A study of the properties of numbers has been around since humans started counting and numbering various objects. Early on the base ten or decimal system become predominate. This is most likely due to the ten fingers. There is a recorded history of various other bases being used. For Instance, the vigesimal system based on twenty was used, possibly due to twenty being the sum of fingers and toes.
A study of the properties of numbers has been around since humans started counting and numbering various objects. Early on the base ten or decimal system become predominate. This is most likely due to the ten fingers. There is a recorded history of various other bases being used. For Instance, the vigesimal system based on twenty was used, possibly due to twenty being the sum of fingers and toes. We still utilize a sexagesimal system based on sixty for time and measurements of angles in minutes and seconds. Today, the binary system is used for computer programming.
The development of numbers required names and symbols and ordering. Many cultures have utilized different forms. We have recorded representations from early Egyptian Numerals, Roman Numerals, Greek Numerals, and more. The numbering that is common today is the Hindu-Arabic Numerals. Historians point to the Brahmi Symbols from 100 B.C. out of India as the likely origin. The Gobar Numerals from 1000 A.D. were introduced in Spain by the Arabs. They bear a great resemblance to the modern numbers we use.
The study of numbers were for the Pythagorean followers an actual basis for religious belief and way of life. Many great philosophers have utilized properties of mathematics as parts of their foundations of belief. Interesting, bizarre, and perhaps powerful patterns and relationships can be found in numbers. For Instance, perfect numbers are those numbers that equal the sum of their divisors. Furthermore, amicable numbers such as 220 and 284 are the sum of each others' divisors. The study of words associated with assigned numbers is called gematry. This has lead to many books and movies; an example of this is the Davinci Code.
Rules have had to be devised to work with numbers. How to add, subtract, multiply, and divide. Orders of operation was established. Upon these basics, theorems, corollaries, and lemmas can be formed. Euclid's division lemma and the Fundamental Theorem of Arithmetic have been two very important discoveries to build on. The great difficulty in proving relatively simple results in number theory prompted no less an authority than Gauss to remark that "it is just this which gives the higher arithmetic that magical charm which has made it the favorite science of the greatest mathematicians, not to mention its inexhaustible wealth, wherein it so greatly surpasses other parts of mathematics." Gauss, often known as the "prince of mathematics," called mathematics the "queen of the sciences" and considered number theory the "queen of mathematics" .
Divisibility is a major subject of study for Number Theory. The Division Algorithm states that for any positive integer a and integer b, there exist unique integers q and r such that b equals the sum of r and the product qa and r is non negative and less than a with r equal to 0 if and only if a divides b. The Linear Diophantine equation is the sum of the products a and x with b and y equal to c. This equation can be used to find general solutions to x and y when a, b, and c are known or let you know that a solution does not exist.
The Fundamental Theorem of Arithmetic states that every integer can be written uniquely as the product of distinct primes each raised to distinct positive integers. Euclid stated in a theorem that there are an infinite number of primes. The proof of this is as follows: Suppose there are a finite number of primes. These primes can be ordered say p1p2...pn . Let N= p1p2...pn +1. By the Fundamental Theorem of Arithmetic , N is divisible by some prime p. This p must be among the listed p, since all the primes are listed, but N is not divisible by any of p, thus a contradiction.
Greatest common divisors and least common multiples are two more very fundamental ideas found in Number Theory. The Euclidean Algorithm uses these concepts . The greatest common divisor of two positive integers a and b can be found without factoring. This is accomplished through recursive use of the division algorithm. To accomplish this start with a=qb + r and recursive work until r=0. The remainder of the previous step is the greatest common divisor of a and b.
Modular arithmetic consists of a positive integer m and two more integers a and b. It is defined a is congruent to b modulo m if m divides the difference of a and b. An important rule is if a is congruent to b modulo m and c is congruent to d modulo m than the sum of a and c is congruent to the sum of b and d modulo m. Also, if a is congruent to b modulo m than the product of a and c is congruent to the product of b and c modulo m. Furthermore, if a is congruent to b modulo m and c is congruent to d modulo m than the product of a and c is congruent to the product of b and d modulo m.
A very well known and very useful theorem in Number Theory is Fermat's Little Theorem (FLT). FLT declares if p is prime and p does not divide a then a raised to the power p-1 is congruent to 1 modulo p. Stated in another way p divides the difference of a raised to the power of p-1 and 1. Wilson's Theorem states that the factorial of p-1 is congruent to -1 modulo p when p is a prime number. Modular arithmetic is great for discovering remainders and determining divisibility.
Permutations and combinations offer many important properties that are useful for Number Theory. For non negative integers n and k, k < or equal to n, the binomial coefficient written as ( n over k) is equal to the factorial of n divided by the product of the factorial of k and the factorial of the difference of n and k. This topic is seen as a primary topic in probability theory. Yet, there are uses in proof techniques for Fermat's and Wilson's Theorems as well as important uses in generating functions.
Number Theory has many sub fields and topics of study. Additional topics include arithmetic functions, primitive roots, and prime numbers. Further study can be done with quadratic residues, sums of squares, partition theory, and geometric number theory. This is just a list from the contents section of George Andrews book, Number Theory, 1994. Oystein Ore in his book, Number Theory and its History, discusses decimal fractions. Continued fractions is a similar topic with very interesting results. Anyone who enjoys algorithms and pattern recognitions would find many interesting topics in Number Theory. A lifetime of research could be done on a topic such as the golden ratio or Mersenne numbers.
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Andrews, G. E. Number Theory. New York: Dover, 1994.
Beiler, A. H. Recreations in the Theory of Numbers: The Queen of Mathematics Entertains, 2nd ed. New York: Dover, 1966 .
"Number Theory." From MathWorld --A Wolfram Web Resource http://mathworld.wolfram.com/NumberTheory
Ore, Ø. Number Theory and Its History. New York: Dover, 1988.