Pure Mathematics – Quantity

1. Pure Mathematics – Quantity • Art. 1. Mathematics is the science of quantity. Any thing which can be multiplied, divided, or measured, is called quantity. Thus, a line is a quantity, because it can be doubled, trebled, or halved; and can be measured, by applying to it another line, as a foot, a yard, or an ell. Weight is a quantity, which can be measured, in pounds, ounces, and grains. Time is a species of quantity, whose measure can be expressed, in hours, minutes, and seconds. But color is not a quantity. It cannot be said, with propriety, that one color is twice as great, or half as great, as another. The operations of the mind, such as thought, choice, desire, hatred, &c. are not quantities. They are incapable of mensuration.

2. Study Of Quantity Quantity The study of quantity starts with numbers, first the familiar natural numbers and integers ("whole numbers") and arithmetical operations on them, which are characterized in arithmetic. The deeper properties of integers are studied in number theory, from which come such popular results as Fermat's Last Theorem. The twin prime conjecture andGoldbach's conjecture are two unsolved problems in number theory. As the number system is further developed, the integers are recognized as a subset of the rational numbers ("fractions"). These, in turn, are contained within the real numbers, which are used to representcontinuousquantities. Real numbers are generalized to complex numbers. These are the first steps of a hierarchy of numbers that goes on to includequarternions and octonions. Consideration of the natural numbers also leads to the transfinite numbers, which formalize the concept of "infinity". Another area of study is size, which leads to the cardinal numbers and then to another conception of infinity: the aleph numbers, which allow meaningful comparison of the size of infinitely large sets.

3. Natural Numbers • The term "natural number" refers either to a member of the set of positive integers 1, 2, 3, ... (Sloane's A000027) or to the set ofnonnegative integers 0, 1, 2, 3, ... (Sloane's A001477; e.g., Bourbaki 1968, Halmos 1974). Regrettably, there seems to be no general agreement about whether to include 0 in the set of natural numbers. In fact, Ribenboim (1996) states "Let  P be a set of natural numbers; whenever convenient, it may be assumed that  0 belongs to P." • The set of natural numbers (whichever definition is adopted) is denoted N. • Due to lack of standard terminology, the following terms and notations are recommended in preference to "counting number," "natural number," and "whole number. • set name symbol ..., , , 0, 1, 2, ... integers Z 1, 2, 3, 4, ... positive integers Z-+ 0, 1, 2, 3, 4, ... nonnegative integers Z-* 0, , , , , ... nonpositive integers , , , , ... negative integers Z--”

4. Integers • One of the numbers ..., , , 0, 1, 2, .... The set of integers forms a ring that is denoted . A given integer  may be negative (), nonnegative (), zero (), or positive (). The set of integers is, not surprisingly, called Integers in Mathematica, and a number  can be tested to see if it is a member of the integers using the command Element[x, Integers]. The commandIntegerQ[x] returns True if  has the Mathematica data type of an integer. • Numbers that are integers are sometimes described as "integral" (instead of integer-valued), but this practice may lead to unnecessary confusions with the integrals of integral calculus. • The ring  of integers has cardinality of aleph0. The generating function for the nonnegative integersis f(x) = x/(1-x)2 =x + 2x2 + 3x3 + 4x4 • There are several symbols used to perform operations having to do with conversion between real numbers and integers. The symbol  ("floor ") means "the largest integer not greater than ," i.e., int(x) in computer parlance. The symbol  means "the nearest integer to " (nearest integer function), i.e., nint(x) in computer parlance. The symbol  ("ceiling ") means "the smallest integer not smaller than ," or -int(-x), where int(x) is the integer part of .

5. Rational Numbers • Rational Numbers • A rational number is a number that can be written as a simple fraction (i.e. as a ratio). i.e. A rational number is a number that can be in the form p/qwhere p and q are integers and q is not equal to zero.

6. Real Numbers • Real Numbers include: • Whole Numbers (like 1,2,3,4, etc) • Rational Numbers (like 3/4, 0.125, 0.333..., 1.1, etc ) • Irrational Numbers (like π, √3, etc ) • Real Numbers can also be positive, negative or zero. So ... what is NOT a Real Number? • √-1 (the square root of minus 1) is not a Real Number, it is anImaginary Number • Infinity is not a Real Number • Why are they called "Real" Numbers? Because they are not Imaginary Numbers. • The Real Numbers did not have a name before Imaginary Numbers were thought of. They got called "Real" because they were not Imaginary. That is the actual answer! •  Real numbers include natural numbers, whole numbers, integers, rational numbers and irrational numbers. Real numbers also include fraction and decimnal numbers.

7. Complex Numbers • A Complex Number is a combination of: • a Real Number • and an Imaginary Number • Imaginary Numbers are special because: whensquared, they give a negative result. • Normally this doesn't happen, because: • when you square a positive number you get a positive result, and • when you square a negative number you also get a positive result (because a negative times a negative gives a positive) • The "unit" imaginary number (like 1 for Real Numbers) is i, which is the square root of -1 • A Complex Number is a combination of a Real Number and an Imaginary Number e.g. a+bi • Complex does not mean complicated. It means the two types of numbers, real and imaginary, together form a complex, just like you might have a building complex (buildings joined together).

8. Number theory • Elementary Number Theory • Elementary number theory is the branch of number theory in which elementary methods (i.e., arithmetic, geometry, and high school algebra) are used to solve equations with integer or rational solutions. An example of a problem which can be solved using elementary number theory is the classification of all Pythagorean triples. • Primes and prime factorization are especially important in number theory, as are a number of functions such as the divisor function, Riemann zeta function, and totient function. Excellent introductions to number theory may be found in Ore (1988) and Beiler (1966). The classic history on the subject (now slightly dated) is that of Dickson (2005abc).

9. Prime Number • A prime number (or prime integer, often simply called a "prime" for short) is a positive integer  that has no positive integer divisorsother than 1 and  itself. (More concisely, a prime number  is a positive integer having exactly one positive divisor other than 1.) For example, the only divisors of 13 are 1 and 13, making 13 a prime number, while the number 24 has divisors 1, 2, 3, 4, 6, 8, 12, and 24 (corresponding to the factorization ), making 24 not a prime number. Positive integers other than 1 which are not prime are called composite numbers. • Prime numbers are therefore numbers that cannot be factored or, more precisely, are numbers  whose divisors are trivial and given by exactly 1 and . • While the term "prime number" commonly refers to prime positive integers, other types of primes are also defined, such as theGaussian primes. • The number 1 is a special case which is considered neither prime nor composite • With 1 excluded, the smallest prime is therefore 2. However, since 2 is the only even prime (which, ironically, in some sense makes it the "oddest" prime), it is also somewhat special, and the set of all primes excluding 2 is therefore called the "odd primes.“ • Euler commented "Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers, and we have reason to believe that it is a mystery into which the mind will never penetrate"  • Prime numbers can be generated by sieving processes (such as the sieve of Eratosthenes), and lucky numbers, which are also generated by sieving, appear to share some interesting asymptotic properties with the primes. 

10. Factoring Primes • Many prime factorization algorithms have been devised for determining the prime factors of a given integer, a process known as factorization or prime factorization. They vary quite a bit in sophistication and complexity.  •  It should be emphasized that although no efficient algorithms are known for factoring arbitrary integers, it has not been proved that no such algorithm exists. It is therefore conceivable that a suitably clever person could devise a general method of factoring which would render the vast majority of encryption schemes in current widespread use, including those used by banks and governments, easily breakable. • Thefundamental theorem of arithmetic states that any positive integer can be represented in exactly one way as a product of primes. • The simplest method of finding factors is so-called "direct search factorization" (a.k.a. trial division). In this method, all possible factors are systematically tested using trial division to see if they actually divide the given number. • More general (and complicated) methods include the elliptic curve factorization method and number field sieve factorization method. • It has been proven that the set of prime numbers is a Diophantine set (Ribenboim 1991, pp. 106-107). http://www.mathsisfun.com/prime-factorization.html

11. divisor function • The  τ function , also called the divisor function, takes a positive integer as its input and gives the number of positive divisors of its input as its output. For example, since 1 , 2 , and 4 are all of the positive divisors of 4 , we have τ(4)=3 . As another example, since 1 , 2 , 5 , and 10 are all of the positive divisors of 10 , we have τ(10)=4 . • The  function behaves according to the following two rules: • 1. If p is a prime and k is a nonnegative integer, then τ(pk)=k+1 . • 2. If gcd(ab)=1 , then τ(ab)=τ(a)τ(b) . • Because these two rules hold for the  function, it is a multiplicative function. • Note that these rules work for the previous two examples. Since 2 is prime, we have τ(4)=(22)=2+1=3 . Since 2 and 5 are distinct primes, we have τ(10)=τ(2*5)=τ(2)τ(5)=(1+1)(1+1)=4 . • If n is a positive integer, the number of prime factors of xn−1 over [x] is (n) . For example,x9−1=(x3−1)(x6+x3+1)=(x−1)(x2+x+1)(x6+x3+1) and τ(9)=3 . • The  τ function is extremely useful for studying cyclic rings. • The sequence τ(n) appears in the OEIS as sequence A000005.

12. Riemann zeta function • Riemann zeta function, function useful in number theory for investigating properties of prime numbers. Written as ζ(x), it was originally defined as the infinite seriesζ(x) = 1 + 2−x + 3−x + 4−x + ⋯.When x = 1, this series is called the harmonic series, which increases without bound—i.e., its sum is infinite. For values of x larger than 1, the series converges to a finite number as successive terms are added. If x is less than 1, the sum is again infinite. The zeta function was known to the Swiss mathematician Leonhard Euler in 1737, but it was first studied extensively by the German mathematician Bernhard Riemann. • In 1859 Riemann published a paper giving an explicit formula for the number of primes up to any preassigned limit—a decided improvement over the approximate value given by the prime number theorem. However, Riemann’s formula depended on knowing the values at which a generalized version of the zeta function equals zero. (The Riemann zeta function is defined for all complex numbers—numbers of the form x + iy, where i =  √(−1) —except for the line x = 1.) Riemann knew that the function equals zero for all negative even integers −2, −4, −6, … (so-called trivial zeros), and that it has an infinite number of zeros in the critical strip of complex numbers between the linesx = 0 and x = 1, and he also knew that all nontrivial zeros are symmetric with respect to the critical line x = 1/2. Riemann conjectured that all of the nontrivial zeros are on the critical line, a conjecture that subsequently became known as the Riemann hypothesis. • In 1900 the German mathematician David Hilbert called the Riemann hypothesis one of the most important questions in all of mathematics, as indicated by its inclusion in his influential list of 23 unsolved problems with which he challenged 20th-century mathematicians. In 1915 the English mathematician Godfrey Hardy proved that an infinite number of zeros occur on the critical line, and by 1986 the first 1,500,000,001 nontrivial zeros were all shown to be on the critical line. Although the hypothesis may yet turn out to be false, investigations of this difficult problem have enriched the understanding of complex numbers.

13. totient function. • Euler's totient function: phi() • (Leonhard Euler was born on 15th April 1707 in Basel, Switzerland and he died on 18th September 1783 in St Petersburg, Russia).If N is a positive integer, then phi(N) is defined to be the number of positive integers less than or equal to N and coprime (do not contain any factor in common with) to N. • Example (1):Prime factor of 8 is only 2 => 8 = 2^3phi(8) = phi(2^3) = phi(2 * 2^2) = (2-1) * 2^2 = 1 * 4 = 4.The numbers of integers less than 8 that coprime with 8 are 4, they are 1, 3, 5 and 7. • Example (2):Prime factor of 10 are 2 and 5 => 10 = (2^1) * (5^1)phi(10) = phi(2) * phi(5) = (2-1) * (5-1) = 1 * 4 = 4.The numbers of integers less than 10 that coprime with 10 are 4, they are 1, 3, 7 and 9. • To calculate Euler totient function, we need to find prime factors, this is the hardest part, for big number it is very hard to find all of its prime factors, this is the security in RSA which is hard to factor a large number. • NOTE: • 1If N is prime then phi(N) = N-1, all integers less than N (1 to N-1) are coprime with N. • 2phi(N) is always even, both for N either composite or prime. • 3Euler totient function is used to find the total of primitive root of N. • 4Euler totient functions also used to find the primitive root of N, only under multiplication (see primitive root).