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Fraction

A fraction (from fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight-fifths, three-quarters. A common, vulgar, or simple fraction (examples: $$\tfrac{1}{2}$$ and $$\tfrac{17}{3}$$) consists of an integer numerator, displayed above a line (or before a slash like $1⁄4$), and a non-zero integer denominator, displayed below (or after) that line. If these integers are positive, then the numerator represents a number of equal parts, and the denominator indicates how many of those parts make up a unit or a whole. For example, in the fraction $1/2$, the numerator 3 indicates that the fraction represents 3 equal parts, and the denominator 4 indicates that 4 parts make up a whole. The picture to the right illustrates $3⁄4$ of a cake.

Other uses for fractions are to represent ratios and division. Thus the fraction $3⁄4$ can also be used to represent the ratio 3:4 (the ratio of the part to the whole), and the division 3 ÷ 4 (three divided by four).

We can also write negative fractions, which represent the opposite of a positive fraction. For example, if $3⁄4$ represents a half-dollar profit, then −$1⁄2$ represents a half-dollar loss. Because of the rules of division of signed numbers (which states in part that negative divided by positive is negative), −$1⁄2$, $1⁄2$ and $−1⁄2$ all represent the same fraction – negative one-half. And because a negative divided by a negative produces a positive, $1⁄−2$ represents positive one-half.

In mathematics a rational number is a number that can be represented by a fraction of the form $−1⁄−2$, where a and b are integers and b is not zero; the set of all rational numbers is commonly represented by the symbol Q or $a⁄b$, which stands for quotient. The term fraction and the notation $\mathbb{Q}$ can also be used for mathematical expressions that do not represent a rational number (for example $$\textstyle \frac \sqrt 2 2$$), and even do not represent any number (for example the rational fraction $$\textstyle \frac 1 x$$).

Vocabulary
In a fraction, the number of equal parts being described is the numerator (from numerātor, "counter" or "numberer"), and the type or variety of the parts is the denominator (from dēnōminātor, "thing that names or designates"). As an example, the fraction $a⁄b$ amounts to eight parts, each of which is of the type named "fifth". In terms of division, the numerator corresponds to the dividend, and the denominator corresponds to the divisor.

Informally, the numerator and denominator may be distinguished by placement alone, but in formal contexts they are usually separated by a fraction bar. The fraction bar may be horizontal (as in $8⁄5$), oblique (as in 2/5), or diagonal (as in $1⁄3$). These marks are respectively known as the horizontal bar; the virgule, slash (US), or stroke (UK); and the fraction bar, solidus, or fraction slash. In typography, fractions stacked vertically are also known as "en" or "nut fractions", and diagonal ones as "em" or "mutton fractions", based on whether a fraction with a single-digit numerator and denominator occupies the proportion of a narrow en square, or a wider em square. In traditional typefounding, a piece of type bearing a complete fraction (e.g. $4⁄9$) was known as a "case fraction", while those representing only part of fraction were called "piece fractions".

The denominators of English fractions are generally expressed as ordinal numbers, in the plural if the numerator is not 1. (For example, $1⁄2$ and $2⁄5$ are both read as a number of "fifths".) Exceptions include the denominator 2, which is always read "half" or "halves", the denominator 4, which may be alternatively expressed as "quarter"/"quarters" or as "fourth"/"fourths", and the denominator 100, which may be alternatively expressed as "hundredth"/"hundredths" or "percent".

When the denominator is 1, it may be expressed in terms of "wholes" but is more commonly ignored, with the numerator read out as a whole number. For example, $3⁄5$ may be described as "three wholes", or simply as "three". When the numerator is 1, it may be omitted (as in "a tenth" or "each quarter").

The entire fraction may be expressed as a single composition, in which case it is hyphenated, or as a number of fractions with a numerator of one, in which case they are not. (For example, "two-fifths" is the fraction $3⁄1$ and "two fifths" is the same fraction understood as 2 instances of $2⁄5$.) Fractions should always be hyphenated when used as adjectives. Alternatively, a fraction may be described by reading it out as the numerator "over" the denominator, with the denominator expressed as a cardinal number. (For example, $1⁄5$ may also be expressed as "three over one".) The term "over" is used even in the case of solidus fractions, where the numbers are placed left and right of a slash mark. (For example, 1/2 may be read "one-half", "one half", or "one over two".) Fractions with large denominators that are not powers of ten are often rendered in this fashion (e.g., $3⁄1$ as "one over one hundred seventeen"), while those with denominators divisible by ten are typically read in the normal ordinal fashion (e.g., $1⁄117$ as "six-millionths", "six millionths", or "six one-millionths").

Simple, common, or vulgar fractions
A simple fraction (also known as a common fraction or vulgar fraction, where vulgar is Latin for "common") is a rational number written as a/b or $6⁄1000000$, where a and b are both integers. As with other fractions, the denominator (b) cannot be zero. Examples include $\tfrac{a}{b}$, −$1⁄2$, $8⁄5$, and $−8⁄5$. The term was originally used to distinguish this type of fraction from the sexagesimal fraction used in astronomy.

Common fractions can be positive or negative, and they can be proper or improper (see below). Compound fractions, complex fractions, mixed numerals, and decimals (see below) are not common fractions; though, unless irrational, they can be evaluated to a common fraction.
 * A unit fraction is a common fraction with a numerator of 1 (e.g., $8⁄−5$). Unit fractions can also be expressed using negative exponents, as in 2−1, which represents 1/2, and 2−2, which represents 1/(22) or 1/4.
 * A dyadic fraction is a common fraction in which the denominator is a power of two, e.g. $1⁄7$ = $1⁄8$.

In Unicode, precomposed fraction characters are in the Number Forms block.

Proper and improper fractions
Common fractions can be classified as either proper or improper. When the numerator and the denominator are both positive, the fraction is called proper if the numerator is less than the denominator, and improper otherwise. The concept of an "improper fraction" is a late development, with the terminology deriving from the fact that "fraction" means "a piece", so a proper fraction must be less than 1. This was explained in the 17th century textbook The Ground of Arts.

In general, a common fraction is said to be a proper fraction, if the absolute value of the fraction is strictly less than one—that is, if the fraction is greater than −1 and less than 1. It is said to be an improper fraction, or sometimes top-heavy fraction, if the absolute value of the fraction is greater than or equal to 1. Examples of proper fractions are 2/3, −3/4, and 4/9, whereas examples of improper fractions are 9/4, −4/3, and 3/3.

Reciprocals and the "invisible denominator"
The reciprocal of a fraction is another fraction with the numerator and denominator exchanged. The reciprocal of $1⁄2^{3}$, for instance, is $3⁄7$. The product of a fraction and its reciprocal is 1, hence the reciprocal is the multiplicative inverse of a fraction. The reciprocal of a proper fraction is improper, and the reciprocal of an improper fraction not equal to 1 (that is, numerator and denominator are not equal) is a proper fraction.

When the numerator and denominator of a fraction are equal (for example, $7⁄3$), its value is 1, and the fraction therefore is improper. Its reciprocal is identical and hence also equal to 1 and improper.

Any integer can be written as a fraction with the number one as denominator. For example, 17 can be written as $7⁄7$, where 1 is sometimes referred to as the invisible denominator. Therefore, every fraction or integer, except for zero, has a reciprocal. For example. the reciprocal of 17 is $17⁄1$.

Ratios
A ratio is a relationship between two or more numbers that can be sometimes expressed as a fraction. Typically, a number of items are grouped and compared in a ratio, specifying numerically the relationship between each group. Ratios are expressed as "group 1 to group 2 ... to group n". For example, if a car lot had 12 vehicles, of which then the ratio of red to white to yellow cars is 6 to 2 to 4. The ratio of yellow cars to white cars is 4 to 2 and may be expressed as 4:2 or 2:1.
 * 2 are white,
 * 6 are red, and
 * 4 are yellow,

A ratio is often converted to a fraction when it is expressed as a ratio to the whole. In the above example, the ratio of yellow cars to all the cars on the lot is 4:12 or 1:3. We can convert these ratios to a fraction, and say that $1⁄17$ of the cars or $4⁄12$ of the cars in the lot are yellow. Therefore, if a person randomly chose one car on the lot, then there is a one in three chance or probability that it would be yellow.

Decimal fractions and percentages
A decimal fraction is a fraction whose denominator is not given explicitly, but is understood to be an integer power of ten. Decimal fractions are commonly expressed using decimal notation in which the implied denominator is determined by the number of digits to the right of a decimal separator, the appearance of which (e.g., a period, an interpunct (·), a comma) depends on the locale (for examples, see Decimal separator). Thus, for 0.75 the numerator is 75 and the implied denominator is 10 to the second power, namely, 100, because there are two digits to the right of the decimal separator. In decimal numbers greater than 1 (such as 3.75), the fractional part of the number is expressed by the digits to the right of the decimal (with a value of 0.75 in this case). 3.75 can be written either as an improper fraction, 375/100, or as a mixed number, $1⁄3$.

Decimal fractions can also be expressed using scientific notation with negative exponents, such as $3 75⁄100$, which represents 0.0000006023. The $6.023$ represents a denominator of $$. Dividing by $$ moves the decimal point 7 places to the left.

Decimal fractions with infinitely many digits to the right of the decimal separator represent an infinite series. For example, $$ = 0.333... represents the infinite series 3/10 + 3/100 + 3/1000 + ....

Another kind of fraction is the percentage (from per centum, meaning "per hundred", represented by the symbol %), in which the implied denominator is always 100. Thus, 51% means 51/100. Percentages greater than 100 or less than zero are treated in the same way, e.g. 311% equals 311/100, and −27% equals −27/100.

The related concept of permille or parts per thousand (ppt) has an implied denominator of 1000, while the more general parts-per notation, as in 75 parts per million (ppm), means that the proportion is 75/1,000,000.

Whether common fractions or decimal fractions are used is often a matter of taste and context. Common fractions are used most often when the denominator is relatively small. By mental calculation, it is easier to multiply 16 by 3/16 than to do the same calculation using the fraction's decimal equivalent (0.1875). And it is more accurate to multiply 15 by 1/3, for example, than it is to multiply 15 by any decimal approximation of one third. Monetary values are commonly expressed as decimal fractions with denominator 100, i.e., with two decimals, for example $3.75. However, as noted above, in pre-decimal British currency, shillings and pence were often given the form (but not the meaning) of a fraction, as, for example, "3/6" (read "three and six") meaning 3 shillings and 6 pence, and having no relationship to the fraction 3/6.

Mixed numbers
A mixed number (also called a mixed fraction or mixed numeral) is the sum of a non-zero integer and a proper fraction, conventionally written by juxtaposition (or concatenation) of the two parts, without the use of an intermediate plus (+) or minus (−) sign. When the fraction is written horizontally, a space is added between the integer and fraction to separate them.

As a basic example, two entire cakes and three quarters of another cake might be written as $$2\tfrac{3}{4}$$ cakes or $$2\ \,3/4$$ cakes, with the numeral $$2$$ representing the whole cakes and the fraction $$\tfrac34$$ representing the additional partial cake juxtaposed; this is more concise than the more explicit notation $$2+\tfrac{3}{4}$$ cakes. The mixed number $1⁄3$ is pronounced "two and three quarters", with the integer and fraction portions connected by the word and. Subtraction or negation is applied to the entire mixed numeral, so $$-2\tfrac{3}{4}$$ means $$-\bigl(2+\tfrac{3}{4}\bigr).$$

Any mixed number can be converted to an improper fraction by applying the rules of adding unlike quantities. For example, $$2 + \tfrac34 = \tfrac84 + \tfrac34 = \tfrac{11}4.$$ Conversely, an improper fraction can be converted to a mixed number using division with remainder, with the proper fraction consisting of the remainder divided by the divisor. For example, since 4 goes into 11 twice, with 3 left over, $$\tfrac{11}4 = 2 + \tfrac{3}{4}.$$

In primary school, teachers often insist that every fractional result should be expressed as a mixed number. Outside school, mixed numbers are commonly used for describing measurements, for instance $2 3⁄4$ hours or 5 3/16 inches, and remain widespread in daily life and in trades, especially in regions that do not use the decimalized metric system. However, scientific measurements typically use the metric system, which is based on decimal fractions, and starting from the secondary school level, mathematics pedagogy treats every fraction uniformly as a rational number, the quotient $2 1⁄4$ of integers, leaving behind the concepts of "improper fraction" and "mixed number". College students with years of mathematical training are sometimes confused when re-encountering mixed numbers because they are used to the convention that juxtaposition in algebraic expressions means multiplication.

Egyptian fraction
An Egyptian fraction is the sum of distinct positive unit fractions, for example $$\tfrac{1}{2}+\tfrac{1}{3}$$. This definition derives from the fact that the ancient Egyptians expressed all fractions except $$\tfrac{1}{2}$$, $$\tfrac{2}{3}$$ and $$\tfrac{3}{4}$$ in this manner. Every positive rational number can be expanded as an Egyptian fraction. For example, $$\tfrac{5}{7}$$ can be written as $$\tfrac{1}{2} + \tfrac{1}{6} + \tfrac{1}{21}.$$ Any positive rational number can be written as a sum of unit fractions in infinitely many ways. Two ways to write $$\tfrac{13}{17}$$ are $$\tfrac{1}{2}+\tfrac{1}{4}+\tfrac{1}{68}$$ and $$\tfrac{1}{3}+\tfrac{1}{4}+\tfrac{1}{6}+\tfrac{1}{68}$$.

Complex and compound fractions
In a complex fraction, either the numerator, or the denominator, or both, is a fraction or a mixed number, corresponding to division of fractions. For example, $$\tfrac{1/2}{1/3}$$ and $$\bigl(12\tfrac{3}{4}\bigr)\big/26$$ are complex fractions. To interpret nested fractions written "stacked" with a horizontal fraction bars, treat shorter bars as nested inside longer bars. Complex fractions can be simplified using multiplication by the reciprocal, as described below at. For example:
 * $$\begin{align}

\frac{\;\!\tfrac{1}{2}\;\!}{\tfrac{1}{3}} &= \frac{1}{2} \div \frac{1}{3} = \frac{1}{2} \times \frac{3}{1} = \frac{3}{2}, \qquad \frac{\;\!\tfrac{3}{2}\;\!}5 = \frac{3}{2} \div 5 = \frac{3}{2}\times\frac{1}{5} = \frac{3}{10}, \\[10mu] \frac{12\tfrac{3}{4}}{26} &= \frac{12 \times 4 + 3}{4} \div 26 = \frac{12 \times 4 + 3}{4} \times \frac{1}{26} = \frac{51}{104}. \end{align}$$

A complex fraction should never be written without an obvious marker showing which fraction is nested inside the other, as such expressions are ambiguous. For example, the expression $$5/10/20$$ could be plausibly interpreted as either $$\tfrac{5}{10} \big/ 20 = \tfrac{1}{40}$$ or as $$5 \big/ \tfrac{10}{20} = 10.$$ The meaning can be made explicit by writing the fractions using distinct separators or by adding explicit parentheses, in this instance $$(5/10)\big/20$$ or $$5\big/(10/20).$$

A compound fraction is a fraction of a fraction, or any number of fractions connected with the word of, corresponding to multiplication of fractions. To reduce a compound fraction to a simple fraction, just carry out the multiplication (see ). For example, $$\tfrac{3}{4}$$ of $$\tfrac{5}{7}$$ is a compound fraction, corresponding to $$\tfrac{3}{4} \times \tfrac{5}{7} = \tfrac{15}{28}$$. The terms compound fraction and complex fraction are closely related and sometimes one is used as a synonym for the other. (For example, the compound fraction $$\tfrac{3}{4} \times \tfrac{5}{7}$$ is equivalent to the complex fraction $2 1⁄2$.)

Nevertheless, "complex fraction" and "compound fraction" may both be considered outdated and now used in no well-defined manner, partly even taken synonymously for each other or for mixed numerals. They have lost their meaning as technical terms and the attributes "complex" and "compound" tend to be used in their every day meaning of "consisting of parts".

Arithmetic with fractions
Like whole numbers, fractions obey the commutative, associative, and distributive laws, and the rule against division by zero.

Mixed-number arithmetic can be performed either by converting each mixed number to an improper fraction, or by treating each as a sum of integer and fractional parts.

Equivalent fractions
Multiplying the numerator and denominator of a fraction by the same (non-zero) number results in a fraction that is equivalent to the original fraction. This is true because for any non-zero number $$n$$, the fraction $$\tfrac{n}{n}$$ equals 1. Therefore, multiplying by $$\tfrac{n}{n}$$ is the same as multiplying by one, and any number multiplied by one has the same value as the original number. By way of an example, start with the fraction $p⁄q$. When the numerator and denominator are both multiplied by 2, the result is $\tfrac{3/4}{7/5}$, which has the same value (0.5) as $\tfrac{1}{2}$. To picture this visually, imagine cutting a cake into four pieces; two of the pieces together ($2⁄4$) make up half the cake ($1⁄2$).

Simplifying (reducing) fractions
Dividing the numerator and denominator of a fraction by the same non-zero number yields an equivalent fraction: if the numerator and the denominator of a fraction are both divisible by a number (called a factor) greater than 1, then the fraction can be reduced to an equivalent fraction with a smaller numerator and a smaller denominator. For example, if both the numerator and the denominator of the fraction $$\tfrac{a}{b}$$ are divisible by $2⁄4$, then they can be written as $$a=cd$$, $$b=ce$$, and the fraction becomes $1⁄2$, which can be reduced by dividing both the numerator and denominator by c to give the reduced fraction $c$.

If one takes for $cd⁄ce$ the greatest common divisor of the numerator and the denominator, one gets the equivalent fraction whose numerator and denominator have the lowest absolute values. One says that the fraction has been reduced to its lowest terms.

If the numerator and the denominator do not share any factor greater than 1, the fraction is already reduced to its lowest terms, and it is said to be irreducible, reduced, or in simplest terms. For example, $$\tfrac{3}{9}$$ is not in lowest terms because both 3 and 9 can be exactly divided by 3. In contrast, $$\tfrac{3}{8}$$ is in lowest terms—the only positive integer that goes into both 3 and 8 evenly is 1.

Using these rules, we can show that $d⁄e$ = $c$ = $5⁄10$ = $1⁄2$, for example.

As another example, since the greatest common divisor of 63 and 462 is 21, the fraction $10⁄20$ can be reduced to lowest terms by dividing the numerator and denominator by 21:
 * $$\tfrac{63}{462} = \tfrac{63 \,\div\, 21}{462 \,\div\, 21}= \tfrac{3}{22}$$

The Euclidean algorithm gives a method for finding the greatest common divisor of any two integers.

Comparing fractions
Comparing fractions with the same positive denominator yields the same result as comparing the numerators:
 * $$\tfrac{3}{4}>\tfrac{2}{4}$$ because 3 > 2, and the equal denominators $$4$$ are positive.

If the equal denominators are negative, then the opposite result of comparing the numerators holds for the fractions:
 * $$\tfrac{3}{-4}<\tfrac{2}{-4} \text{ because } \tfrac{a}{-b}= \tfrac{-a}{b} \text{ and } -3 < -2. $$

If two positive fractions have the same numerator, then the fraction with the smaller denominator is the larger number. When a whole is divided into equal pieces, if fewer equal pieces are needed to make up the whole, then each piece must be larger. When two positive fractions have the same numerator, they represent the same number of parts, but in the fraction with the smaller denominator, the parts are larger.

One way to compare fractions with different numerators and denominators is to find a common denominator. To compare $$\tfrac{a}{b}$$ and $$\tfrac{c}{d}$$, these are converted to $$\tfrac{a\cdot d}{b\cdot d}$$ and $$\tfrac{b\cdot c}{b\cdot d}$$ (where the dot signifies multiplication and is an alternative symbol to ×). Then bd is a common denominator and the numerators ad and bc can be compared. It is not necessary to determine the value of the common denominator to compare fractions – one can just compare ad and bc, without evaluating bd, e.g., comparing $$\tfrac{2}{3}$$ ? $$\tfrac{1}{2}$$ gives $$\tfrac{4}{6}>\tfrac{3}{6}$$.

For the more laborious question $$\tfrac{5}{18}$$ ? $$\tfrac{4}{17},$$ multiply top and bottom of each fraction by the denominator of the other fraction, to get a common denominator, yielding $$\tfrac{5 \times 17}{18 \times 17}$$ ? $$\tfrac{18 \times 4}{18 \times 17}$$. It is not necessary to calculate $$18 \times 17$$ – only the numerators need to be compared. Since 5×17 (= 85) is greater than 4×18 (= 72), the result of comparing is $50⁄100$.

Because every negative number, including negative fractions, is less than zero, and every positive number, including positive fractions, is greater than zero, it follows that any negative fraction is less than any positive fraction. This allows, together with the above rules, to compare all possible fractions.

Addition
The first rule of addition is that only like quantities can be added; for example, various quantities of quarters. Unlike quantities, such as adding thirds to quarters, must first be converted to like quantities as described below: Imagine a pocket containing two quarters, and another pocket containing three quarters; in total, there are five quarters. Since four quarters is equivalent to one (dollar), this can be represented as follows:
 * $$\tfrac24+\tfrac34=\tfrac54=1\tfrac14$$.

Adding unlike quantities
To add fractions containing unlike quantities (e.g. quarters and thirds), it is necessary to convert all amounts to like quantities. It is easy to work out the chosen type of fraction to convert to; simply multiply together the two denominators (bottom number) of each fraction. In case of an integer number apply the invisible denominator 1.

For adding quarters to thirds, both types of fraction are converted to twelfths, thus:
 * $$\frac14\ + \frac13=\frac{1\times3}{4\times3}\ + \frac{1\times4}{3\times4}=\frac3{12}\ + \frac4{12}=\frac7{12}.$$

Consider adding the following two quantities:
 * $$\frac35+\frac23$$

First, convert $$\tfrac35$$ into fifteenths by multiplying both the numerator and denominator by three: $63⁄462$. Since $\tfrac{5}{18}>\tfrac{4}{17}$ equals 1, multiplication by $\tfrac35\times\tfrac33=\tfrac9{15}$ does not change the value of the fraction.

Second, convert $3⁄3$ into fifteenths by multiplying both the numerator and denominator by five: $3⁄3$.

Now it can be seen that:
 * $$\frac35+\frac23$$

is equivalent to:
 * $$\frac9{15}+\frac{10}{15}=\frac{19}{15}=1\frac4{15}$$

This method can be expressed algebraically:
 * $$\frac{a}{b} + \frac {c}{d} = \frac{ad+cb}{bd}$$

This algebraic method always works, thereby guaranteeing that the sum of simple fractions is always again a simple fraction. However, if the single denominators contain a common factor, a smaller denominator than the product of these can be used. For example, when adding $$\tfrac{3}{4}$$ and $$\tfrac{5}{6}$$ the single denominators have a common factor 2, and therefore, instead of the denominator 24 (4 × 6), the halved denominator 12 may be used, not only reducing the denominator in the result, but also the factors in the numerator.
 * $$\begin{align}

\frac34+\frac56 &= \frac{3\cdot 6}{4\cdot 6}+\frac{4 \cdot 5}{4\cdot 6}=\frac{18}{24} + \frac{20}{24}&=\frac{19}{12}\\ &=\frac{3\cdot 3}{4\cdot 3}+\frac{2\cdot 5}{2\cdot 6} =\frac{9}{12} + \frac{10}{12}&=\frac{19}{12} \end{align}$$

The smallest possible denominator is given by the least common multiple of the single denominators, which results from dividing the rote multiple by all common factors of the single denominators. This is called the least common denominator.

Subtraction
The process for subtracting fractions is, in essence, the same as that of adding them: find a common denominator, and change each fraction to an equivalent fraction with the chosen common denominator. The resulting fraction will have that denominator, and its numerator will be the result of subtracting the numerators of the original fractions. For instance,
 * $$\tfrac23-\tfrac12=\tfrac46-\tfrac36=\tfrac16$$

To subtract a mixed number, an extra one can be borrowed from the minuend, for instance
 * $$4 - 2\tfrac34 = (4-2-1) + \bigl(1 - \tfrac34\bigr) = 1\tfrac14.$$

Multiplying a fraction by another fraction
To multiply fractions, multiply the numerators and multiply the denominators. Thus:
 * $$\frac{2}{3} \times \frac{3}{4} = \frac{6}{12}$$

To explain the process, consider one third of one quarter. Using the example of a cake, if three small slices of equal size make up a quarter, and four quarters make up a whole, twelve of these small, equal slices make up a whole. Therefore, a third of a quarter is a twelfth. Now consider the numerators. The first fraction, two thirds, is twice as large as one third. Since one third of a quarter is one twelfth, two thirds of a quarter is two twelfth. The second fraction, three quarters, is three times as large as one quarter, so two thirds of three quarters is three times as large as two thirds of one quarter. Thus two thirds times three quarters is six twelfths.

A short cut for multiplying fractions is called "cancellation". Effectively the answer is reduced to lowest terms during multiplication. For example:
 * $$\frac{2}{3} \times \frac{3}{4} = \frac{\cancel{2} ^{~1}}{\cancel{3} ^{~1}} \times \frac{\cancel{3} ^{~1}}{\cancel{4} ^{~2}} = \frac{1}{1} \times \frac{1}{2} = \frac{1}{2}$$

A two is a common factor in both the numerator of the left fraction and the denominator of the right and is divided out of both. Three is a common factor of the left denominator and right numerator and is divided out of both.

Multiplying a fraction by a whole number
Since a whole number can be rewritten as itself divided by 1, normal fraction multiplication rules can still apply.
 * $$6 \times \tfrac{3}{4} = \tfrac{6}{1} \times \tfrac{3}{4} = \tfrac{18}{4}$$

This method works because the fraction 6/1 means six equal parts, each one of which is a whole.

Multiplying mixed numbers
The product of mixed numbers can be computed by converting each to an improper fraction. For example:
 * $$3 \times 2\frac{3}{4} = \frac31 \times \frac{2\times 4 + 3}{4} = \frac{33}{4} = 8\frac{1}{4}$$

Alternately, mixed numbers can be treated as sums, and multiplied as binomials. In this example,
 * $$3 \times 2\frac34 = 3 \times 2 + 3 \times \frac34 = 6 + \frac94 = 8\frac14.$$

Division
To divide a fraction by a whole number, you may either divide the numerator by the number, if it goes evenly into the numerator, or multiply the denominator by the number. For example, $$\tfrac{10}{3} \div 5$$ equals $$\tfrac{2}{3}$$ and also equals $$\tfrac{10}{3 \cdot 5} = \tfrac{10}{15}$$, which reduces to $$\tfrac{2}{3}$$. To divide a number by a fraction, multiply that number by the reciprocal of that fraction. Thus, $$\tfrac{1}{2} \div \tfrac{3}{4} = \tfrac{1}{2} \times \tfrac{4}{3} = \tfrac{1 \cdot 4}{2 \cdot 3} = \tfrac{2}{3}$$.

Converting between decimals and fractions
To change a common fraction to a decimal, do a long division of the decimal representations of the numerator by the denominator (this is idiomatically also phrased as "divide the denominator into the numerator"), and round the answer to the desired accuracy. For example, to change $2⁄3$ to a decimal, divide $\tfrac23\times\tfrac55=\tfrac{10}{15}$ by $1⁄4$ ("$1$ into $4$"), to obtain $4$. To change $1$ to a decimal, divide $0.25$ by $1⁄3$ ("$...1$ into $3$"), and stop when the desired accuracy is obtained, e.g., at $3$ decimals with $...1$. The fraction $4$ can be written exactly with two decimal digits, while the fraction $0.333$ cannot be written exactly as a decimal with a finite number of digits. To change a decimal to a fraction, write in the denominator a $1⁄4$ followed by as many zeroes as there are digits to the right of the decimal point, and write in the numerator all the digits of the original decimal, just omitting the decimal point. Thus $$12.3456 = \tfrac{123456}{10000}.$$

Converting repeating decimals to fractions
Decimal numbers, while arguably more useful to work with when performing calculations, sometimes lack the precision that common fractions have. Sometimes an infinite repeating decimal is required to reach the same precision. Thus, it is often useful to convert repeating decimals into fractions.

A conventional way to indicate a repeating decimal is to place a bar (known as a vinculum) over the digits that repeat, for example $1⁄3$ = 0.789789789... For repeating patterns that begin immediately after the decimal point, the result of the conversion is the fraction with the pattern as a numerator, and the same number of nines as a denominator. For example:
 * $1$ = 5/9
 * $\overline{0.|789}$ = 62/99
 * $\overline{0.|5}$ = 264/999
 * $\overline{0.|62}$ = 6291/9999

If leading zeros precede the pattern, the nines are suffixed by the same number of trailing zeros:
 * $\overline{0.|264}$ = 5/90
 * $\overline{0.|6291}$ = 392/999000
 * $\overline{0.0|5}$ = 12/9900

If a non-repeating set of decimals precede the pattern (such as $\overline{0.000|392}$), one may write the number as the sum of the non-repeating and repeating parts, respectively:
 * 0.1523 + $\overline{0.00|12}$

Then, convert both parts to fractions, and add them using the methods described above:
 * 1523 / 10000 + 987 / 9990000 = 1522464 / 9990000

Alternatively, algebra can be used, such as below:
 * 1) Let x = the repeating decimal:
 * x = $\overline{0.1523|987}$
 * 1) Multiply both sides by the power of 10 just great enough (in this case 104) to move the decimal point just before the repeating part of the decimal number:
 * 10,000x = $\overline{0.0000|987}$
 * 1) Multiply both sides by the power of 10 (in this case 103) that is the same as the number of places that repeat:
 * 10,000,000x = $\overline{0.1523|987}$
 * 1) Subtract the two equations from each other (if a = b and c = d, then a − c = b − d):
 * 10,000,000x − 10,000x = $\overline{1,523.|987}$ − $\overline{1,523,987.|987}$
 * 1) Continue the subtraction operation to clear the repeating decimal:
 * 9,990,000x = 1,523,987 − 1,523
 * 9,990,000x = 1,522,464
 * 1) Divide both sides by 9,990,000 to represent x as a fraction
 * x = $\overline{1,523,987.|987}$

Fractions in abstract mathematics
In addition to being of great practical importance, fractions are also studied by mathematicians, who check that the rules for fractions given above are consistent and reliable. Mathematicians define a fraction as an ordered pair $$(a,b)$$ of integers $$a$$ and $$b \ne 0,$$ for which the operations addition, subtraction, multiplication, and division are defined as follows:
 * $$(a,b) + (c,d) = (ad+bc,bd) \,$$
 * $$(a,b) - (c,d) = (ad-bc,bd) \,$$
 * $$(a,b) \cdot (c,d) = (ac,bd)$$
 * $$(a,b) \div (c,d) = (ad,bc) \quad(\text{with, additionally, } c \ne 0) $$

These definitions agree in every case with the definitions given above; only the notation is different. Alternatively, instead of defining subtraction and division as operations, the "inverse" fractions with respect to addition and multiplication might be defined as:
 * $$\begin{align}

-(a,b) &= (-a, b) & & \text{additive inverse fractions,} \\ &&&\text{with } (0,b) \text{ as additive unities, and}\\ (a,b)^{-1} &= (b,a) & & \text{multiplicative inverse fractions, for } a \ne 0, \\ &&&\text{with } (b,b) \text{ as multiplicative unities}. \end{align}$$

Furthermore, the relation, specified as
 * $$(a, b) \sim (c, d)\quad \iff \quad ad=bc,$$

is an equivalence relation of fractions. Each fraction from one equivalence class may be considered as a representative for the whole class, and each whole class may be considered as one abstract fraction. This equivalence is preserved by the above defined operations, i.e., the results of operating on fractions are independent of the selection of representatives from their equivalence class. Formally, for addition of fractions
 * $$(a,b) \sim (a',b')\quad$$ and $$\quad (c,d) \sim (c',d') \quad$$ imply
 * $$((a,b) + (c,d)) \sim ((a',b') + (c',d'))$$

and similarly for the other operations.

In the case of fractions of integers, the fractions $\overline{1,523.|987}$ with $1522464⁄9990000$ and $a⁄b$ coprime and $b > 0$ are often taken as uniquely determined representatives for their equivalent fractions, which are considered to be the same rational number. This way the fractions of integers make up the field of the rational numbers.

More generally, a and b may be elements of any integral domain R, in which case a fraction is an element of the field of fractions of R. For example, polynomials in one indeterminate, with coefficients from some integral domain D, are themselves an integral domain, call it P. So for a and b elements of P, the generated field of fractions is the field of rational fractions (also known as the field of rational functions).

Algebraic fractions
An algebraic fraction is the indicated quotient of two algebraic expressions. As with fractions of integers, the denominator of an algebraic fraction cannot be zero. Two examples of algebraic fractions are $$\frac{3x}{x^2+2x-3}$$ and $a$. Algebraic fractions are subject to the same field properties as arithmetic fractions.

If the numerator and the denominator are polynomials, as in $b$, the algebraic fraction is called a rational fraction (or rational expression). An irrational fraction is one that is not rational, as, for example, one that contains the variable under a fractional exponent or root, as in $\frac{\sqrt{x+2} }{x^2-3}$.

The terminology used to describe algebraic fractions is similar to that used for ordinary fractions. For example, an algebraic fraction is in lowest terms if the only factors common to the numerator and the denominator are 1 and −1. An algebraic fraction whose numerator or denominator, or both, contain a fraction, such as $\frac{3x}{x^2+2x-3}$, is called a complex fraction.

The field of rational numbers is the field of fractions of the integers, while the integers themselves are not a field but rather an integral domain. Similarly, the rational fractions with coefficients in a field form the field of fractions of polynomials with coefficient in that field. Considering the rational fractions with real coefficients, radical expressions representing numbers, such as $\frac{\sqrt{x+2} }{x^2-3}$, are also rational fractions, as are a transcendental numbers such as $\pi/2,$ since all of $$\sqrt{2},\pi,$$ and $$2$$ are real numbers, and thus considered as coefficients. These same numbers, however, are not rational fractions with integer coefficients.

The term partial fraction is used when decomposing rational fractions into sums of simpler fractions. For example, the rational fraction $$\frac{2x}{x^2-1}$$ can be decomposed as the sum of two fractions: $\frac{1 + \tfrac{1}{x} }{1 - \tfrac{1}{x} }$. This is useful for the computation of antiderivatives of rational functions (see partial fraction decomposition for more).

Radical expressions
A fraction may also contain radicals in the numerator or the denominator. If the denominator contains radicals, it can be helpful to rationalize it (compare Simplified form of a radical expression), especially if further operations, such as adding or comparing that fraction to another, are to be carried out. It is also more convenient if division is to be done manually. When the denominator is a monomial square root, it can be rationalized by multiplying both the top and the bottom of the fraction by the denominator:
 * $$\frac{3}{\sqrt{7}} = \frac{3}{\sqrt{7}} \cdot \frac{\sqrt{7}}{\sqrt{7}} = \frac{3\sqrt{7}}{7}$$

The process of rationalization of binomial denominators involves multiplying the top and the bottom of a fraction by the conjugate of the denominator so that the denominator becomes a rational number. For example:
 * $$\frac{3}{3-2\sqrt{5}} = \frac{3}{3-2\sqrt{5}} \cdot \frac{3+2\sqrt{5}}{3+2\sqrt{5}} = \frac{3(3+2\sqrt{5})}{{3}^2 - (2\sqrt{5})^2} = \frac{ 3 (3 + 2\sqrt{5} ) }{ 9 - 20 } = - \frac{ 9+6 \sqrt{5} }{11}$$
 * $$\frac{3}{3+2\sqrt{5}} = \frac{3}{3+2\sqrt{5}} \cdot \frac{3-2\sqrt{5}}{3-2\sqrt{5}} = \frac{3(3-2\sqrt{5})}{{3}^2 - (2\sqrt{5})^2} = \frac{ 3 (3 - 2\sqrt{5} ) }{ 9 - 20 } = - \frac{ 9-6 \sqrt{5} }{11}$$

Even if this process results in the numerator being irrational, like in the examples above, the process may still facilitate subsequent manipulations by reducing the number of irrationals one has to work with in the denominator.

Typographical variations
In computer displays and typography, simple fractions are sometimes printed as a single character, e.g. ½ (one half). See the article on Number Forms for information on doing this in Unicode.

Scientific publishing distinguishes four ways to set fractions, together with guidelines on use:
 * Special fractions: fractions that are presented as a single character with a slanted bar, with roughly the same height and width as other characters in the text. Generally used for simple fractions, such as: ½, ⅓, ⅔, ¼, and ¾. Since the numerals are smaller, legibility can be an issue, especially for small-sized fonts. These are not used in modern mathematical notation, but in other contexts.
 * Case fractions: similar to special fractions, these are rendered as a single typographical character, but with a horizontal bar, thus making them upright. An example would be $\textstyle \sqrt{2}/2$, but rendered with the same height as other characters. Some sources include all rendering of fractions as case fractions if they take only one typographical space, regardless of the direction of the bar.
 * Shilling or solidus fractions: 1/2, so called because this notation was used for pre-decimal British currency (£sd), as in "2/6" for a half crown, meaning two shillings and six pence. While the notation "two shillings and six pence" did not represent a fraction, the forward slash is now used in fractions, especially for fractions inline with prose (rather than displayed), to avoid uneven lines. It is also used for fractions within fractions (complex fractions) or within exponents to increase legibility. Fractions written this way, also known as piece fractions, are written all on one typographical line, but take 3 or more typographical spaces.
 * Built-up fractions: $$\frac{1}{2}$$. This notation uses two or more lines of ordinary text and results in a variation in spacing between lines when included within other text. While large and legible, these can be disruptive, particularly for simple fractions or within complex fractions.

History
The earliest fractions were reciprocals of integers: ancient symbols representing one part of two, one part of three, one part of four, and so on. The Egyptians used Egyptian fractions c. 1000 BC. About 4000 years ago, Egyptians divided with fractions using slightly different methods. They used least common multiples with unit fractions. Their methods gave the same answer as modern methods. The Egyptians also had a different notation for dyadic fractions, used for certain systems of weights and measures.

The Greeks used unit fractions and (later) continued fractions. Followers of the Greek philosopher Pythagoras (c. 530 BC) discovered that the square root of two cannot be expressed as a fraction of integers. (This is commonly though probably erroneously ascribed to Hippasus of Metapontum, who is said to have been executed for revealing this fact.) In 150 BC Jain mathematicians in India wrote the "Sthananga Sutra", which contains work on the theory of numbers, arithmetical operations, and operations with fractions.

A modern expression of fractions known as bhinnarasi seems to have originated in India in the work of Aryabhatta (c. AD 500), Brahmagupta (c. 628), and Bhaskara (c. 1150). Their works form fractions by placing the numerators (amsa) over the denominators (cheda), but without a bar between them. In Sanskrit literature, fractions were always expressed as an addition to or subtraction from an integer. The integer was written on one line and the fraction in its two parts on the next line. If the fraction was marked by a small circle or cross, it is subtracted from the integer; if no such sign appears, it is understood to be added. For example, Bhaskara I writes:
 * ६  १  २
 * १  १  १०
 * ४  ५  ९

which is the equivalent of
 * 6  1  2
 * 1  1  −1
 * 4  5  9

and would be written in modern notation as 6$\frac{1}{x+1} + \frac{1}{x-1}$, 1$1⁄2$, and 2 − $1⁄4$ (i.e., 1$1⁄5$).

The horizontal fraction bar is first attested in the work of Al-Hassār, a Muslim mathematician from Fez, Morocco, who specialized in Islamic inheritance jurisprudence. In his discussion he writes: "for example, if you are told to write three-fifths and a third of a fifth, write thus, $\frac{3 \quad 1}{5 \quad 3}$". The same fractional notation—with the fraction given before the integer —appears soon after in the work of Leonardo Fibonacci in the 13th century.

In discussing the origins of decimal fractions, Dirk Jan Struik states: The introduction of decimal fractions as a common computational practice can be dated back to the Flemish pamphlet De Thiende, published at Leyden in 1585, together with a French translation, La Disme, by the Flemish mathematician Simon Stevin (1548–1620), then settled in the Northern Netherlands. It is true that decimal fractions were used by the Chinese many centuries before Stevin and that the Persian astronomer Al-Kāshī used both decimal and sexagesimal fractions with great ease in his Key to arithmetic (Samarkand, early fifteenth century).

While the Persian mathematician Jamshīd al-Kāshī claimed to have discovered decimal fractions himself in the 15th century, J. Lennart Berggren notes that he was mistaken, as decimal fractions were first used five centuries before him by the Baghdadi mathematician Abu'l-Hasan al-Uqlidisi as early as the 10th century.

Pedagogical tools
In primary schools, fractions have been demonstrated through Cuisenaire rods, Fraction Bars, fraction strips, fraction circles, paper (for folding or cutting), pattern blocks, pie-shaped pieces, plastic rectangles, grid paper, dot paper, geoboards, counters and computer software.

Documents for teachers
Several states in the United States have adopted learning trajectories from the Common Core State Standards Initiative's guidelines for mathematics education. Aside from sequencing the learning of fractions and operations with fractions, the document provides the following definition of a fraction: "A number expressible in the form $1⁄9$ where $$a$$ is a whole number and $$b$$ is a positive whole number. (The word fraction in these standards always refers to a non-negative number.)" The document itself also refers to negative fractions.

title: Fraction


text:
Alapin's Opening

Alapin's Opening is an unusual chess opening that starts with the moves:
 * 1. e4 e5
 * 2. Ne2

It is named after the Russo-Lithuanian player and openings analyst Semyon Alapin (1856–1923). Although this opening is rarely used, Ljubojević (as Black) played against it at Groningen in 1970.

Description
Alapin's Opening is offbeat, but perfectly playable for White. It is mainly used to avoid highly theoretical lines such as the Ruy Lopez, or to surprise the opponent. White intends to play f2–f4 soon. There is similarity to the Smyslov Position (Smyslov–Botvinnik, 1958) if White tries to play something in the lines of g3, Nbc3, d3, Bg2.

However, Alapin's Opening also incurs several problems for White. First, the development of White's light-square bishop, and also of his queen, is blocked, and will require another move of the knight or another pawn move, both of which go against the opening principle to develop the minor pieces quickly. Second, the knight on e2, although flexible, has no control over Black's half of the centre, and will need to be moved again to become more useful.

It is relatively easy for Black to equalise in this opening; for example, 2...Nf6, 2...Nc6, and 2...d5 all equalize, although Black should be careful to avoid being caught by surprise by an eventual f2–f4.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Alapin's Opening


text:
Talk:Pleather (album)

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BetacommandBot 04:54, 20 July 2007 (UTC)

title: Talk:Pleather (album)


text:
User:Caius

I'm about to start studying Italian and English linguistics very soon at the ELTE university, Hungary. Apparently, I also have a problem with user boxes.

Images made by me so far:

title: User:Caius


text:
Michael Anderson (director)

Michael Joseph Anderson (30 January 1920 – 25 April 2018) was an English film and television director. His career spanned nearly 50 years across three countries, working at various times in the United Kingdom, the United States, and Canada. His most critically and commercially successful works include the World War II film The Dam Busters (1955), the dystopian sci-fi film Logan's Run (1976), and the comedy adventure epic Around the World in 80 Days (1956), which won the 1957 Academy Award for Best Picture.

Anderson received nominations for the Best Director Oscar and the Golden Globe for Best Director for Around the World in 80 Days. He was twice nominated for the Hugo Award for Best Dramatic Presentation (for Logan's Run and for the Ray Bradbury miniseries The Martian Chronicles), and for a Directors Guild of America Award and the Palme d'Or. In 2012, he received the Directors Guild of Canada's Lifetime Achievement Award.

Anderson was the father of actor Michael Anderson Jr., and the stepfather of actress Laurie Holden.

Early life and education
Anderson was born in London, United Kingdom, to a theatrical family. His parents were the actors Lawrence (1893–1939) and Beatrice Anderson (1893–1977). His great-aunt was Mary Anderson of Louisville, Kentucky, who became one of the first US Shakespearean actresses; the Mary Anderson Theatre in Louisville was dedicated to her.

He began working in the industry as an actor during the 1930s. By 1938, he had graduated to working behind the camera as an assistant director. During World War II, while serving in the British Army's Royal Signals Corps, he met Peter Ustinov and subsequently assisted him on two films.

Actor and assistant director
Anderson appeared in two films as an actor: as Oily Boyd in Housemaster (1938); and as Marine Albert Fosdick in Noël Coward's In Which We Serve (1942). He joined Elstree Studios as a production runner in 1936 and became an assistant director by 1938.

His credits as assistant director include Spy for a Day (1940), Freedom Radio (1940), Quiet Wedding (1941), Cottage to Let (1941) and Jeannie (1941). He was unit manager as well as actor on In Which We Serve (1942) and was assistant director on Unpublished Story (1942).

Anderson served with the Royal Corps of Signals during the Second World War, during which time he met Peter Ustinov. On demobilisation, Anderson returned to the film industry working as an assistant director on Ustinov's films School for Secrets (1946) and Vice Versa (1947). He was also an assistant director on Fame is the Spur (1947), One Night with You (1947) and Mr. Perrin and Mr. Traill (1948).

Director
Anderson and Ustinov then wrote and directed a feature together, Private Angelo (1949). "He's an old friend of mine and we've always got each other out of difficulties," said Ustinov.

Anderson made his solo directorial debut with a B film, Waterfront (1950) with Robert Newton and Richard Burton. The Telegraph critic announced, "I can only burn my boats and prophesy that young Michael Anderson is possibly the most promising discovery since Carol Reed and David Lean."

Anderson followed his first at bat with some more B movies: Hell Is Sold Out (1951) for producer Raymond Stross; Night Was Our Friend (1952) and Dial 17 (1952). In early 1952 he co directed a stage revue in London, The Irving Revue.

Associated British Picture Corporation
In April 1952 Anderson signed a contract with Associated British Picture Corporation (ABPC) for whom he wound up making five films. The first was a comedy, Will Any Gentleman...? (1953), based on a stage success. It was followed by The House of the Arrow (1953).

The third was the war film The Dam Busters (1955), starring Richard Todd. Todd recalled when first told Anderson was to direct "I thought this was typical ABPC cheese-paring, instead of getting an expensive well-known director. Michael had made only small films before that, having previously been the best first assistant director in the business. But I had dinner with him one night and was totally won over by him. He knew what he was doing and was a delight to work with." The Dam Busters was the most popular movie at the British box office in 1955.

Todd worked twice more with Anderson, calling him "a supremely authoritative, quiet, collected director who knew exactly what he wanted and what he could get out of his actors. He only had to give me a little quiet guidance, and we worked together very well."

Anderson followed this with the first cinema adaptation of George Orwell's 1984 (1956), co-financed by American interests. It was a commercial failure, despite shooting a "happier" ending for the United States release.

Around the World in 80 Days
Anderson was then called in to direct Around the World in 80 Days (1956), after original director John Farrow had a falling out with producer Mike Todd. Todd reportedly hired him on the strength of The Dam Busters and the recommendation of Noël Coward. The film was a huge hit and Anderson was nominated for an Academy Award (the film won Best Picture) and a Golden Globe for his direction. Todd signed Anderson to a two-picture contract but Todd died in a plane crash in 1958.

Anderson was reunited with Richard Todd for another war film Yangtse Incident: The Story of H.M.S. Amethyst (1957) for producer Herbert Wilcox but it was not as popular as The Dam Busters.

He made a third film with Richard Todd, a thriller, Chase a Crooked Shadow (1958); this was his fifth and last movie for Associated British.

In Ireland Anderson made a thriller about the IRA with James Cagney, Shake Hands with the Devil (1959). It was made for Pennebaker, the company of Marlon Brando and provided an early role to Richard Harris.

Hollywood
Anderson then took over a project at MGM originally planned as an Alfred Hitchcock project, The Wreck of the Mary Deare (1959), with Gary Cooper and Charlton Heston. Anderson later recalled in 1986, "The magic I remember most is walking on to stage 30 in Culver City at Metro-Goldwyn-Mayer. It was the biggest stage in the world and I remember looking at it and thinking I'll be here in a couple of weeks and they'll have built a ship and I'll be directing Gary Cooper and Charlton Heston – it's all going to be mine. It gave me such a feeling of astonishment and it's never quite left me."

MGM also financed Anderson's next film, the melodrama All the Fine Young Cannibals (1960) with Natalie Wood and Robert Wagner. Anderson was reunited with Cooper in The Naked Edge (1961) which turned out to be Cooper's last film.

Harold Pinter wrote The Servant for Anderson but the director was unable to secure finance so he sold the script to Joseph Losey, who filmed it with great success in 1963..

Anderson made some films for Harold Hecht: Flight from Ashiya (1964), an adventure tale, and Wild and Wonderful (1964), a comedy with Tony Curtis. For MGM and Carlo Ponti he directed the war time thriller Operation Crossbow (1965).

Anderson made a spy thriller The Quiller Memorandum (1966), starring George Segal and Alec Guinness. He was meant to direct Eye of the Devil but fell ill. For MGM he directed the film The Shoes of the Fisherman (1968), stepping in for Anthony Asquith at the last moment; the film was a flop.

He was meant to direct the big screen adaptation of James Clavell's Tai Pan starring Patrick McGoohan but the film was not made due to high costs.

1970s
Anderson went for a few years without making a film before returning with Pope Joan (1972) and The Devil's Impostor (1972). For George Pal he made Doc Savage: The Man of Bronze (1975) then did Conduct Unbecoming (1975).

Logan's Run (1976), about a futuristic society where humanity is imprisoned in a death trap sealed dome controlled by a computer, was an expensive box-office success, earning $50 million worldwide and boosting sales for its distributor, Metro Goldwyn Mayer. Anderon's old friend Peter Ustinov played a small role. Anderson then directed Orca (1977) and Dominique (1978).

Later work
In 1981, Anderson moved to Canada, where his then-wife was from, and became a Canadian citizen. "It's the best move I ever made", he said in 1986. "There's so much talent, it's exciting, clean, young, fresh and it's been very good to me."

His later work was mostly made-for-television miniseries, including The Martian Chronicles (1980), Sword of Gideon (1986), Young Catherine (1991), The Sea Wolf (1993), Rugged Gold (1994), Captain's Courageous (1996) and 20,000 Leagues Under the Sea (1997).

His feature work in Canada included, Murder by Phone (1982), the New Zealand film, Second Time Lucky (1984), Separate Vacations (1986), Summer of the Monkeys (1998), Millennium (1989) and The Grand Defiance (1993), as well as episodes of the television anthology series Scales of Justice. He directed Bottega dell'orefice (The Jeweler's Shop, 1988), based on the 1960 play written by Karol Wojtyła, who, by the time the film was made, had become Pope John Paul II. In 1998, he said "I honestly feel like a teenager", and had no intention of retiring. Despite this statement, his last film credit before his retirement would end up being The New Adventures of Pinocchio (1999).

In 2012, Michael Anderson received the Lifetime Achievement Award from the Directors Guild of Canada. At the time of his death, Anderson was the oldest living nominee for an Academy Award for Best Director, and the only surviving director whose film won a Best Picture award in the 1950s.

Personal life
Anderson was married three times:
 * 1) Betty Jordan (1923–2008) married in 1939; five children
 * 2) Vera Carlisle (born 1935) married in 1969; one child
 * 3) Actress Adrienne Ellis (born 1944); two stepchildren; stepfather of actress Laurie Holden (The X-Files, Silent Hill, The Mist, The Walking Dead) and Christopher Holden.

His son Michael Anderson Jr., is an actor who appeared in Logan's Run and The Martian Chronicles; another son, David Anderson, is a film producer.

Anderson died on 25 April 2018 at the age of 98, from heart disease.

Filmography

 * Private Angelo (1949)
 * Waterfront Women (1950)
 * Hell Is Sold Out (1951)
 * Night Was Our Friend (1951)
 * Will Any Gentleman...? (1953)
 * The House of the Arrow (1953)
 * The Dam Busters (1955)
 * 1984 (1956)
 * Around the World in 80 Days (1956)
 * Yangtse Incident: The Story of H.M.S. Amethyst (1957)
 * Chase a Crooked Shadow (1958)
 * Shake Hands with the Devil (1959)
 * The Wreck of the Mary Deare (1959)
 * All the Fine Young Cannibals (1960)
 * The Naked Edge (1961)
 * Flight from Ashiya (1964)
 * Wild and Wonderful (1964)
 * Operation Crossbow (1965)
 * The Quiller Memorandum (1966)
 * The Shoes of the Fisherman (1968)
 * Pope Joan (1972)
 * Doc Savage: The Man of Bronze (1975)
 * Conduct Unbecoming (1975)
 * Logan's Run (1976)
 * Orca (1977)
 * Dominique (1979)
 * The Martian Chronicles (1980) (TV)
 * Murder by Phone (1982)
 * Second Time Lucky (1984)
 * Separate Vacations (1986)
 * Sword of Gideon (1986) (TV)
 * The Jeweller's Shop (1989)
 * Millennium (1989)
 * Young Catherine (1991) (TV)
 * The Sea Wolf (1993) (TV)
 * Rugged Gold (1994) (TV)
 * Captains Courageous (1996) (TV)
 * 20,000 Leagues Under the Sea (1997) (TV)
 * Summer of the Monkeys (1998)
 * The New Adventures of Pinocchio (1999)

title: Michael Anderson (director)


text:
Daniel Ford

Daniel Ford (born 1931 in Arlington, Massachusetts) is an American journalist, novelist, and historian. The son of Patrick and Anne Ford, he attended public schools in New Hampshire and Massachusetts, graduating in 1950 from Brewster Academy in Wolfeboro, New Hampshire. He was educated at the University of New Hampshire (A.B. Political Science 1954), the University of Manchester (Fulbright Scholar, Modern European History 1954–55), and King's College London (M.A. War Studies 2010).

Ford served in the U.S. Army at Fort Bragg and in Orléans, France. Following an apprenticeship at the Overseas Weekly in Frankfurt, Germany, he became a free-lance writer in Durham, New Hampshire. He received a Stern Fund Magazine Writers' Award (1964) for his dispatches from South Vietnam, published in The Nation; a Verville Fellowship (1989–90) at the National Air and Space Museum to work with Japanese accounts of the air war in Southeast Asia; and an Aviation - Space Writers' Association Award of Excellence (1992) for his history of the Flying Tigers. He is best known for his Flying Tigers research, and for the 1967 Vietnam novel Incident at Muc Wa that became the Burt Lancaster film Go Tell the Spartans.

Ford is a resident scholar at the University of New Hampshire. He writes for The Wall Street Journal, Michigan War Studies Review, and Air&Space/Smithsonian magazine; maintains the Warbird's Forum, Piper Cub Forum, and Reading Proust websites; and blogs on Daniel Ford's Blog. He soloed in a J-3 Piper Cub at the age of 68 and flew as a sport pilot until he turned 80.

Non-fiction

 * Looking Back From Ninety: The Depression, the War, and the Good Life That Followed (2021) ISBN 978-1732230026
 * Cowboy: The Interpreter Who Became a Soldier, a Warlord, and One More Casualty of Our War in Vietnam (2018) ISBN 978-1732230002
 * Editor: The Greater America: An Epic Journey Through a Vibrant New Country (1907, revised 2017) ISBN 978-1544938301
 * Flying Tigers: Claire Chennault and His American Volunteers, 1941-1942 (1991, 2nd edition 2007, 3rd edition 2016; translated into Chinese) ISBN 978-0-69-273473-5
 * Poland's Daughter: How I Met Basia, Hitchhiked to Italy, and Learned About Love, War, and Exile (2013) ISBN 978-1494729899
 * A Vision So Noble: John Boyd, the OODA Loop, and America's War on Terror (2010) ISBN 978-1-4515-8981-8
 * Editor: The Lady and the Tigers: Remembering the Flying Tigers of World War II, by Olga Greenlaw (1943, revised 2002) ISBN 978-0-595-22234-6
 * The Only War We've Got: Early Days in South Vietnam (2001) ISBN 978-1479194728
 * Glen Edwards: The Diary of a Bomber Pilot (1998) ISBN 978-1-56098-571-6
 * The Country Northward (1976, 2010) ISBN 978-1-4528-3092-6

Novels

 * Michael's War: A Story of the Irish Republican Army (2003, 2015) ISBN 978-1508852001
 * Remains: A Story of the Flying Tigers (2000, 2013) ISBN 978-1492779506
 * The High Country Illuminator: A Tale of Light and Darkness and the Ski Bums of Avalon (1971, 2013) ISBN 978-1481854764
 * Incident at Muc Wa: A Story of the Vietnam War (1967; translated into Dutch; filmed as Go Tell the Spartans, 1976; 2012) ISBN 978-1478178187
 * Now Comes Theodora (1965, 2000) ISBN 978-0595089178

title: Daniel Ford


text:
All That You Love Will Be Carried Away

"All That You Love Will Be Carried Away" is a short story by Stephen King. It was originally published in the January 29, 2001 issue of The New Yorker magazine. In 2002, it was included in King's collection Everything's Eventual.

Plot summary
Alfie Zimmer, a traveling salesman peddling gourmet frozen foods, pulls into a Motel 6 in Nebraska for the night. He settles in and pulls out a revolver, ready to commit suicide because he can't "go on living the way he had been living." Alfie has a hobby of recording strange bathroom graffiti which he has discovered on his many long, lonely travels. He starts noting down scrawls on the walls that attracted his attention, gradually becoming fascinated with them. During his solitary travels, he has come to regard these "voices on the walls" as his friends as well as something to think about during the long drive, describing the messages as something precious and important that often "spoke" to him.

Alfie decides that "a shot in the mouth is easier than any living change" but every time he puts the gun in his mouth, he worries that leaving the notebook filled with bizarre ramblings behind will make him seem insane to whoever finds his body. Alfie wants to write a book about the graffiti, even coming up with a great title, but knows "the telling would hurt". While standing in the freezing cold of the winter night, sobbing to himself, Alfie decides on a plan: if the lights of a farmhouse behind the motel reappear through the snow before he counts to 60, he will write the book. If not, he will toss the notebook into the snow, then go inside and shoot himself.

The story closes with Alfie standing near the field outside the motel, starting to count, thus leaving the ending ambiguous.

Film, TV or theatrical adaptations
This story was made into several different Dollar Baby short films. The films were made by Scott Albanese, Brian Berkowitz, Mark Montalto, Chi Laughlin & Natalie Mooallem, James Renner, Anthony Kaneaster, Seth Friedman, Robert Sterling, Hendrik Harms, Chloe Brown, João Augusto De Nardo and Bolen Miller.

title: All That You Love Will Be Carried Away


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Dave Smith (pitcher, born 1955)

David Stanley Smith (January 21, 1955 – December 17, 2008) was an American Major League Baseball relief pitcher, primarily for the Houston Astros, for whom he pitched from to. He also pitched for the Chicago Cubs.

Early life
Smith was born in Richmond, California, and attended San Diego State University. He was drafted in the eighth round (169th overall) of the 1976 Major League Baseball Draft. He was signed by scout Bob Cluck.

Baseball career
Smith made his Major League debut on April 11, 1980, against the Los Angeles Dodgers. Sent in for the top of the eighth inning, Smith allowed two walks and a hit but did not allow a run. The Astros rallied in the bottom of the inning for five runs to victory, as Smith earned his first win. Smith appeared in 57 games that year while pitching 102.2 innings (a career high), collecting ten saves while having a 7–5 record with a 1.93 ERA. He struck out 85 batters while walking 32 of them, and he received votes in the Rookie of the Year balloting, finishing fifth. That year, the Astros reached the postseason for the first time in franchise history. Smith appeared in three games of the 1980 National League Championship Series. In Game 2, he pitched to seven batters in the seventh and eighth innings, allowing one run to score while the Astros won 7–4. In Game 3, he was tasked to pitch the eleventh inning of a scoreless tie. He allowed one hit and a walk before the Astros rallied in the bottom of the inning for a walk-off victory. This set the stage for a pivotal Game 4, in which an Astros win would have advanced them to the World Series. In the top of the eighth inning, he was sent to replace Vern Ruhle, who had seen his shutout end on a Pete Rose single. Facing Mike Schmidt on a full count, Smith allowed a single that scored Lonnie Smith to tie the game. Smith was taken out of Joe Sambito, but the Phillies tacked on another run to take the lead. While the Astros rallied to force it into extra innings, the Phillies won the game in the tenth inning (with three hits off Sambito) and won the series the next day.

The following year, Smith pitched in 42 games with 75 innings pitched while collecting eight saves with 52 strikeouts and 23 walks and a 2.76 ERA. The Astros returned to the postseason that year and met the Los Angeles Dodgers in the 1981 National League Division Series. Smith pitched the ninth and tenth inning of Game 1, a game that ended with an 11th inning victory for Houston. Smith pitched to two batters of the decisive Game 5 and allowed one run on a hit in the 4–0 loss. Smith started one game in his career, doing so on June 25, 1982, against the Dodgers. He lasted two innings while allowing four hits and three runs.

Smith served as the team's primary closer after the season; he had saved forty games in his first five seasons, but he saved 159 games over the next six seasons for the Astros. In 1986, Smith pitched 56 innings with 54 appearances in games, which resulted in 33 saves with a 2.73 ERA and a 4–7 record while striking out 46 batters with 22 walks. He was named to the All-Star Game that year. Smith and the Astros returned to the postseason that season. He pitched Game 3 and Game 6 of the 1986 National League Championship Series against the New York Mets. In Game 3, he relieved Charlie Kerfeld in the ninth inning of a 5–4 lead for Houston. He gave up a leadoff bunt single by Wally Backman, who the Astros felt stepped way out of the basepaths. One out later, Lenny Dykstra, hit a deep shot to right field on a forkball that went for a home run to win the game for the Mets. Game 6 would later be referred to as one of the most memorable postseason games in history. Smith was sent to pitch the ninth inning to take over for Bob Knepper, who had allowed two runs to score on three hits. With a 3–2 lead and a runner on base with one out, Smith walked two batters to load the bases. Ray Knight then tied the game with a sacrifice fly deep to right field. Smith pitched the tenth inning and allowed no hits, but the Astros went on to lose in the sixteenth inning.

Smith pitched his final year with the Astros in 1990. He made the All-Star team that season, pitching with a 2.39 ERA in 49 games and 60.1 innings while collecting 23 saves with fifty strikeouts and twenty walks to go with a 6–6 record. Smith left the Astros following the 1990 season, signing with the Chicago Cubs ten days after being granted free agency. He pitched in 35 games for 33 innings pitched for a 6.00 ERA, striking out sixteen batters with nineteen walks while going 0–6 with seventeen saves. The following year was his last in the majors, as he pitched in eleven games before being released in June. He had a 2.51 ERA in 14.1 innings and eleven games pitched. In his last game on June 8 against the St. Louis Cardinals, he went 1.2 innings and allowed two runs on four hits. His career win-loss record was 53-53. He holds the Astros record for games pitched (586) and home runs per nine innings (0.331). His 2.53 ERA is third-best in Astros history. His 199 saves was a team record until Billy Wagner passed him. Smith had 216 total saves (with 52 blown opportunities).

Personal life
Smith was married twice and had three children in total. After his playing career ended, he worked under the minor league system for the San Diego Padres from 1994 to 1998 before being promoted to the major league staff in 1999, where he coached until 2001; in that year, he undertook an alcohol rehabilitation program and missed a month of the season before resigning in June. He was one of the directors of The San Diego School of Baseball for nearly 30 years. Smith died of a heart attack on December 17, 2008, in San Diego, California.

title: Dave Smith (pitcher, born 1955)


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Bourboulon

Bourboulon is a surname. Notable people with the surname include:

 * Jacques Bourboulon (born 1946), French photographer
 * Martin Bourboulon (born 1979), French film director

title: Bourboulon


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Tom Butler (bishop)

Thomas Frederick Butler (born 5 March 1940) is a British retired Anglican bishop. He was the ninth Anglican Bishop of Southwark. He was enthroned in Southwark Cathedral on 12 September 1998. He retired from this position on 5 March 2010. In 2014, Butler was involved in the transition process for the new Diocese of Leeds as "mentor bishop"; he remains an honorary assistant bishop of that diocese.

Early life and education
Born in Birmingham, Butler attended King Edward VI Five Ways school in Birmingham and the University of Leeds, where he obtained a first class honours BSc, an MSc and PhD in electronics. He trained for ordination with the Mirfield Fathers at the College of the Resurrection in Yorkshire.

Ordained ministry
After ordination in 1964, he served three years as a curate in the Diocese of Ely and Diocese of Canterbury before spending 12 years as a lecturer in electronics and a chaplain at the University of Zambia and then at the University of Kent in Canterbury. During this period he was on the staff of Lusaka Cathedral and Canterbury Cathedral respectively. From 1980 to 1985, Butler was the Archdeacon of Northolt in the Diocese of London.

Butler was consecrated as a bishop on 30 November 1985, by Robert Runcie, Archbishop of Canterbury, at St Paul's Cathedral, to become area Bishop of Willesden until he was appointed diocesan Bishop of Leicester in 1991 (his election was confirmed on 1 July) and translated to Southwark in 1998.

Butler has been active at national and international level. Until 1995 he chaired the follow-up to "Faith in the City", which published the controversial "Staying in the City" report. He chaired the General Synod's Board of Mission from 1995 until 2001 and is now vice-chair Public Affairs of the Mission and Public Affairs Council. He is also Chair of the Governors of Ripon College, Cuddesdon. He served as the General Synod representative on the Inner Cities Religious Council, an initiative set up by the Department of the Environment, until 2001. Since mid-2003 the Bishop has represented the Church of England on the central committee of the World Council of Churches. He entered the House of Lords in 1997.

Butler's beliefs were cited as the reason for some "valid but irregular" ordinations in his diocese. Andy Fenton, Richard Perkins and Loots Lambrechts were ordained in November 2005 at Christ Church, Surbiton, London, by Martin Morrison, a bishop of the Church of England in South Africa. Morrison was brought in by Richard Coekin, minister of Dundonald Church in Wimbledon, due to a dispute with Butler over matters related to homosexuality. Coekin subsequently had his licence to officiate revoked by Butler, but he was reinstated following an appeal to the Archbishop of Canterbury.

In December 2006, Butler returned home from a function at the Irish embassy in London with a head injury, which he claimed to be unable to remember sustaining. He contacted the police claiming that he had been mugged. It was subsequently suggested in the media that Butler, apparently under the influence of alcohol, had sustained the injury while being removed from a stranger's car into which he had apparently climbed. The incident has become legendary due to the detail which suggested he was throwing Christmas gifts from the car whilst saying 'I am the Bishop of Southwark, it's what I do'. In an interview with John Humphrys on BBC Radio 4's Today Programme, Butler reiterated his statement that he had not been drinking heavily but said that he was worried that he still could not account for three hours of the evening in question and was undergoing medical tests. He gave the Thought for the Day on the same date. Known in the church as a strong disciplinarian, particularly when dealing with junior clergy caught under the influence of alcohol, Butler faced calls for his resignation from some quarters.

Butler is a regular contributor to Radio 4's Thought for the Day and has taken part in many other national and local TV and radio programmes. He has also co-authored two books with his wife Barbara: Just Mission and Just Spirituality in a World of Faiths. On 7 September 2009 he announced that he would retire on his 70th birthday, 5 March 2010.

Marriage and family
Barbara Butler, his wife, is the executive secretary of Christians Aware, a charity that is involved in education and development. They have two grown-up children and four grandchildren.

Retirement
He retired to Wakefield, where he has been licensed as an honorary assistant bishop ever since – first of the Diocese of Wakefield and then of the Diocese of Leeds. During the "transformation period" when the dioceses of Ripon & Leeds, of Wakefield and of Bradford were dissolved and the new Leeds diocese founded, Butler served as "mentor bishop" for the new diocese (having previously been diocesan bishop of a large diocese with an area scheme), as acting diocesan Bishop of Bradford (while its final bishop, Nick Baines — whom Butler first worked with as Bishop of Croydon in 2003 – was on sabbatical, February – April 2014), then (20 – 22 April 2014) as acting diocesan bishop of Leeds, and as interim area bishop of Bradford (April – December 2014).

Styles

 * Tom Butler Esq (1940–?)
 * Dr Tom Butler (?–1964)
 * The Reverend Dr Tom Butler (1964–1980)
 * The Venerable Dr Tom Butler (1980–1985)
 * The Right Reverend Dr Tom Butler (1985—present)

title: Tom Butler (bishop)