + 10 + 11 + 20 + 22 + 44 + 55 + 110 = 284 and the sum of all the divisors (other than itself) of 284 are 1 + 2 + 4 + 71 + 142 = 220. I have taken a great interest in amicable numbers and have so far discovered 300 pairs of amicable numbers. Check out this file amicable.txt for the latest pair of amicable numbers. 

Perfectly Divisible Numbers

How do you tell if a number is perfectly divisible by a small number? Here are the methods. Take an example of the number 1,081,080.

Perfectly divisible by 1

All integers are perfectly divisible by 1 so 1,081,080 is perfectly divisible by 1.

Perfectly divisible by 2

Any even number is perfectly divisible by 2 so 1,081,080 is perfectly divisible by 2.

Perfectly divisible by 3

If the sum of all the digits is perfectly divisible by 3, the number is perfectly divisible by 3.  The sum of the digits in 1,081,080 is 1 + 0 + 8 + 1 + 0 + 8 + 0 = 18. Continue to add up all the digits. 1 + 8 = 9. 9 is perfectly divisible by 3 so 1,081,080 is perfectly divisible by 3.

Perfectly divisible by 4

If the last 2 digits is perfectly divisible by 4, the number is perfectly divisible by 4. In the number 1,081,080 the last 2 digits is 80 and it is perfectly divisible by 4 so 1,081,080 is perfectly divisible by 4.

Perfectly divisible by 5

If the last digit ends with 5 OR 0, the number is perfectly divisible by 5 so 1,081,080 is perfectly divisible by 5.

Perfectly divisible by 6

If the number is perfectly divisible by 2 AND 3, the number is perfectly divisible by 6 so 1,081,080 is perfectly divisible by 6.

Perfectly divisible by 7

This one is tough. Split the number into groups of 3 starting from the end and insert alternating (-) AND (+). 1,081,080 would be split into 1 AND 081 AND 080. So + 1 - 081 + 080 = 0. If the sum is 0 OR perfectly divisible by 7, the number is perfectly divisible by 7 so 1,081,080 is perfectly divisible by 7.

Unless it is a very large number, it would probably be faster to actually do the division to find out if it is divisible by 7.

Perfectly divisible by 8

If the last 3 digits is perfectly divisible by 8, the number is perfectly divisible by 8. In the number 1,081,080 the last 3 digits is 080 and is perfectly divisible by 8 so the number is perfectly divisible by 8.

Perfectly divisible by 9

Same method as the factor of 3. If the sum of the digits is perfectly divisible by 9, the number is perfectly divisible by 9. The sum of the digits in 1,081,080 is 1 + 0 + 8 + 1 + 0 + 8 + 0 = 18. Continue to add up all the digits. 1 + 8 = 9. 9 is perfectly divisible by 9 so 1,081,080 is perfectly divisible by 9. 

One of the most interesting fact about this is that if the sum of all digits is not 9, then that sum is the remainder of the number if it had divided by 9. Eg. 23,456 divided by 9 gives you a remainder of 2 (2 + 3 + 4 + 5 + 6 = 20; 2 + 0 = 2; so remainder is 2).

Perfectly divisible by 10

If the last digit ends with 0, the number is perfectly divisible by 10 so 1,081,080 is perfectly divisible by 10.

Perfectly divisible by 11

Another tough one. Split alternate digits and put them in 2 groups. So 1,081,080 would be split into 
         1  8  0  0 
AND  0  1   8 
        ________
         1081080
        =======
Now add up all the digits in the 2 groups. 1 + 8 + 0 + 0 = 9; 0 + 1 + 8 = 9. Take the difference between the 2 sums. If the difference of the 2 sums is 0 OR 11, the number is perfectly divisible by 11 so 1,081,080 is perfectly perfectly divisible by 11.

Perfectly divisible by 12

If the number is perfectly divisible by 3 AND 4, the number is perfectly divisible by 12 so 1,081,080 is perfectly divisible by 12.

Perfectly divisible by 13

Same method as the factor of 7. Split the number into groups of 3 starting from then end and insert alternating (-) AND (+). 1,081,080 would be split into 1 AND 081 AND 080. So + 1 - 081 + 080 = 0. If the sum is 0 OR perfectly divisible by 13, the number is perfectly divisible by 13 so 1,081,080 is perfectly divisible by 13.

Unless it is a very large number, it would probably be faster to actually do the division to find out if it is divisible by 13.

Magic Square

To qualify as a magic square, each row, column and corner-to-corner (diagonally) must add up to the same number AND each number in each box can only be limited to the number of boxes available (ie. 3 x 3 = 9 so numbers available 1-9; 4 x 4 = 16 so numbers available 1-16 etc.) AND can only be used once.

3 x 3 Square (Total = 15)

4 x 4 Square (Total = 34)

5 x 5 Square (Total = 65)

6 x 6 Square (Total = 111)

7 x 7 Square (Total = 175)

8 x 8 Square (Total = 260)

9 x 9 Square (Total = 369)

10 x 10 Square (Total = 505)

Anyone who has magic square for 6 x 6 and 9 x 9, please e-mail me at hifever@hotmail.com. Please feel free with any comments to the e-mail above as well.

Time, Dates & Years  

How many days are there in a year? 
365? Close.
365 ¼? Closer.
365.24219878? Yeah! Now that’s precision! That’s 365 days 5 hours 48 minutes and 45.974592 seconds.

Many people thinks that a leap year comes once every 4 years right? Wrong! 1900 was not a leap year and neither was 1800. 2000 is but 2100 will not be one. Why? Well, in a course of the last 2,000 years, how many days were there supposed to be? Let’s do some calculations.

2,000 x 365.24219878 = 730,484.39756 days

If we have a leap year every 4 years how many days would there in the last 2,000 years?
1,500 (normal years) x 365 = 547,500
500 (leap years)        x 366 = 183,000
Total days                           = 730,500 

That’s 15.60244 days more. How do we compensate for it? Well, we take every 4th year to be a leap year but not any century year that is not divisible by 400 years. That means 1896 is a leap year because it is divisible by 4, 1900 is not a leap year because although it is divisible by 4, it is a century year and it is not divisible by 400. 2000 is a leap year because it is both divisible by 4 and 400. That’s why 2100 is not a leap year. Let’s do the calculations again with the corrected method.

1,515 (normal years) x 365 = 552,975
485 (leap years)        x 366 = 177,510
Total days                           = 730,485

This is more accurate but what about the 0.60244 days? This is compensated by adding a second every few years. The first leap second was on June 30, 1972 and there have been 20 leap seconds added since then. The one that I know of occurred on December 31, 1995.

Our calendars wasn’t always 365 days. The Babylonians’ year was only 360 days. The Romans improve on it. By the time of Julius Caesar, the people had learned that a year consisted of 365 ¼ days. That’s why we called it a Julian calendar, named after Julius Caesar. He decreed that the year would consist of 365 days and each of the 6 hours in each year would be disregarded for 3 years and that an entire day would be added in the forth year to become a “leap year”.

By 1582, the calendar was 10 days out of sync with the seasons. To fix this, Pope Gregory XIII decreed that 10 days (5 -14 October 1582) would be erased from the calendar. 4 October 1582 was called 15 October 1582. Most of the world’s Catholic countries immediately adopted this Gregorian calendar. The British however did not. They clung to the Julian calendar until 1752. By this time, the sync was out by 11 days until finally, it was agreeded to strike 11 days from 3 - 13 September 1752. 2 September 1752 became 14 September 1752. This is the reason why George Washington’s birthday is sometimes in dispute. He was born before 1752. Should history take the actual date of birth or the adjusted date?

The name of months were named after Roman gods and Latin words for numbers. 

• January - from Janus, the 2 faced god who looks forward and backwards at the same time.
• February - from a Roman festival, Februa.
• March - from Mars the Roman god of War.
• April - from the Latin Aprillis, month of Venus.
• May - from Maia, the goddess of increase.
• June - from the name of a prominent Roman family, Junius.
• July - the month of Julius Caesar.
• August - the month of Augustus Caesar.
• September - the Roman’s seventh (septem) month.
• October - the Roman’s eighth (octo) month.
• November - the Romans’s ninth (novem) month.
• December - the Roman’s tenty (decem) month.

The Roman year began in Spring which started in March so September, October, November and December  were the 7th, 8th, 9th, and 10th month.

The years after the supposed year of Jesus Christ’s birth are tagged A.D. from the Latin anno domini which means “in the year of our Lord”. To be strictly correct, A.D. should be placed before the year as in A.D. 2000. Years before Christ was born should be marked B.C., which comes after the date as in 100 B.C. Since there was no year 0 as the year that Christ was born was year 1, the first century (100 years) ended in year 100 and so the 2nd century begins in year 101. Similarly, the 2nd millennium (1,000 years) would end on year 2000 thus the 3rd millenium only start in year 2001. Still I am one of the suckers that will go for the 31 December 1999 celebration simply because this is the only time in the last 1,000 years and the next 1,000 years to come where all the digits of time will change from 1999 December 31 23:59:59 to 2000 January 1 00:00:00.

Fibonacci Number

Standing not far away from the Leaning Tower of Pisa is a small statue of Leonardo Fibonacci, a 13^th century Italian mathematician. Fibonacci published 3 major works, the best known was Liber Abaci (Book of Calculations). In Liber Abaci, the Fibonacci sequence of numbers is first presented as a solution to mathematical problem involving the reproduction rate of rabbits! This sequence is commonly referred to as the Fibonacci number.

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, … 

Take the 1^st number, 1 and add it to the last sequence number (nil) and you will get the next sequence number, 1. Then take the current sequence number, 1 and add it to the last sequence number, 1 and you will get the next sequence number, 2. Continue this method to get the rest of the sequence numbers. This sequence of number has been observed to occur in nature in the most marvelous manner. The most popular is the coordination of flowers’ petals which are usually in that sequence.

Take the first number and divide it with the next sequence number, you will get a ratio of 1 (1/1 = 1). Do it a few more times and you will get the following results
1/1 = 1
1/2 = 0.50
2/3 = 0.67
3/5 = 0.6
5/8 = 0.625
8/13 = 0.615
13/21 = 0.619
21/34 = 0.618
34/55 = 0.618
All ratios after this is around 0.618.

If you do the reverse, that is to take the latter sequence to divide the earlier sequence, you will get a ratio of the 1.618 or the inverse of 0.618
55/34 = 1.618
34/21 = 1.619
21/13 = 1.615
13/8 = 1.625
8/5 = 1.6
5/3 = 1.66
3/2 = 1.5
2/1 = 2

The ratio of 1.618 and 0.618 was known to the ancient Greek and Egyptian mathematicians and Fibonacci rediscovered it. This ratio is known as the Golden Ratio or Golden Mean. It is known to have application in music, art, architecture, and biology. The Greeks used the Golden Mean in constructing the Parthenon. The Egyptians used the Golden Ratio in building the Great Pyramid of Gizeh. The properties of the ratio were known to Pythagoras, Plato and Leonardo da Vinci. In the financial sector, the Golden Ratio is frequently use to predict the retracement of a bull market and is well known to all technical analysts. It should be acknowledged that the Fibonacci relationships do seem to recur though out nature and in virtually all areas of human activities. The recurring of the Fibonacci number seem endless and continue to pop-out from scientific studies and research.

You Can Me At hifever@hotmail.com
Or Me At 11022292.