Theory of Numbers Introduction

What is Number Theory? Number theory is a branch of mathematics mainly concerned with the integers that has been a topic of study for centuries. Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10th Edition, © 2012

Number theory opens the doors to many mathematical conjectures. Number theory is a prime source to show that numbers can be fascinating. Number theory opens the doors to many mathematical conjectures. Number theory provides an avenue to extend and practice mathematical skills. Number theory offers a source of recreation. Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10th Edition, © 2012

Number Theory and Patterns Where are the perfect squares? Where are the odd numbers? Find the sum of the upright diamonds (such as 1 and 3, 2 and 6). If you do this in order, what is the pattern of the sums? Find the sum of the numbers in each row. Find a short cut for finding the sum. Find another pattern and describe it. Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10th Edition, © 2012

A factor of a number divides that number with no remainder A factor of a number divides that number with no remainder. What are the factors of this rectangle? A multiple of a number is the product of that number and any other whole number. Multiples of 5 are 5,10,15,20… Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10th Edition, © 2012

A composite number is any number with more than two factors. A prime number is a whole number greater than 1 that has exactly two factors, 1 and itself. A composite number is any number with more than two factors. Notice that the number 5 is a prime number since it can only be represented as either a 5x1 or a 1x5 rectangle. Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10th Edition, © 2012

A number is divisible by another number if there is no remainder. Today, divisibility rules provide opportunities to discover why a rule works or to discover a rule. Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10th Edition, © 2012

Two numbers are relatively prime if they have no common factors other than 1. Star patterns can be used to investigate relatively prime and not relatively prime pairs of numbers: (12, 5) Star Continue connecting every 5 points with a straight line. 12 points, connect top point (12 o/clock) to 5 points, clockwise. Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10th Edition, © 2012

Polygonal or figurate numbers are numbers related to geometric shapes. Notice the perfect squares. Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10th Edition, © 2012

In some elementary books, modular arithmetic is called clock arithmetic. In mod 8 (see the “clock” below), we use the numbers 0, 1, 2, …7. What do you think the sum of 6 and 7 would be? Try it on the clock and think of starting at 6 o’clock and adding 7 hours. What time would it be? Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10th Edition, © 2012

Other Number Theory Topics: Pascal’s Triangle What would be the numbers in Row 7? Row 8? Row 9? What patterns do you see? 3. Find the sum of the numbers in each row beginning with row 1 and ending with row 6. What do you think would be the sum of row 7, of row 8, of the row 20? 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 Row 0 Row 1 Pascal’s Triangle is named for the Frenchman who lived in the 17th Century. Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10th Edition, © 2012

A Pythagorean triple is a triple of numbers (a, b, c) such that a² + b² = c². Pythagoras, the person given credit for the Pythagorean Theorem, lived in Greece in about 500 BC. Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10th Edition, © 2012

Do you see how it is generated? Fibonacci was an Italian who lived in the 13th Century. His famous sequence also was known early in India. 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, . . . Do you see how it is generated? Look at the picture of the finger bone. Have you seen these numbers before? Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10th Edition, © 2012

Famous Mathematicians in Number Theory

Tell me and I will Forget, Show me and I will remember, Involve me and I will understand. Confucius

Confucius

Confucius Kong Qui or K’ung Fu-tzu better known as Confucius He was born in August 27, 551 B.C. in Tuo, China His teachings focused on: creating ethical model of family and pubic interaction and setting educational standards

Confucius Philosophy & Teachings His Social Philosophy was based primarily on the principle of: a. Ren or loving others while exercising self-discipline He believed that Ren could put into action using the Golden Rule “What you do not wish for yourself, do not do to others”

Confucius’s Philosophy & Teachings His Political Beliefs was based on the concept of SELF-DISCIPLINE According to Confucius: “Leaders could motivate their subjects to follow the law by teaching them virtue & the unifying force of ritual propriety

Confucius’s Philosophy & Teachings His Philosophy of Education focused on the SIX ARTS: a. Archery d. Music b. Calligraphy e. Chariot Driving c. Computation f. Ritual To Confucius, the main objective of being an educator was to teach people to live with integrity. Through his teachings, he strove to resurrect the traditional values of benevolence, propriety and ritual in Chinese society.

Confucius Major Works: He was credited in writing & Editing influential Traditional Chinese Classics Rearrangement Book of Odes A Historical Book He did a revision Book of Documents He compiled a historical account of the 12 dukes of Lu called Spring & Autumn Annals Lunyu Sets forth Confucius’ Philosophical & Political beliefs Compiled by his disciples Translated in English as The Analects of Confucius , which means a collection of sayings and ideas

Died on November 21, 479 B.C. in Qufu, China Death & Legacy Died on November 21, 479 B.C. in Qufu, China 4th Century B.C. He was regarded as a sage who had deserved greater recognition in his time 2nd Century BC during China’s first Han Dynasty, his ideas became the foundation of the state ideology Today he is widely considered one of the most influential teachers in Chinese history Confucius

The outstanding German mathematician Karl Friedrich Gauss (1777–1855) once said, “Mathematics is the queen of the sciences and arithmetic is the queen of mathematics.”

“Arithmetic,” in the sense Gauss uses it, is number theory, which, along with geometry, is one of the two oldest branches of mathematics. Number theory, as a fundamental body of knowledge, has played a pivotal role in the development of mathematics.

The study of number theory is elegant, beautiful, and delightful The study of number theory is elegant, beautiful, and delightful. A remarkable feature of number theory is that many of its results are within the reach of amateurs. These results can be studied, understood, and appreciated without much mathematical sophistication.

Number theory provides a fertile ground for both professionals and amateurs. We can also find throughout number theory many fascinating conjectures whose proofs have eluded some of the most brilliant mathematicians. We find a great number of unsolved problems as well as many intriguing results.

Another interesting characteristic of number theory is that although many of its results can be stated in simple and elegant terms, their proofs are sometimes long and complicated.

Generally speaking, we can define “number theory” as the study of the properties of numbers, where by “numbers” we mean integers and, more specifically, positive integers.

Studying number theory is a rewarding experience for several reasons Studying number theory is a rewarding experience for several reasons. First, it has historic significance. Second, integers, more specifically, positive integers, are the building blocks of the real number system, so they merit special recognition.

Third, the subject yields great beauty and offers both fun and excitement. Finally, the many unsolved problems that have been daunting mathematicians for centuries provide unlimited opportunities to expand the frontiers of mathematical knowledge.

Although number theory was originally studied for its own sake, today it has intriguing applications to such diverse fields as computer science and cryptography (the art of creating and breaking codes).

The foundations for number theory as a discipline were laid out by the Greek mathematician Pythagoras and his disciples (known as the Pythagoreans). The Pythagorean brotherhood believed that “everything is number” and that the central explanation of the universe lies in number.

They also believed some numbers have mystical powers They also believed some numbers have mystical powers. The Pythagoreans have been credited with the invention of amicable numbers, perfect numbers, figurate numbers, and Pythagorean triples. They classified integers into odd and even integers, and into primes and composites.

Another Greek mathematician, Euclid (ca. 330–275 B. C Another Greek mathematician, Euclid (ca. 330–275 B.C.), also made significant contributions to number theory.

Who is Pythagoras?

Pythagoras (ca. 572–ca. 500 B.C.), a Greek philosopher and mathematician, was born on the Aegean island of Samos. After extensive travel and studies, he returned home around 529 B.C. only to find that Samos was under tyranny, so he migrated to the Greek port of Crontona, now in southern Italy.

There he founded the famous Pythagorean school among the aristocrats of the city. Besides being an academy for philosophy, mathematics, and natural science, the school became the center of a closely knit brotherhood sharing arcane rites and observances.

The brotherhood ascribed all its discoveries to the master The brotherhood ascribed all its discoveries to the master. A philosopher, Pythagoras taught that number was the essence of everything, and he associated numbers with mystical powers.

He also believed in the transmigration of the soul, an idea he might have borrowed from the Hindus. Suspicions arose about the brotherhood, leading to the murder of most of its members.

The school was destroyed in a political uprising The school was destroyed in a political uprising. It is not known whether Pythagoras escaped death or was killed.

Who is Euclid?

Little is known about Euclid’s life Little is known about Euclid’s life. He was on the faculty at the University of Alexandria and founded the Alexandrian School of Mathematics.

When the Egyptian ruler King Ptolemy I asked Euclid, the father of geometry, if there were an easier way to learn geometry than by studying The Elements, he replied, “There is no royal road to geometry.”

Leopold Kronecker (1823–1891) was born in 1823 into a well-to-do family in Liegnitz, Prussia (now Poland). After being tutored privately at home during his early years and then attending a preparatory school, he went on to the local gymnasium, where he excelled in Greek, Latin, Hebrew, mathematics, and philosophy.

There he was fortunate to have the brilliant German mathematician Ernst Eduard Kummer (1810–1893) as his teacher. Recognizing Kronecker’s mathematical talents, Kummer encouraged him to pursue independent scientific work. Kummer later became his professor at the universities of Breslau and Berlin.

Tell me and I will Forget, Show me and I will remember, Confucius Assignment!!! 1 whole sheet of paper! State what you understand or your thoughts about the quote from Confucius! Word count minimum is 100 words! This is worth 50 points! Two letter word like “is, to etc are not counted! Tell me and I will Forget, Show me and I will remember, Involve me and I will understand. Confucius

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