Topics About Special Numbers
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Following are some items relating to special numbers discussed in the history of mathematics.

Contents of this Page
Perfect Numbers
Square Numbers
The Nature of 
Prime Numbers
Amicable Numbers
The Golden Ratio
Animals and Numbers
Ideal and Irational Numbers
Fibonacci Numbers
The History of Zero
 
 
 
 
 
 
 
 
 
 
 
 
 


Perfect Numbers

 

The Pythagoreans produced a theory of numbers comprised of numerology and scientific speculation. In their numerology, even numbers were feminine and odd numbers masculine. The numbers also represented abstract concepts such as 1 stood for reason, 2 stood for opinion, 3 stood for harmony, 4 stood for justice, and so on. Their arithmetica had a theory of special classes of numbers. There were “perfect” numbers of two kinds. The first kind included only 10, which was basic to the decimal system and the sum of the first four numbers 1 + 2 + 3 + 4 = 10. The second kind of “perfect” numbers were those equal to the sum of their proper divisors.

A perfect number is a positive integer that is equal to the sum of it divisors. However, for the case of a perfect number, the number itself is not included in the sum. The Greeks called a number such as 6 or 28 a “perfect” number because the sum of the proper divisors in each case is equal to the number; the proper divisors of 6 are 1, 2, and 3, and their sum is 6.

Although perfect numbers are regarded as arithmetical curiosities, their study has helped to develop the theory of numbers. Euclid proved that a number n of the form (2n-1)*2n-1 is a perfect number if the factor 2n-1 is prime. For example, if n assumes the value 2, 3, 5, or 7, the expression 2n-1 takes on the value 3, 7, 31, or 127, all of which are prime. For these values of n we obtain the perfect numbers 6, 28, 496, and 8,128.

The Neoplatonists Nicomachus of Gerasa and Iamblichus of Chalcis listed these perfect numbers and concluded that they follow a pattern: They alternately end in a 6 or an 8, and there is one perfect number for each interval from 1 to 10, 10 to 100, 100 to 1,000 and 1,000 to 10,000. They conjectured that both parts of the pattern would continue, but in this they were wrong. The fifth perfect number, which was discovered in the fifth century, corresponds to n = 13 and is 33,550,336, with eight digits rather than six. In addition, the sixth perfect number, like the fifth, ends with a six.

In 1961, the twentieth perfect number was found. It contains 2,663 digits in the decimal representation and corresponds to the case where n = 4,423. Today, thirty-seven perfect numbers are known. The prime for the largest of these is 23,021,377, which is 909,526 digits in length, and the largest perfect number is 1,819,050 digits in length It is not known whether there are an infinite number of perfect numbers.

In 1757, Leonhard Euler probed that every even perfect number must be of Euclid’s form. It has also been proven that every even perfect number must end in six or eight and if it ends in six, the digit preceding it must be odd. No one has as yet discovered an odd perfect number, but it is known that none exist below 1020.

Contributed by Kristen Shelton

References

Calinger, Ronald, Classics of Mathematics, Prentice Hall, New Jersey 1995.

Boyer, Carl B., A History of Mathematics, John Wiley & Sons, Inc., New York 1989.


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Square Numbers

Square array of dots, probably formed with pebbles, led the Greeks to numbers that were perfect squares- that is to numbers which, when expressed in a various of ways as the products of two numbers, would have two equal factors.


The most complete discussion of square numbers was given by a Greek, Nicomachus of Gerasa (c. A.D. 100) in Introdictio Arithmetica, the earliest extant manuscript, dating back to the tenth century. Nicomachus was not the original mathematician, but he did organize previous generations of mathematics in a clear and precise manner. The first 10 square numbers-

1, 4, 9, 16, 25, 36, 49, 64, 81, 100

Each is a result of multiplying a number by itself-

1*1, 2*2, 3*3, 4*4, 5*5……..

- which can also be written-

12 , 22 , 32 , 42 ………

Interesting facts:

If you add any 2 consecutive triangular numbers, you will always make a square number. The triangular numbers are formed, geometrically, like the square numbers

Triangular numbers can also be computed by the formula

n(n + 1)/2, for n = 1, 2, 3, 4, ...
As we see from the above figure, the first three triangular numbers are 1, 3, and 6.

All square numbers end in either a 0, 1, 4, 5, 6 or a 9. Never a 2, 3, 7, or an 8.

If you subtract 2 consecutive square numbers, you will get an odd number- 3, 5, 7, 9……

Contributed by Andrea Reynolds

Bibliography:

The National Council of Teachers of Mathematics. Historical Topics for the Mathematics Classroom. Thirty- first Yearbook: Washington D.C. 1969.

Reference:

www.hoxie.org/math/count/square.htm


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The Nature of

What is (Pi)?

(Pi) is the ratio of the distance around a circle to the circles' diameter. Mostly (Pi) is written as a fraction:

(distance around a circle)/(distance across and through the center of a circle)

or more simply C/D

The result of this fraction is a famous nonending, nonrepeating infinite string of decimals that mathematicians have been studying ever since Archimedes (287-212 B.C.E.). Archimedes is credited with providing a method of calculating (Pi). This number goes something like this: 3.14159265358979323 and so on. Your calculator ( or most calculators) go ten places. This is considered more than enough to ensure accuracy in measurements. Measuring is one of the main things you can do with this number.

What does this number mean?

What mathematicians discovered is the existence in nature of ratios. A ratio is a relationsip between two things. For example 1/2 is a ratio that gives us information about how the 1 and 2 are related. In this case 1 is divided by 2. Not only do these relationhisp exist but some of them 'always' have the same answer. They don't change. In other words the ratio of a circles' diameter to its circumference is always the same. It doesn't matter what these two numbers are. Their ratio is always equal to that very long number with all the decimals.

What is (Pi) good for?

You can find the area of a circle, the circumference of a cirlce, and the volume of a cylinder using the value of (Pi). Remember finding the area of a quadrilateral is the length(l) times the width(w).
l x w

In the figure below the area is equal to the sum of the 'squares' found in the figure.


Now try to find the number of 'squares' in a circle.


How do you account for shaded areas that are not square? Some math people call these shapes lunes. You can calculate area much easier by using your formula for (Pi): (1/2)*(Pi)*r2.

In this case the radius equals 1. The value for (Pi) is 3.14. The formula now looks like this:

(1/2)(3.14)(12) = area of the circle.

History of (Pi)

Since (Pi) is a nonending,nonrepeating, infinite decimal it has stood as a monument to futility and utter uselessness. Various people have sat around calculating (without calculators, calculus, or even algebra) this number.:

Ptolemy (ca.150 A.D.) 3.1416
Tsu Ch'ung Chi (430-501 A.D.) 355/113
Al Khwarizmi (c. 800 A.D.) 3.1416
Al ' Kashi (c. 1430) 14 places
Viete ( 1540-1603) 9 places
Roomen (1561-1615) 17 places
Van Ceulen (c. 1600) 35 places

Today if you so desire you may download this number up to 50 million places on to your computer. You to can add your name to the list!!

Contributed by David Leib

References:

Beckmann, Petr. A History of (Pi). St. Martins Press, New York 1971.

(Pi) sites:
The Miraculous Bailey-Borwein-Plouffe Pi Algorithm
The Pi Project
Pi search page
www.exploratorium.edu/learning-studio/pi/pi.html


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Prime Numbers

A natural number that possesses only two factors, itself and 1, is called a prime number. You may think that it would be fairly easy to figure out all of these numbers. Well, I am sure that you could find the prime numbers within the first 20 natural numbers without much trouble, and you may be able to go even further without the use of a calculator. However, it does become much harder to figure out all of the prime numbers, especially in your head.

It was an early Greek mathematician that is most famous for his work with prime numbers. His name was Eratosthenes. It was this Greek mathematician that created what is known as the Sieve of Eratosthenes. This Sieve allows anyone to find all of the prime numbers quite easily. This is how the Sieve works . . .

Eratosthenes wrote all of the natural numbers down and proceeded to sieve out all of the numbers that were not primes. The number one has only one factor, so it is not a prime number. We move to number two which is a prime number. Strike every second number after the number two (or multiple of two). Move to the next prime number. It is three. Now we strike every third number after the number three (or multiple of three) and so on. We continue to strike out every nth number after the number n.

Now you are probably wondering just how many prime numbers there are. Well, there are infinitely many primes. In other words, there are more than we could ever count. We know that there is an infinite number of primes because if you were to multiply all of the known primes together and add 1, then you would get a number that must be divisible by at least one new prime number. You can keep doing this forever and never find the end of the prime number list.

One more interesting thing about prime numbers. Prime numbers are considered the "building blocks" of the natural numbers because every single natural number, excluding the number 1, is either a prime number or a product of prime numbers. In other words, every number, for example, 160 is either a prime or can be factored into a product of primes. To factor 160, we observe that 160 is even and hence is divisible by 2 (which happens to be the only even prime), so we divide 160 by 2, giving 160 = 2*80. Now 80 is even and nece divisible by 2, so 80 = 2*40 and 160 = 2*2*40. But 40 is also divisible by 2, 40 = 2*20, and 160 = 2*2*2*20. Continuing to divide by 2 until we can no longer do so, we get 160 = 2*2*2*2*2*5. Now 5 is a prime, so we would write 160 as a product of primes like this: 2*2*2*2*2*5 = 25*5. If we were to multiply all of these prime numbers, the answer would be 160. Thus, since all numbers other than 1 are prime numbers or products of prime numbers, we can use a process like the above to find the prime divisors of any number.

Enjoy your prime number hunting!

Contributed by Courtney Ast

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Amicable Numbers

Throughout history there have been many different interesting numbers or types of numbers. One of these types is amicable numbers. Amicable numbers are a pair of numbers with the following property: the sum of all of the proper divisors of the first number (not including itself) exactly equals the second number while the sum of all of the proper divisors of the second number (not including itself) likewise equals the first number.

For example let's show that 220 & 284 are amicable numbers:
First we find the proper divisors of 220:

1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110

If you add up all of these numbers you will see that they sum to 284.

Now find the proper divisors of 284:

1, 2, 4, 71, 142

These sum to 220, and therefore 220 & 284 are amicable numbers.

The set of 220 and 284 was the first known set of amicable numbers. Pythagoras discovered the relationship and coined the term amicable because he considered the numbers to be a symbol of friendship. No other pairs were known until 1636 when Fermat discovered 17,296 and 18,416 as a second pair. This pair was actually discovered over three hundred years earlier by the Arab mathematician al-Banna, but it was never known in the West until Fermat's findings. Then in 1638, Descartes discovered a third pair of 9,363,584 and 9,437,056.

The First Thirteen Amicable Pairs
1 220 284
2 1,184 1,210
3 2,620 2,924
4 5,020 5,564
5 6,232 6,368
6 10,774 10,856
7 12,285 14,595
8 17,296 18,416
9 63,020 76,084
10 66,928 66,992
11 67,095 71,145
12 69,615 87,633
13 79,750 88,730

It wasn't until 1747 when Leonhard Euler turned his attention to amicable numbers that progress began to take place. Euler was able to produce 58 pairs over the next three years. How did he find so many, so quickly? Euler developed a formula that would produce amicable pairs. The only problem was that the formula didn't generate every amicable pair. Today there are over 5000 known pairs, the largest of which was found by Mariano Garcia on October 4, 1997 contained 4829 digits in each pair.

There are still many questions left to be answered about amicable numbers. Are there an infinite amount of pairs? Do the pairs always appear so that they are both even or both odd? What about the existence of amicable triples or quadruples? These are all questions that mathematicians are searching & researching today. Who knows, maybe you could discover some of these answers and put yourself in the history books alongside Euler and Fermat.

Contributed by Damian Smithhisler

References

Web Addresses:

  1. http://www.askdrmath.com/problems/sparks11.12.98.html
  2. http://ellerbruch.nmu.edu/classes/Ma48…tudents/carlson/amicable.html
  3. http://forum.swarthmore.edu/dr.math/problems/keelan7.23.98.html
Books:
1. Dunham, William (1990). Journey Though Genius. New York: Wiley Science Editions.

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The Golden Ratio: Phi

The ratio of 1:1.61803 has unique and storied history. It is also know as Phi, named after the Greek sculptor Phidias. The ratio is found in nature, art, architecture, poetry, music, and of course math.

As stated, the Golden Ratio is also known as Phi. Phidias lived from c. 500 - 432 BC. He incorporated the ratio into many of his sculptures, one of which is the ancient Greek temple, the Parthenon. This was a rectangle shaped temple whose sides are in the Agolden proportion@.

The number Phi is as unique as the number pi. It is a number whose decimal never ends and never has a pattern repeat itself. Also, there are only two numbers that remain the same when you square them, 0 and 1. To fine the value of Phi squared you simple add one to Phi. There are only two numbers that have this property. It is Phi and a number closely related to it: 0.61803. This number is also known as phi (note the lower case p).

Phi is supposedly the >most pleasing= rectangle proportion. It is found in not only Greek architecture, but other architecture as well. A couple examples are the pyramids and the United Nations building.

The Golden Ratio is also seen in nature. It can be found in many different ways on the human body. For instance, measure from the top of the head to the base of the neck and also from the base of the neck to the belly button. You will find this ratio is close to the Golden Ratio. Other examples can be found in flowers, ferns, eggs, and sea shell to name a few.

It is found in art, also. Many books about painting will point out that it is better to position objects to Aabout one-third@ of the way across and not in the center of the picture. This seems to make the picture more pleasing to the eye.

Another interesting thing about the Golden Ratio has to do with the Fibonacci sequence. If you find the ratio of two successive numbers in the sequence you will find as the numbers get larger the ratio gets closer and closer the Phi. For instance, 5/3 = 1.667. 233/144 = 1.6181. 610/377 =1.6181.

Examples of the Golden Ratio can be seen everywhere. It is a unique number that has some very interesting properties. I have only touched on the surface of what is out there about it. Please check some of the web sites listed below to find out more about the Golden Ratio or Phi.

Contributed by James Means

References

Web Addresses:

  1. The Golden Ratio
  2. The Golden section ratio: Phi
  3. The Golden Section in Art, Architecture, and Music
Books:
1. Dunham, William (1990). Journey Though Genius. New York: Wiley Science Editions.

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Animals and Numbers

Can animals count or even have a concept of numbers? Many specialists in animal behavior have conducted experiments that have shown animals to possess an apparent sense of rudimentary perception of quantities. This is sometimes called the "number " sense. The number sense allows an animal to determine the differences in size between two small collections of similar objects. It also determines that a collection isn't the same after some objects are removed. Mothers of domestic animals and other animals have shown that they can definitely determine and perceive when one of their young is missing from the group.

Birds have shown that they can be trained to determine the number of seeds in different piles of seeds up to the number five. An example of their ability to count up to five is contained in the story of the crow. In this story, a squire was very frustrated with a particular crow. This crow had made its' nest in the watchtower of the squire's estate. The squire continuously tried to surprise the crow, but had no such luck. Each time the squire approached the nest, the crow would leave. It would not return until the squire had left. From a distance the bird would watch for the man's departure. With this in mind, the squire came up with a plan to fool the bird. He took another man into the tower with him. Hoping to outwit the bird, one man left the tower. However, the bird would still not return to the tower until both men had departed. This experiment was repeated in the succeeding days with two, three, and even four men. Yet, the crow was not to be fooled. Finally, five men were sent into the tower. As before, all entered the tower. Only one remained as the other four left the area. At this point, the crow lost count and returned to it's nest. Not being able to distinguish above the number four cost the crow it's life, as the squire rid the tower of the crow forever.

Birds are not the only creatures with a concept of numbers. The behavior of insects, such as the solitary wasp, indicates some basic mathematical understanding. The mother wasp lays her eggs in individual cells and provides each egg with a number of live caterpillars, on which the young will feed on when they hatch. The number of caterpillars is different among species, but it is always the same for each sex of eggs. The male is smaller, so the mother supplies him with only five caterpillars. The female, who is large, receives ten caterpillars in her cell. This shows that the mother can not only distinguish which cell contains a male or a female, but she can also distinguish between the numbers five and ten in the caterpillars she is providing.

Other animals, such as animals in circuses, have been hypothesized to have mathematical talents. Animals such as horses and elephants are displayed to have the abilities to add, subtract, and count. However, these talents are more of a trained response triggered by a cue from the trainer. The skills demonstrated are not an innate response, as seen with the crow and the wasp.

Contributed by Lloyd Holt

References

George Ifrah, From One to Zero. New York. Viking, l996.


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Ideal & Irrational Numbers

The Pythagoreans were people who followed the mathematician, Pythagoras, best known for the right triangle theorem which is named after him. They believed that the universe could be explained by numbers. They were especially aware of the numbers 1,2,3,4 which they called the tetractys. They swore to the tetractys and saw "fourness" in many things. The geometric elements point, line, surface and solid are examples of fourness.

To Pythagoras, numbers had personalities - masculine or feminine, perfect or incomplete, beautiful or ugly. Ten was the best: containing one, two, three, and four - and written in dot formation it formed a perfect triangle. Because the sum of 1,2,3 and 4 is 10; ten was considered the ideal number.

Whereas four represented nature, they believed that ten represented the universe. They used the eight planets which had been discovered at this time plus two invented planets to support this belief. The Pythagoreans were unable to see these two planets because the part of the Earth where they lived "faced away from these planets."

Pythagoras looked at ratios of whole numbers to explain nature. Upon his death, his followers used the method of indirect proof to show that the square root of 2 cannot be written as a ratio of two whole numbers. In using this proof, irrational numbers - numbers which cannot be written as the ratio of two whole numbers - were discovered.

PROOF:
 sqr (2) = m/n (a fraction reduced to lowest terms)
  2 = m2/n2 
  m2 = 2n2 (so, m is a multiple of 2, call it 2q)
  4q2 = 2n2 
  2q2 = n2 (so, n is a multiple of 2)

So, both m & n are multiples of 2, which is impossible, because m/n was reduced to lowest terms. Thus the proof that the square root of 2 cannot be expressed as a fraction, ie it is an irrational number.

Contributed by Carol Conrad

References

  1. http://forum.swarthmore.edu//library.browse/static.topic/r...
  2. www.history.mes.st~andrews.ac.uk.history.mathematic/pythagoras.html
  3. Smith, Sanderson. Agnesi to Zeno. California: Key Curriculum Press, 1996.

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Fibonacci Numbers

The Fibonacci Number Sequence was first presented in Leonardo Pisano's book, "Liber abaci" or "Book of Calculating". It is a sequence that I find to be very fascinating, and suprisingly it is a part of every day nature.

The Fibonacci sequence can be found in sea shell spirals, branching plants, petals on flowers, and in pine cones. I will explain this sequence to you in 3 different ways: the basic sequence, the rabbit problem, and the bees.

The Basic Sequence
The first twenty numbers of the sequence are as follows:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181

The numbers are obtained by adding two numbers to get the next. For example:
0 + 1 = 15 + 8 = 13 55 + 89 = 144 610 + 987 = 1597
1 + 1 = 2 8 + 13 = 21 89 + 144 = 233 987 + 1597 = 2584
1 + 2 = 3 13 + 21 = 34144 + 233 = 377 1597 + 2584 = 4181
2 + 3 = 5 21 + 34 = 55233 + 377 = 610 and so on……….
3 + 5 = 834 + 55 = 89 377 + 610 = 987  

The Rabbit Problem
This is another problem stated in Pisano's book. The question posed wanted you to figure out, under ideal circumstances, if you placed 2 rabbits together how many pairs of rabbits could be produced from the 1st pair in one year. Assume that the rabbits never die and that the female always produces one new pair (1 male/1 female) every month from the 2nd month on.

The Bees
This example is more realistic than that of the rabbits because there is a chance that they will have more than 2 offspring at a time. It is also possible that the offspring won't be a male and a female. The breeding pattern of bees is much more practical.

Many female worker bees are sterile. The queen bee's unfertilized eggs produce male bees. This means that they will have no "father". Female bees are produced when the queen actually mates with a male. They have both parents. Most females are workers but some are selected to become queen bees. From these facts we find the Fibonacci numbers:
The male bee has 1 parent
The female bee has 2 parents, 3 grand-parents, 5 great-grand-parents, 8 great-great-grand-parents, and 13 great-great-great-grand-parents

Contributed by Alayna R. Cobb

References

  1. Vorobev, N. Fibonacci Numbers. New York, Blaisdell Publishing Co. <1961>.
  2. The Fibonacci Quarterly. Santa Clara CA: Fibonacci Association.

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The History of Zero

In today's modern mathematics, we have become accustomed to zero as a number. It's hard to believe that most ancient number systems didn't include zero. The Mayan civilization may have been among the first to have a symbol for zero. The Mayas flourished in the Yucatan peninsula of Mexico about 1300 years ago. They used the as a placeholder, in a vertical place-value system. It is considered one of their cultures greatest achievements.

The ancient Egyptians, Romans, and Greeks alike had no symbol for zero. In Greek geometry, zero and irrational numbers were impossible. The Greeks made great strides in mathematics, but it was all done with a number system without zero. The Greek astronomer Ptolemy (ca. A.D. 150) was the first to write a zero at the end of a number. For this he used a circular symbol.

In ancient Babylonian history there was no use of the zero. In the later Babylonian or during the Seleucid period a special symbol, which was also used as a separation mark between sentences, came into use for a zero. There's a definite possibility that the Babylonians used this mark for a zero within a number, as early as the end of the eighth century B.C. Up until the time of Aristotle, there seems to be no evidence that the Babylonians ever regarded zero as a number. Aristotle discussed division by zero in connection with speed through a vacuum.

Throughout the Dark Ages, Western mathematics was held back by the Roman's traditional numbering system. The first to think differently was Leonardo Fibonacci. He was a merchant's son, born in the Italian city-state Pisa, late in the twelfth century. In Pisa, he studied the work of Euclid and other Greek mathematicians. When he was still a boy, he moved to the Muslim city of Bugia, in North Africa. There he examined leather and furs before they were shipped back to Pisa. Leonardo got an education in Arabic culture as he traveled around the Mediterranean to Constantinople, Egypt and Syria. He recognized that the Hindu-Arabic numerals, the numerals we use today, were superior to the Roman numerals he had grown up with in the West.

In the sixth century, mathematicians in India developed a place-value system. They introduced the concept of zero to keep their symbols in their proper places. In the seventh century, Hindu scholars introduced to Islam the ideas of zero and place-value. These ideas spread rapidly throughout the Arabic world. Six centuries later, Fibonacci was so impressed with the ease of Hindu-Arabic numerals that he wrote a book entitled Liber abaci.

The Pisan local merchants, the trading class, ignored Fibonacci's book. They were wallowing in prosperity and did not want to be bothered with giving up Roman numerals and adopting a zero. Ferbonacci's mathematician friends liked the new number system and slowly over time gave up the Roman numerals. By the fifteenth century, the numerals were showing up on coins and gravestones. Western mathematics had emerged from the Dark Ages, and was flourishing into a new number system with a zero, the Hindu-Arabic numerals. The immediate advances in mathematics after that time are proof of the importance of, the zero.

Contributed by Pam Nye

References

  1. Kaplan, Robert. The Nothing That Is: A Natural History of Zero, Oxford University Press, 2000
  2. Wells, David. The Penguin Dictionary of Curious and Interesting Numbers, Penguin Books, 1987
  3. Hoffman, Paul. The Man Who Loved Only Numbers, Hyperion, New York, 1998
  4. Burton, David M. The History of Mathematics, 3rd Addition, The McGraw -Hill Companies Inc., 1997

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