The use of percentages is a way of expressing ratios in terms of whole numbers. A ratio or fraction is converted to a percentage by multiplying by 100 and appending a "percentage sign" %. For example, if an investment grows from to , then is times as much as , i.e., 173.08% of . So it is also true that the investment has grown by . A change of a certain percent is sometimes said to be a change of percentage points.
A unit fraction is a fraction with numerator 1. Examples of unit fractions include 1/2, 1/3, 1/12, and 1/123456. The famous Rhind papyrus, dated to around 1650 BC, discusses unit fractions and contains a table of representations of as a sum of distinct unit fractions for odd between 5 and 101. As a result, sums of unit fractions are now known as Egyptian fractions.
A fraction containing each of the digits 1 through 9 is called a pandigital fraction. The following table gives the number of pandigital fractions which represent simple unit fractions. The numbers of pandigital fractions for 1/1, 1/2, 1/3, ... are 0, 12, 2, 4, 12, 3, 7, 46, 3, ... (OEIS A054383).#fractions12,2412,37,46,,,,,,,3004
A mixed fraction is an improper fraction written in the form . In common usage such as cooking recipes, is often written as (e.g., 1 ), much to the chagrin of mathematicians, to whom means , which is quite a different beast from .(The author of this work discovered this fact early in his mathematical career after having points marked off a calculus exam for using the recipe-like notation. Future mathematicians are therefore encouraged to avoid mixed fractions, except perhaps in the kitchen.)
A number is called an Egyptian number if it is the sum of the denominators in some unit fraction representation of a positive whole number not consisting entirely of 1s. For example,so is an Egyptian number. The numbers that are not Egyptian are 2, 3, 5, 6, 7, 8, 12, 13, 14, 15, 19, 21, and 23 (OEIS A028229; Konhauser et al. 1996, p. 147).If is the sum of denominators of a unit fraction representation composed of distinct denominators which are not all 1s, then it is called a strictly Egyptian number. For example, by virtue of is Egyptian, but it is not strictly Egyptian. Graham (1963) proved that every number is strictly Egyptian. Numbers that are strictly Egyptian are 11, 24, 30, 31, 32, 37, 38, 43, ... (OEIS A052428), and those which are not are 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, ... (OEIS A051882)...
An Egyptian fraction is a sum of positive (usually) distinct unit fractions. The famous Rhind papyrus, dated to around 1650 BC contains a table of representations of as Egyptian fractions for odd between 5 and 101. The reason the Egyptians chose this method for representing fractions is not clear, although André Weil characterized the decision as "a wrong turn" (Hoffman 1998, pp. 153-154). The unique fraction that the Egyptians did not represent using unit fractions was 2/3 (Wells 1986, p. 29).Egyptian fractions are almost always required to exclude repeated terms, since representations such as are trivial. Any rational number has representations as an Egyptian fraction with arbitrarily many terms and with arbitrarily large denominators, although for a given fixed number of terms, there are only finitely many. Fibonacci proved that any fraction can be represented as a sum of distinct unit fractions (Hoffman..
If the period of a repeating decimal for , where is prime and is a reduced fraction, has an even number of digits, then dividing the repeating portion into halves and adding gives a string of 9s. For example, , and .
If a fixed fraction of a given amount of money is lost, and then the same fraction of the remaining amount is gained, the result is less than the original and equal to the final amount if a fraction is first gained, then lost. This can easily be seen from the fact that(1)(2)
The least common denominator of a collection of fractionsis the least common multiple of their denominators.
An irreducible fraction is a fraction for which , i.e., and are relatively prime. For example, in the complex plane, is reducible, while is not.The figure above shows the irreducible fractions plotted in the complex plane (Pickover 1997; Trott 2004, p. 29).
A common fraction is a fraction in which numerator and denominator are both integers, as opposed to fractions. For example, is a common fraction, whileis not. Common fractions are sometimes also called vulgar fractions (Derbyshire 2004, p. 171).
A reducible fraction is a fraction such that , i.e., can be written in reduced form. A fraction that is not reducible is said to be irreducible.For example, in the complex plane, is reducible, while is not.
The ratio of two numbers and is written , where is the numerator and is the denominator. The ratio of to is equivalent to the quotient . Betting odds written as correspond to . A number which can be expressed as a ratio of integers is called a rational number.
Fractran is an algorithm applied to a given list , , ..., of fractions. Given a starting integer , the FRACTRAN algorithm proceeds by repeatedly multiplying the integer at a given stage by the first element that yields an integer product. The algorithm terminates when there is no such .The listwith starting integer generates a sequence 2, 15, 825, 725, 1925, 2275, 425, 390, 330, 290, 770, ... (OEIS A007542). Conway (1987) showed that this sequence has an amazing connection with prime numbers, and in fact is a generator for the primes. In particular, the only powers of two (other than 2 itself) that occur in this sequence are those with prime exponent: , , , , ....
If is (directly) proportional to , then is a constant. The relationship is written , which impliesfor some constant known as the constant of proportionality.
The Farey sequence for any positive integer is the set of irreducible rational numbers with and arranged in increasing order. The first few are(1)(2)(3)(4)(5)(OEIS A006842 and A006843). Except for , each has an odd number of terms and the middle term is always 1/2.Let , , and be three successive terms in a Farey series. Then(6)(7)These two statements are actually equivalent (Hardy and Wright 1979, p. 24). For a method of computing a successive sequence from an existing one of terms, insert the mediant fraction between terms and when (Hardy and Wright 1979, pp. 25-26; Conway and Guy 1996; Apostol 1997). Given with , let be the mediant of and . Then , and these fractions satisfy the unimodular relations(8)(9)(Apostol 1997, p. 99).The number of terms in the Farey sequence for the integer is(10)(11)where is the totient function and is the summatory function of , giving 2, 3, 5, 7, 11, 13, 19, ... (OEIS A005728). The asymptotic limit..
Anomalous cancellation is a "canceling" of digits of and in the numerator and denominator of a fraction which results in a fraction equal to the original. Note that if there are multiple but differing counts of one or more digits in the numerator and denominator there is ambiguity about which digits to cancel, so it is simplest to exclude such cases from consideration.There are exactly four anomalous cancelling proper fractions having two-digit base-10 numerator and denominator:(1)(2)(3)(4)(c.f. Boas 1979). The first few 3-digit anomalous cancelling numbers are(5)(6)and the first few with four digits are(7)(8)The numbers of anomalously cancelling proper fractions having digits in both numerator and denominator for , 2, ... are 0, 4, 161, 1851, ....The numbers of anomalously cancelling proper fractions having or fewer digits in both numerator and denominator for , 2, ... are 0, 4, 190, 2844, ....The concept of anomalous cancellation..
The use of permil (a.k.a. parts per thousand) is a way of expressing ratios in terms of whole numbers. Given a ratio or fraction, it is converted to a permil-age by multiplying by 1000 and appending a "mil sign" . For example, if an investment grows from a number to a number , then is times as much as , or 1730.8.
Pick any two relatively prime integers and , then the circle of radius centered at is known as a Ford circle. No matter what and how many s and s are picked, none of the Ford circles intersect (and all are tangent to the x-axis). This can be seen by examining the squared distance between the centers of the circles with and ,(1)Let be the sum of the radii(2)then(3)But , so and the distance between circle centers is the sum of the circle radii, with equality (and therefore tangency) iff . Ford circles are related to the Farey sequence (Conway and Guy 1996).If , , and are three consecutive terms in a Farey sequence, then the circles and are tangent at(4)and the circles and intersect in(5)Moreover, lies on the circumference of the semicircle with diameter and lies on the circumference of the semicircle with diameter (Apostol 1997, p. 101)...
A rational number expressed in the form (in-line notation) or (traditional "display" notation), where is called the numerator and is called the denominator. When written in-line, the slash "/" between numerator and denominator is called a solidus.A mathematical joke states that 4/3 of people don't understand fractions.A proper fraction is a fraction such that , and a reduced fraction is a fraction with common terms canceled out of the numerator and denominator.The Egyptians expressed their fractions as sums (and differences) of unit fractions. Conway and Guy (1996) give a table of Roman notation for fractions, in which multiples of 1/12 (the uncia) were given separate names.The rules for the algebraic combination of fractions are given by(1)(2)(3)(4)Note however that the above results will not necessarily be reducedfractions...
The function defined by Minkowski for the purpose of mapping the quadratic surds in the open interval into the rational numbers of in a continuous, order-preserving manner. takes a number having continued fraction to the number(1)The function satisfies the following properties (Salem 1943). 1. is strictly increasing. 2. If is rational, then is of the form , with and integers. 3. If is a quadratic surd, then the continued fraction is periodic, and hence is rational. 4. The function is purely singular (Denjoy 1938). can also be constructed as(2)where and are two consecutive irreducible fractions from the Farey sequence. At the th stage of this definition, is defined for values of , and the ordinates corresponding to these values are for , 1, ..., (Salem 1943).The function satisfies the identity(3)A few special values include(4)(5)(6)(7)(8)(9)(10)(11)where is the golden ratio...
Given a Pythagorean triple , the fractions and are called Pythagorean fractions. Diophantus showed that the Pythagorean fractions consist precisely of fractions of the form .
The extension ring obtained from a commutative unit ring (other than the trivial ring) when allowing division by all non-zero divisors. The ring of fractions of an integral domain is always a field.The term "ring of fractions" is sometimes used to denote any localization of a ring. The ring of fractions in the above meaning is then referred to as the total ring of fractions, and coincides with the localization with respect to the set of all non-zero divisors.When defining addition and multiplication of fractions, all that is required of the denominators is that they be multiplicatively closed, i.e., if , then ,(1)(2)Given a multiplicatively closed set in a ring , the ring of fractions is all elements of the form with and . Of course, it is required that and that fractions of the form and be considered equivalent. With the above definitions of addition and multiplication, this set forms a ring.The original ring may not embed in this ring of..
Archimedes' axiom, also known as the continuity axiom or Archimedes' lemma, survives in the writings of Eudoxus (Boyer and Merzbach 1991), but the term was first coined by the Austrian mathematician Otto Stolz (1883). It states that, given two magnitudes having a ratio, one can find a multiple of either which will exceed the other. This principle was the basis for the method of exhaustion, which Archimedes invented to solve problems of area and volume.Symbolically, the axiom states thatiff the appropriate one of following conditions is satisfied for integers and : 1. If , then . 2. If , then . 3. If , then . Formally, Archimedes' axiom states that if and are two line segments, then there exist a finite number of points , , ..., on such thatand is between and (Itô 1986, p. 611). A geometry in which Archimedes' lemma does not hold is called a non-Archimedean Geometry...