MATHEMATICAL EXCURSIONS

input { width: 100%; font-size: medium; } body { font-family: Verdana, Arial, Helvetica, sans-serif; font-size: x-small; margin: 0px; padding: 0px; } h1 { font-size: medium; font-weight: bold; } td { text-align: center; vertical-align: middle; } table { margin: 4px; border: thick solid #0099FF; }

The solids and their regularities were discovered by the Pythagoreans and were called originally Pythagorean solids. The Greek philosopher Plato described the solids in detail later in his book "Timaeus" and assigned the items to the Platonic conception of the world, hence today they are well-known under the name "Platonic Solids."

Loops "Mathematicians have their own version of stunt flying. They use numbers instead of airplanes. (Wouldn't you know it!) They get their thrills from numerical loopings and from discovering patterns while they're doing their tricks. One person's pattern is another's loop-the-loop." Here are some interesting looping techniques (involving starting with some number and repeating some process on it over and over again) that you might want to try out: Start with any number you like and follow the following rules: If your number is even, divide it by two. Otherwise, multiply it by 3, then add 1. Repeat step 1. And again, and again... that's the looping part. Here's a sample. Start with 10. It's even, so take half. That gives you 5. That's odd, so multiply it by 3 and add 1: (5x3)+1 = 16. Back to even, so take half and get 8. Half again gives you 4. Half again gets you to 2, and half again gives you 1. Since 1 is odd, multiply by 3 and add 1 to get 4. Half of 4 gives you 2, and half of that gets you back to 1. You're in a loop now and will be forever if you keep at it. Try the same procedure for 33 and see if you get the following: 33, 100, 50, 25, 76, 38, 19, 58, 29, 88, 44, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1, 4, 2, 1, ... A calculator might come in handy here. Have some fun trying some more, but beware: If you start with 27, it takes 109 steps! Other time-consuming yet small numbers are 31, 41, 47, 55, 62, 63, 71, and 73. Of course, any of these numbers multiplied by two or four (or any other power of two) will give you a long sequence too. With a calculator, you should be able to poke right along; without one, you'll still get there - it will just take a while longer. If you have a certain kind of mind, you may find this system interesting. After you've tried it several times, you may wonder whether every number loops into that 4-2-1 pattern at the end. I have personally tested this for every number between 1 and 4000 (with a computer program, of course; I might post the code here shortly), and each of those numbers fall into that loop. Obviously or surprisingly, depending on whether you consider yourself a mathematician or not, this doesn't satisfy mathematicians. The question they have is: Will every number loop into that pattern? It's hard to tell. It's not easy to figure out a proof, and so far no-one has come up with one. No-one has found a number that doesn't work, but no-one has been able to prove that every number works. Here's another looping trick similar to the first one. This one was developed by Clifford Pickover, who calls it the Juggler Sequence. The difference between this sequence and the loop described above is that, to get the next number, you take the square root of the current number if it is even, and multiply by the square root of the current number if it is odd. Most sequences end . . ., 6, 2, 1. Some never seem to end, though. You might want to try a few numbers. For the next looping procedure, start with any two numbers from 0 to 9 and follow this rule: Add the two numbers and write down just the digit that is in the ones place. Here's an example: Suppose you start with 8 and 9. Adding them gives you 17. Keep just the 7, which is in the ones place. Add the last two numbers, the 9 and the 7. That gives 16; keep the 6, then you have 8-9-7-6. Keep going, adding the last two numbers in the series each time, keeping only the digit in the ones place. Do this until you get 8 and 9 again. Then the loop starts all over. The 8-9 pattern has twelve numbers in the loop before it repeats. The pattern is: 8-9-7-6-3-9-2-1-3-4-7-1-8-9. If going around in a numerical circle appeals to you, you may have the makings of a terrific mathematician. Hang in there. But beware. If you start with the same two numbers, but in the opposite order, and follow the same rule: 9-8-7-5, and so on, it will take 60 numbers before it starts to repeat! Don't tackle that one unless you're sure you have the time. For a quickie, try 2 and 6. Here's some questions you might want to think about: How many different possible pairs of numbers are there to start with? (It's okay to start with two numbers that are the same). Do all pairs of numbers eventually return to the starting point? What's the shortest loop you can find? Is there a pattern of odds and evens in the loop? Here is a numerical example using words. Start with any number (in the example below, 39). Write it as a word: thirty-nine. Then continue as shown. Start with any number 39 Write it as a word thirty-nine Count the letters 10 Write that as a word ten Count the letters 3 Write that as a word three Count the letters 5 Write that as a word five Count the letters 4 Write that as a word four Count the letters 4 You'll get 4 forever and ever. In fact, you will always end up with 4, no matter what number you start with originally. Try a different number and see. Convince yourself with some examples, then see if you can figure out why you'll always get to four. A good starting point would be to look at how to write out big numbers. You may also want to determine what the result would be in other languages.

Forces & Gravity A Perfect Circle? Because of Earth's rotation, our planet is not a perfect sphere. It is a little flatter at the poles, and there is a bulge at the equator. This means that if you were standing at the poles, you're actually closer to the center of the earth than if you're standing on the equator. Since the force of gravity between two objects is controlled in part by the distance to the center of the mass, the force of gravity at the poles is slightly more than at the equator. If you were to stand on a bathroom scale at the equator, you would actually weigh less than at the poles.

Number & Math Play Average Speed Louise runs the first half of a race at 5 miles per hour. Then she picks up her pace and runs the last half of the race at 10 miles per hour. What is her average speed on the course? The answer is not 7.5 miles per hour. Assume the course is 20 miles long. Louise needs 2 hours to run the first 10 miles (the first half of the course) and 1 hour to run the last 10 miles (the last half), meaning she needs 3 hours to run the total of 20 miles. Therefore, her average speed is 6.7 (or 6 2/3) miles per hour.

Alan Turing was born on June 23, 1912 in London. His father was a British civil servant on India. His mother went back to England to await the birth of Alan. After he was born his parents returned to India leaving him in care of foster parents. His mother thought that the environment in India was unsuitable for him. Excerpt for occasional visits back home by his parent, they rarely saw Alan. He had a lonely childhood. When he was thirteen he entered the Sherbourne School. At the school he was criticized for his handwriting, struggles at English, and even mathematics. He was too interested with his own ideas to produce solutions using methods taught by his teachers. He did win almost every possible mathematics prizes at Sherbourne. He declared, while at the school, that he was an atheist. Despite the difficult school, Alan entered Kings College., in Cambridge in 1931. When he graduated from Kings College, he was offered a teaching position. In 1936, Turing announced his invention of the Turing Machine. It was a computer program that would give instructions to a machine. He was eager to put his program into practical applications by the use of electronics. It was his idea of creating a machine that turned the thought process of data transmitted into electronic boxes and screens. This was the forerunner to the modern computer that we know of in our present time. Information would be fed into the machine by electronic tape. In 1936, Turning went to Princeton University as a graduate student. He returned to England in 1938. While at Princeton, Turing had played with the idea of the construction of a computer. With the outbreak of World War 11, Turing was asked by the British government to break the German codes by deciphering encrypted German communications. The German had a device called Enigma that constantly changed the codes that the Germans were using. It was thought to be impossible to break the Germans codes. Turing created a code-breaking device. Breaking the codes was of great assistance to the winning of World War 11 by the allies. The creation of the machine was the first step toward a digital computer. The code breaking was called the Clossus. The Allies knew every political and military move the Germans used. Only a handful of people knew of its existence From 1942 till March 1943, Turing was in the United States helping with decoding issues. At the end of the war, Turing was invited by the National Physical Laboratory in London to design a computer. Turing returned to Cambridge for the academic year 1947-1948. His interests ranged on many topics that were far removed from computers and mathematics. He studied neurology. He wrote a paper called Intelligent Machinery that was published in 1969. The concept of artificial intelligence was brought forth in the paper. Turing believed that machines could be created that would mimic the thought process of the brain. He said that machines would be able to play chess and do anything else Turing remains a hero to the believers of artificial intelligence. Turning can be said to be a non-conformist all during his life. In the early fifties the British government started to consider him a security risk. Turing died of potassium cyanide poisoning while conducting experiments. The cyanide was found on a half eaten apple beside him. An inquest concluded that it was self-administered, but his mother always maintained that it was an accident. The official pronouncement by the British government was A moment of mental in-balance Some people call him the father of the modern computer

FACTS ABOUT THE INCA NUMBER SYSTEM AND OTHER SCIENTIFIC ACHIEVEMENTS QUIPOS WERE VERY IMPORTANT IN THE EVERY DAY LIFE OF THE INCAS. THE QUIPU WAS A ROPE THAT HAD SEVERAL KNOTS TIED ON IT. THE INCA NUMBER SYSTEM WAS THE SAME AS OURS. IT WAS WRITTEN TO THE BASE 10. THE SPACES ON THE ROPE REPRESENTED THE NUMER ZERO.THEY WERE VERY SKILLED IN SCIENCE. THEY DID NOT HAVE ANY FORMAL WRITING SYSTEM. THE AMERICAN INDIAN AND THE INCAS HAD ONE THING IN COMMON; THEY BOTH DID NOT HAVE A FORMALIZED WRITING SYSTEM. ALL COMMUNCIATION WAS DONE ORALLY.INFORMATION WAS HANDED DOWN FROM GENERATION TO GENERATION BY TELLING STORIES. Excerpt from the "History of Mathematics Through the Ages" copyright 2003 Frank Bell

THE HISTORY OF PI THROUGH THE AGES Pi was known to the ancient Egyptians. They gave the value to be approximately 4/3 to the 4th power which gives a number approximately equal to 3.1604. The reference to Pi appeared in the Ahmes papyrus. Archmides found that Pi to be about 3.14. He named Pi in a book called The Measurement of a Circle In the book he states that Pi is between 3 10/31 and 3 1/7. He got this number by making a polygon of 96 sides and inscribing a circle inside the polygon. The first person to use the Greek letter Pi for the representation of the number was an English Mathematician by the name of William Jones. Pi is the most famous ratio in mathematic. The number Pi goes on forever as a decimal representation. This tpe of number is called an irrational number.Pi has been taken to the following decimal 3.141592265358793This is the decimal expansion of the number. There is no pattern of repeating digits in a certain block of numbers. A verse in the Bible says and he made a molten sea, ten cubits from the one brim to the other; it was round all about and his height was five cubits did compass it about [1 Kings 7, 23] The phrase was talking about the temple of Solomon, built around 950BC It gives Pi equal to 3 A way to calculate Pi is to find the circumference of a circle and the diameter of that circle then divide the circumference by the diameter. The was an Englishman named Shanks who in 1873 calculated Pi to 707 places Mind you he did it without the aid of a calculator or a computer. In 1949 a computer calculated Pi to 2,000 decimal places. Some interesting facts about the number Pi Pi is involved in DNA procedures Pi is in a rainbow When rain drop falls Pi emerges in the spreading circles that are formes Pi appears in music Pi can be found in waves Pi is found in tables used by actuaries in the insurance business Pi will be with us till the end of time Some other facts about pi Pi is the first letter in Greek word perimeter meaning distance around Euler, the famous mathematician, used the symbol for pi to be equal to the ratio of the circumference to the diameter of a circle. Ptolemy in 150 A.D. found pi to be approximately 337/120 or 3.1416. In 480 A. D. in China, pi was found to be approximately equal to 355/113 or 3.2425929. John Walls equation for pi in 1650 A.D. pi/2 =(2) (2) (4) (4) (6) (6) (8).. ____________________ (1) (3) (3) (5) (5) (7) (7) _.. Leibniz equation for pi pi/4= 1 1/3 + 1/5 1/7 + .. In 1949 ENIAC, the first modern computer, spent 70 hours to compute pi to 2,037 places In 1997 pi is computed to 51,539,600, 000 decimal places In September 2002 pi is computer to 1,240,000,000,000 decimal places It took over 400 hours to compute with supercomputer Excerpt from " The History of Mathematics Through the Ages" Copyright 2003 by Frank Bell

In my many years of teaching mathematics, I have found that many students have difficulty in solving math word stories. The stories are mathmatical equations expressed in English sentences. Some examples of these math word stories are:

Why math is taught in school! I was riding to work yesterday when I observed a female driver cut right in front of a pickup truck, causing him to have to drive on to the shoulder to avoid hitting her. This evidently angered the driver enough that he hung his arm out his window and "flipped" the woman off. "Man, that guy is stupid," I thought to myself. I always smile nicely and wave in a sheepish manner whenever a female does anything to me in traffic, and here's why: I drive 48 miles each way every day to work. That's 96 miles each day. Of these, 16 miles each way is bumper-to-bumper. Most of the bumper-to-bumper is on an 8 lane highway. There are 7 cars every 40 feet for 32 miles. That works out to be 982 cars every mile, or 31,424 cars. Even though the rest of the 32 miles is not bumper-to-bumper, I figure I pass at least another 4000 cars. That brings the number to something like 36,000 cars that I pass every day. Statistically, females drive half of these. That's 18,000 women drivers! In any given group of females, 1 in 28 has P.M.S. That's 642. According to Cosmopolitan, 70% describe their love life as dissatisfying or unrewarding. That's 449. According to the National Institutes of Health, 22% of all females have seriously considered suicide or homicide. That's 98. And 34% describe men as their biggest problem. That's 33. According to the National Rifle Association, 5% of all females carry weapons, and this number is increasing. That means that every single day, I drive past at least one female that has a lousy love life, thinks men are her biggest problem, has seriously considered suicide or homicide, has P.M.S., and is armed. Flip one off? ...I think not !!!!

ARCHIMEDES Born: 287 BC in Syracuse, Sicily Died: 212 BC in Syracuse, Sicily Archimedes' father was an astronomer.His fathers name was Phidas and was related to Hieron 11, the king of Syracuse. It is not known whether he was married or had any other children. Nothing more is known about his father. Archimedes was born in 287 BC in Syracuse, a Greek seaport colony in Sicily. Archimedes was a native of Syracuse, Sicily. He is considered as one of the three greatest mathematicians along with Newton and Gauss. Except for his studies at Euclids school in Alexandria, Egypt, he spent his entire life in his birthplace of Syracuse. Much of Archimedes fame comes from his relationship with Hieron, the king of Syracuse and the kings son. Archimedes made a habit of solving the kings most complicated problems. to the amazement of the king. Archimedes uncovered a fraud against King Hieron of Syracuse using his principle of buoyancy. The king suspected that a solid gold crown he ordered was partly made of silver. Archimedes first took two equal weights of gold and silver and compared their weights when immersed in water. The difference between these two comparisons reveled that the crown was not solid gold.This is now called the Archimedes Principle. He was so excited at the solution of the problem that he ran through the streets of Syracuse shouting EUREKA! EUREKA![I have found it] Some of Archimedes discoveries and inventions: The water screw, still in use in Egypt for irrigation, the Hydrostate Principle [a body immersed in a fluid is subject to the overflow equal to the weight of the fluid displaced by the body], the measurement of a circle ,discovery of the irrational number pi, ,the Sand Reckoner problem, the intergal calculus [ finding the area under a curve]

Excerpt from: The History of Mathematics Through the Ages

Excerpt from" The History of Mathematics Through the Ages"

THE USE OF MATHENATICAL SYMBOLS The idea of numbers and counting can be traced back to the beginning of mankind. It can be assumed that early man did not use words or symbols when he counted. He probably used objects like pebbles, knots on a rope or marks on the ground to represent the numbers. From this prehistoric beginning a type of counting system with numbers, symbols and words evolved through time as an evolutionary process. There were different systems of numbers. The Mayans for instance used numbers that were to the base 20. The Romans used Roman numerals that are still used today. The Hindu-Arabic numerals system uses 10 as the base This is the present day number system. PERFECT NUMBERS The Pythagoreans in their number system called the even numbers female and the odd numbers male. Their numbers also stood for an abstract idea.The 1 stood for reason, the 2 stood for opinion the 3 stood for harmony the 4 stood for justice and so on. The Pythagorean arithmetic has special classes of numbers. They were called perfect numbers. There were two kinds of perfect numbers 1. The first kind included 10, which is the sum of the first four positive whole numbers1+2+3+4=10 2.The second kind of perfect numbers were those equal to the sum of their proper divisors. Example: the Pythagoreans called 6 a perfect number. The sum of the proper divisors is equal to the number. In other words what numbers will go into evenly 6 without a remainder? The answer is 1,2,and 3 The numbers will go into 6 evenly and the sum of the three is 6 Is 28 a perfect number? What are the numbers that will go into 28 evenly? What is the sum of those numbers? Do they add to the number 28? Notice that the number itself is not included in the answer. PRIME NUMBERS If a natural number has only two factors, itself and one, then the number is a prime number. Some may think that it is easy to find all of the prime numbers. We can find some of the prime numbers by starting with the first 20 natural numbers 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20 The prime numbers are 2,3,5,7,11,13,17,19 why is 4 not a prime number, because the are other factors for beside 1 and 4 like 2 which is a factor for 4 A Greek mathematician named Eratostenes is most famous for his investigation into prime numbers. He created what is known as the Sieve of Eratosthenes. This Sieve allows anyone to find any prime number. The number one is not prime. It has only one factor. This is how the Sieve works: Strike every second number the number 2 or multiple of 2. Move to the next number three. Strike after the number three every third number and so on or multiples of three Example using the sieve: Take the first 20 natural numbers 1,2,3,4,5,6,7,8,9,10,11,12,13,14.15,16,17,18,19,20 Strike number 1 out. Go to number 2 .two is prime. Strike every out every second number after 2 or multiples of 2 So 4,6,8,10, 12, 14, 16, 18, and 20 are eliminated We have the following numbers left 2, 3,5,7,9,11,13,17,19.Go to number 3 . 3 is a prime number by the definition. Strike out every third number after 3 or are multiples of 3The numbers that we have left from the original given numbers are2,3,5,7,1,13,17,19 The question is , how many prime numbers are there in the number system? The answer is infinite number of primes FIBONACCI NUMBERS The first twenty numbers in this sequence are 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 previous numbers in order to get the next number. 0+1=1, 1+1=2, 2+3=5, 3+5=8, 5+8=13, 8+13=21 and so on ZERO IN THE HISTORY OF MATHEMATICS Who discovered zero? We really do not know that answer to this question. I guess you can say the number zero evolved through the history of mathematics. We know that there are two important uses of zero. One is being used as the empty place indicator in our number system like the number 3106, the zero is used to separate the the number three from the number one. The numbers 316 and 3106 are two different numbers The second use of zero is the number itself. The Babylonians around 400BC put two wedge shape figures into a number value where we would put zero today. The Greeks how great in developing mathematics did not use zero as a place value in their numbers. In the geometry developed by the Greeks they said that zero and the irrational numbers were impossible. The Mayan, 1300 years ago, had some concept of zero. They used this design to represent zero in their number system. Around 650 AD the use of zero came into vogue by the introduction of Indian mathematics. The Indians used a place-value system and zero was used to show the empty space. AMICABLE NUMBERS There have been throughout the history of numbers many interesting numbers. One of these is called the amicable number. What is an amicable number? It is a pair of two numbers that have the following property: The sum of all proper divisors of the first number [not including the number itself] exactly equals the second number while the sum of all the proper divisors of the second number [not including the number itself] likewise equals the first number. If this sounds a little confusing to you an example of the property just stated will help you grasp the meaning of the statement. Are the numbers 220 and 284 amicable numbers? We first find the proper divisors of 220. What do we mean by the phrase proper divisors? A proper divisor is that when you divide it into an other number you get no remainder For the number 220 the proper divisors are as follows: 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110 All of these numbers are the proper divisors of 220. If you add up all these numbers you get the second number 284. Take the second number 284. What are the proper divisors for this number? They are 1, 2, 4, 71, 142 Add up all of these numbers and you get the first number 220. So the answer to the question is yes, they are amicable numbers. No other numbers but 220 and 284 were found to be amicable numbers until 1636 when Fermat discovered a second pair 17,296 and 18,416. In 1638 Descartes found a third pair 9,363,584 and 9,437,056. The First Thirteen Amicable Numbers 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 1`8,416 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 Euler produced 58 pairs of amicable numbers in1747. He developed a formula that proves amicable pairs. Ther are over 5,000 known amicable pairs. The largest was found in 1997 and ot contained 4,829 digits in each pair. Some questions still to be answered 1. Are there an infinite amount of pairs of amicable numbers? 2. do the pairs appear so that they are both even or they are both odd? 3. Are there triplets amicable numbers Who knows maybe you will discover the answers to these questions. THE GOLDEN RATIO Phi The ratio of 1;1.61803 is unique and has many stories in the history of mathematics. Do not confuse the word Pi with Phi. They are two different numbers. Phi was named after the Greek sculpture Phidas. This ratio is found in nature, art, architecture, poetry , music and of course mathematic. The number phi is as unique as the number Pi. It is a decimal that never ends and never has a pattern of repeating itself. Sound familiar. The number falls into the classification as a irrational number. This golden ratio is found in the pyramids in Giza, Egypt. It is found in the United Nations Building in New York City. It is found in many different ways. Another interesting fact about the golden ratio is with the Fibonacci sequence. If you find the ratio of two successive numbers in this sequence you will find as the numbers get larger the ratio gets closer and closer to Phi Example: Take the numbers 3 and 5 in the Fibonacci sequence Express the numbers as a ratio 5/3= 1.618 Take 233 and 144 in the sequence 233/144= 1.6181 Take 610 and 377 in the sequence 6109/377= 1.6181 FRACTIONS THROUGHOUT HISTORY The Babylonians wrote numbers in a system which was like a place value was like a place value system.The numbers were written to the base 60 rather than our system which is to the base ten.Their place value system made it easy to writ fractions The number above represents the square root of two in Babylonian notation The Babylonian number system did not use zero The ancient Egyptians and the Greeks used unit fractions. By this I mean that the numerator of all the fractions was one. An example of this using our number system, 1/7, 1/8, 1/4., 1/3, and so on. They did not have fractions to represent Say 7/8 or 15/16 The Romans did not use numerals to represent fractions but used words to describe the fractions The table below shows an example of our fractions compared to the Roman fraction Our representation Roman representation 11/12 deony 9/12 dodrans 8/12 bes 1/24 semuncia It the ancient world it was common to write a fraction like ¾ as 3,4. They used a comma to distinguish the numbers.The horizontal bar was introduced by the Arabs in 1200. Fibonacci was the European mathematician to use the fraction bar [/] The use of a decimal fraction was introduced around 929 AD. An Arab mathematician wrote the earliest known text offering this presentation. In 1586 Simon Steven published two texts called The Tenth and The Decimal which explained the use of decimal fractions .An example from the text, 5.912 Some mathematicians around this tome did not use the dot but wrote a fraction as 0/56 to represent 0.56 USE OF THE SYMBOLS OF OPERATION IN MATHEMATICS The plus[+] and minus[-] signs were written in texts between 1365 and 1366.The symbols appears in manuscripts. These manuscripts are in the Deseden library that shows the uses of the plus and minus signs. The plus and minus signs came into general use in England in 1557. The multiplication symbol X was used about 1628 and was published in a mathematical text in 1631. The dot was used as a symbol for multiplication by Leibniz . He wrote a letter to the mathematician John Bernoulli stating that he did not like X to represent the word for multiplication as it may be confused with using the letter x which represents a value in a equation. He introduced the dot. The closed parenthesis [ ] was used around 1486 to 1567 The colon : was introduced in a text written by an English mathematician by the name of Johnson. Leibnz used the colon for both the ratio and division. The division sign :/: sign It is claimed to be used in a book by John Pell. The positive exponents first appeared in the 14th century . In 1637 Renee Descartes introduced the modern notation of exponents used today. The summation symbol is and was first used by Euler or marks on the ground to represent the numbers. From this prehistoric beginning a type of counting system with numbers, symbols and words evolved through time as an evolutionary process. There were different systems of numbers. The Mayans for instance used numbers that were to the base 20. The Romans used Roman numerals that are still used today. The Hindu-Arabic numerals system uses 10 as the base This is the present day number system. PERFECT NUMBERS The Pythagoreans in their number system called the even numbers female and the odd numbers male. Their numbers also stood for an abstract idea.The 1 stood for reason, the 2 stood for opinion the 3 stood for harmony the 4 stood for justice and so on. The Pythagorean arithmetic has special classes of numbers. They were called perfect numbers. There were two kinds of perfect numbers 2. The first kind included 10, which is the sum of the first four positive whole numbers1+2+3+4=10 2.The second kind of perfect numbers were those equal to the sum of their proper divisors. Example: the Pythagoreans called 6 a perfect number. The sum of the proper divisors is equal to the number. In other words what numbers will go into evenly 6 without a remainder? The answer is 1,2,and 3 The numbers will go into 6 evenly and the sum of the three is 6 Is 28 a perfect number? What are the numbers that will go into 28 evenly? What is the sum of those numbers? Do they add to the number 28? Notice that the number itself is not included in the answer. PRIME NUMBERS If a natural number has only two factors, itself and one, then the number is a prime number. Some may think that it is easy to find all of the prime numbers. We can find some of the prime numbers by starting with the first 20 natural numbers 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20 The prime numbers are 2,3,5,7,11,13,17,19 why is 4 not a prime number, because the are other factors for beside 1 and 4 like 2 which is a factor for 4 A Greek mathematician named Eratostenes is most famous for his investigation into prime numbers. He created what is known as the Sieve of Eratosthenes. This Sieve allows anyone to find any prime number. The number one is not prime. It has only one factor. This is how the Sieve works: Strike every second number the number 2 or multiple of 2. Move to the next number three. Strike after the number three every third number and so on or multiples of three Example using the sieve: Take the first 20 natural numbers 1,2,3,4,5,6,7,8,9,10,11,12,13,14.15,16,17,18,19,20 Strike number 1 out. Go to number 2 .two is prime. Strike every out every second number after 2 or multiples of 2 So 4,6,8,10, 12, 14, 16, 18, and 20 are eliminated We have the following numbers left 2, 3,5,7,9,11,13,17,19.Go to number 3 . 3 is a prime number by the definition. Strike out every third number after 3 or are multiples of 3The numbers that we have left from the original given numbers are2,3,5,7,1,13,17,19 The question is , how many prime numbers are there in the number system? The answer is infinite number of primes FIBONACCI NUMBERS The first twenty numbers in this sequence are 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 previous numbers in order to get the next number. 0+1=1, 1+1=2, 2+3=5, 3+5=8, 5+8=13, 8+13=21 and so on ZERO IN THE HISTORY OF MATHEMATICS Who discovered zero? We really do not know that answer to this question. I guess you can say the number zero evolved through the history of mathematics. We know that there are two important uses of zero. One is being used as the empty place indicator in our number system like the number 3106, the zero is used to separate the the number three from the number one. The numbers 316 and 3106 are two different numbers The second use of zero is the number itself. The Babylonians around 400BC put two wedge shape figures into a number value where we would put zero today. The Greeks how great in developing mathematics did not use zero as a place value in their numbers. In the geometry developed by the Greeks they said that zero and the irrational numbers were impossible. The Mayan, 1300 years ago, had some concept of zero. They used this design to represent zero in their number system. Around 650 AD the use of zero came into vogue by the introduction of Indian mathematics. The Indians used a place-value system and zero was used to show the empty space. AMICABLE NUMBERS There have been throughout the history of numbers many interesting numbers. One of these is called the amicable number. What is an amicable number? It is a pair of two numbers that have the following property: The sum of all proper divisors of the first number [not including the number itself] exactly equals the second number while the sum of all the proper divisors of the second number [not including the number itself] likewise equals the first number. If this sounds a little confusing to you an example of the property just stated will help you grasp the meaning of the statement. Are the numbers 220 and 284 amicable numbers? We first find the proper divisors of 220. What do we mean by the phrase proper divisors? A proper divisor is that when you divide it into an other number you get no remainder For the number 220 the proper divisors are as follows: 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110 All of these numbers are the proper divisors of 220. If you add up all these numbers you get the second number 284. Take the second number 284. What are the proper divisors for this number? They are 1, 2, 4, 71, 142 Add up all of these numbers and you get the first number 220. So the answer to the question is yes, they are amicable numbers. No other numbers but 220 and 284 were found to be amicable numbers until 1636 when Fermat discovered a second pair 17,296 and 18,416. In 1638 Descartes found a third pair 9,363,584 and 9,437,056. The First Thirteen Amicable Numbers 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 1`8,416 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 Euler produced 58 pairs of amicable numbers in1747. He developed a formula that proves amicable pairs. Ther are over 5,000 known amicable pairs. The largest was found in 1997 and ot contained 4,829 digits in each pair. Some questions still to be answered 4. Are there an infinite amount of pairs of amicable numbers? 5. do the pairs appear so that they are both even or they are both odd? 6. Are there triplets amicable numbers Who knows maybe you will discover the answers to these questions. THE GOLDEN RATIO Phi The ratio of 1;1.61803 is unique and has many stories in the history of mathematics. Do not confuse the word Pi with Phi. They are two different numbers. Phi was named after the Greek sculpture Phidas. This ratio is found in nature, art, architecture, poetry , music and of course mathematic. The number phi is as unique as the number Pi. It is a decimal that never ends and never has a pattern of repeating itself. Sound familiar. The number falls into the classification as a irrational number. This golden ratio is found in the pyramids in Giza, Egypt. It is found in the United Nations Building in New York City. It is found in many different ways. Another interesting fact about the golden ratio is with the Fibonacci sequence. If you find the ratio of two successive numbers in this sequence you will find as the numbers get larger the ratio gets closer and closer to Phi Example: Take the numbers 3 and 5 in the Fibonacci sequence Express the numbers as a ratio 5/3= 1.618 Take 233 and 144 in the sequence 233/144= 1.6181 Take 610 and 377 in the sequence 6109/377= 1.6181 FRACTIONS THROUGHOUT HISTORY The Babylonians wrote numbers in a system which was like a place value was like a place value system.The numbers were written to the base 60 rather than our system which is to the base ten.Their place value system made it easy to writ fractions The number above represents the square root of two in Babylonian notation The Babylonian number system did not use zero The ancient Egyptians and the Greeks used unit fractions. By this I mean that the numerator of all the fractions was one. An example of this using our number system, 1/7, 1/8, 1/4., 1/3, and so on. They did not have fractions to represent Say 7/8 or 15/16 The Romans did not use numerals to represent fractions but used words to describe the fractions The table below shows an example of our fractions compared to the Roman fraction Our representation Roman representation 11/12 deony 9/12 dodrans 8/12 bes 1/24 semuncia It the ancient world it was common to write a fraction like ¾ as 3,4. They used a comma to distinguish the numbers.The horizontal bar was introduced by the Arabs in 1200. Fibonacci was the European mathematician to use the fraction bar [/] The use of a decimal fraction was introduced around 929 AD. An Arab mathematician wrote the earliest known text offering this presentation. In 1586 Simon Steven published two texts called The Tenth and The Decimal which explained the use of decimal fractions .An example from the text, 5.912 Some mathematicians around this tome did not use the dot but wrote a fraction as 0/56 to represent 0.56 USE OF THE SYMBOLS OF OPERATION IN MATHEMATICS The plus[+] and minus[-] signs were written in texts between 1365 and 1366.The symbols appears in manuscripts. These manuscripts are in the Deseden library that shows the uses of the plus and minus signs. The plus and minus signs came into general use in England in 1557. The multiplication symbol X was used about 1628 and was published in a mathematical text in 1631. The dot was used as a symbol for multiplication by Leibniz . He wrote a letter to the mathematician John Bernoulli stating that he did not like X to represent the word for multiplication as it may be confused with using the letter x which represents a value in a equation. He introduced the dot. The closed parenthesis [ ] was used around 1486 to 1567 The colon : was introduced in a text written by an English mathematician by the name of Johnson. Leibnz used the colon for both the ratio and division. The division sign :/: sign It is claimed to be used in a book by John Pell. The positive exponents first appeared in the 14th century . In 1637 Renee Descartes introduced the modern notation of exponents used today. The summation symbol is and was first used by Euler

Excerpts from The History of Mathematics Through the Ages

Scientific calculator