67-568: The Leelavati Award is an award for outstanding contribution to public outreach in mathematics. It is named after the 12th-century mathematical treatise " Lilavati " devoted to arithmetic, algebra, and the decimal system written by the Indian mathematician Bhāskara II , also known as Bhaskara Achārya. In the book the author posed, in verse form, a series of problems in (elementary) arithmetic to one Leelavati (perhaps his daughter) and followed them up with hints to solutions. This work appears to have been
134-515: 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, 1 / 2 1 / 3 {\displaystyle {\tfrac {1/2}{1/3}}} and ( 12 3 4 ) / 26 {\displaystyle {\bigl (}12{\tfrac {3}{4}}{\bigr )}{\big /}26} are complex fractions. To interpret nested fractions written "stacked" with
201-404: 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
268-470: A basic example, two entire cakes and three quarters of another cake might be written as 2 3 4 {\displaystyle 2{\tfrac {3}{4}}} cakes or 2 3 / 4 {\displaystyle 2\ \,3/4} cakes, with the numeral 2 {\displaystyle 2} representing the whole cakes and the fraction 3 4 {\displaystyle {\tfrac {3}{4}}} representing
335-473: A book in her name, one that would remain till the end of time as a good name is akin to a second life. Many of the problems are addressed to Līlāvatī herself, who must have been a very bright young woman. For example "Oh Līlāvatī, intelligent girl, if you understand addition and subtraction, tell me the sum of the amounts 2, 5, 32, 193, 18, 10, and 100, as well as [the remainder of] those when subtracted from 10000." and "Fawn-eyed child Līlāvatī, tell me, how much
402-430: A cake into four pieces; two of the pieces together ( 2 / 4 ) make up half the cake ( 1 / 2 ). 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
469-480: A common fraction. In Unicode, precomposed fraction characters are in the Number Forms block. 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
536-511: A common man could understand. Excerpt from Lilavati (Appears as an additional problem attached to stanza 54, Chapter 3. Translated by T N Colebrook) Whilst making love a necklace broke. A row of pearls mislaid. One sixth fell to the floor. One fifth upon the bed. The young woman saved one third of them. One tenth were caught by her lover. If six pearls remained upon the string How many pearls were there altogether? Bhaskaracharya's conclusion to Lilavati states: Joy and happiness
603-488: A cup with a small hole at the bottom of a vessel filled with water, arranged so that the cup would sink at the beginning of the propitious hour. He put the device in a room with a warning to Lilavati to not go near it. In her curiosity, though, she went to look at the device. A pearl from her bridal dress accidentally dropped into it, thus upsetting it. The auspicious moment for the wedding thus passed unnoticed leaving Bhaskara II devastated. Thus, he promised his daughter to write
670-841: 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 § Division . For example: 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 {\displaystyle 5/10/20} could be plausibly interpreted as either 5 10 / 20 = 1 40 {\displaystyle {\tfrac {5}{10}}{\big /}20={\tfrac {1}{40}}} or as 5 / 10 20 = 10. {\displaystyle 5{\big /}{\tfrac {10}{20}}=10.} The meaning can be made explicit by writing
737-427: A line (or before a slash like 1 ⁄ 2 ), 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 3 / 4 , the numerator 3 indicates that the fraction represents 3 equal parts, and
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#1732801231749804-418: 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
871-482: 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: 1 2 {\displaystyle {\tfrac {1}{2}}} and 17 3 {\displaystyle {\tfrac {17}{3}}} ) consists of an integer numerator , displayed above
938-405: A person randomly chose one car on the lot, then there is a one in three chance or probability that it would be yellow. 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
1005-409: A piece of type bearing a complete fraction (e.g. 1 / 2 ) 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, 2 / 5 and 3 / 5 are both read as
1072-436: A smaller numerator and a smaller denominator. For example, if both the numerator and the denominator of the fraction a b {\displaystyle {\tfrac {a}{b}}} are divisible by c {\displaystyle c} , then they can be written as a = c d {\displaystyle a=cd} , b = c e {\displaystyle b=ce} , and
1139-534: A sum of unit fractions in infinitely many ways. Two ways to write 13 17 {\displaystyle {\tfrac {13}{17}}} are 1 2 + 1 4 + 1 68 {\displaystyle {\tfrac {1}{2}}+{\tfrac {1}{4}}+{\tfrac {1}{68}}} and 1 3 + 1 4 + 1 6 + 1 68 {\displaystyle {\tfrac {1}{3}}+{\tfrac {1}{4}}+{\tfrac {1}{6}}+{\tfrac {1}{68}}} . In
1206-432: 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 5 / 10 = 1 / 2 = 10 / 20 = 50 / 100 , for example. As another example, since the greatest common divisor of 63 and 462 is 21, the fraction 63 / 462 can be reduced to lowest terms by dividing
1273-402: 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 / 7 ), its value is 1, and the fraction therefore
1340-403: Is 4 to 2 and may be expressed as 4:2 or 2:1. 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 4 / 12 of the cars or 1 / 3 of the cars in the lot are yellow. Therefore, if
1407-407: 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
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#17328012317491474-436: 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
1541-443: 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 17 / 1 , 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 1 / 17 . A ratio
1608-524: Is indeed ever increasing in this world for those who have Lilavati clasped to their throats, decorated as the members are with neat reduction of fractions , multiplication and involution , pure and perfect as are the solutions, and tasteful as is the speech which is exemplified. The translations or editions of the Lilavati into English and other languages include: Fraction (mathematics) A fraction (from Latin : fractus , "broken") represents
1675-475: 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 1 / 3 ), oblique (as in 2/5), or diagonal (as in 4 ⁄ 9 ). These marks are respectively known as
1742-465: Is the fraction 2 / 5 and "two fifths" is the same fraction understood as 2 instances of 1 / 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, 3 / 1 may also be expressed as "three over one".) The term "over"
1809-536: Is the number [resulting from] 135 multiplied by 12, if you understand multiplication by separate parts and by separate digits. And tell [me], beautiful one, how much is that product divided by the same multiplier?" The word Līlāvatī itself means playful or one possessing play (from Sanskrit, Līlā = play, -vatī = female possessing the quality). The book contains thirteen chapters, mainly definitions, arithmetical terms, interest computation, arithmetical and geometrical progressions, plane geometry , solid geometry ,
1876-461: 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 1 2 {\displaystyle {\tfrac {1}{2}}} . When the numerator and denominator are both multiplied by 2, the result is 2 / 4 , which has the same value (0.5) as 1 / 2 . To picture this visually, imagine cutting
1943-506: Is to find a common denominator. To compare a b {\displaystyle {\tfrac {a}{b}}} and c d {\displaystyle {\tfrac {c}{d}}} , these are converted to a ⋅ d b ⋅ d {\displaystyle {\tfrac {a\cdot d}{b\cdot d}}} and b ⋅ c b ⋅ d {\displaystyle {\tfrac {b\cdot c}{b\cdot d}}} (where
2010-446: 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
2077-633: Is used as a synonym for the other. (For example, the compound fraction 3 4 × 5 7 {\displaystyle {\tfrac {3}{4}}\times {\tfrac {5}{7}}} is equivalent to the complex fraction 3 / 4 7 / 5 {\displaystyle {\tfrac {3/4}{7/5}}} .) 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
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2144-426: 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., 1 / 117 as "one over one hundred seventeen"), while those with denominators divisible by ten are typically read in
2211-473: The Golādhyāya . Bhaskara II's book on arithmetic is the subject of interesting legends that assert that it was written for his daughter, Lilavati. As the story goes, the author had studied Lilavati's horoscope and predicted that she would remain both childless and unmarried. To avoid this fate, he ascertained an auspicious moment for his daughter's wedding. To alert his daughter at the correct time, he placed
2278-468: The rational fraction 1 x {\displaystyle \textstyle {\frac {1}{x}}} ). In a fraction, the number of equal parts being described is the numerator (from Latin : numerātor , "counter" or "numberer"), and the type or variety of the parts is the denominator (from Latin : dēnōminātor , "thing that names or designates"). As an example, the fraction 8 / 5 amounts to eight parts, each of which
2345-564: The Leelavati Prize was initiated as a one-time international award for outstanding public outreach work for mathematics. The award was so well received at the conference and in the mathematical press that the IMU decided to turn the prize into a recurring four-yearly award and the award ceremony a regular feature of every ICM closing ceremony. The Leelavati prize is not intended to reward mathematical research but rather outreach activities in
2412-431: 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. The reciprocal of a fraction is another fraction with the numerator and denominator exchanged. The reciprocal of 3 / 7 , for instance, is 7 / 3 . The product of a non-zero fraction and its reciprocal
2479-744: The additional partial cake juxtaposed; this is more concise than the more explicit notation 2 + 3 4 {\displaystyle 2+{\tfrac {3}{4}}} cakes. The mixed number 2 + 3 / 4 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 3 4 {\displaystyle -2{\tfrac {3}{4}}} means − ( 2 + 3 4 ) . {\displaystyle -{\bigl (}2+{\tfrac {3}{4}}{\bigr )}.} Any mixed number can be converted to an improper fraction by applying
2546-428: The attributes "complex" and "compound" tend to be used in their every day meaning of "consisting of parts". 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. Multiplying
2613-588: The broadest possible sense. It carries a cash prize of 1,000,000 Indian Rupees (14,000 US dollars) together with a citation, and is sponsored by Infosys since 2014. Lilavati Līlāvatī is a treatise by Indian mathematician Bhāskara II on mathematics, written in 1150 AD. It is the first volume of his main work, the Siddhānta Shiromani , alongside the Bijaganita , the Grahaganita and
2680-1037: The convention that juxtaposition in algebraic expressions means multiplication. An Egyptian fraction is the sum of distinct positive unit fractions, for example 1 2 + 1 3 {\displaystyle {\tfrac {1}{2}}+{\tfrac {1}{3}}} . This definition derives from the fact that the ancient Egyptians expressed all fractions except 1 2 {\displaystyle {\tfrac {1}{2}}} , 2 3 {\displaystyle {\tfrac {2}{3}}} and 3 4 {\displaystyle {\tfrac {3}{4}}} in this manner. Every positive rational number can be expanded as an Egyptian fraction. For example, 5 7 {\displaystyle {\tfrac {5}{7}}} can be written as 1 2 + 1 6 + 1 21 . {\displaystyle {\tfrac {1}{2}}+{\tfrac {1}{6}}+{\tfrac {1}{21}}.} Any positive rational number can be written as
2747-415: 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, 1 / 3 = 0.333... represents the infinite series 3/10 + 3/100 + 3/1000 + .... Another kind of fraction is the percentage (from Latin : per centum , meaning "per hundred", represented by the symbol %), in which
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2814-525: 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 p / q 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
2881-555: The denominator ( b ) cannot be zero. Examples include 1 / 2 , − 8 / 5 , −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
2948-427: The denominator 4 indicates that 4 parts make up a whole. The picture to the right illustrates 3 / 4 of a cake. Fractions can be used to represent ratios and division . Thus the fraction 3 / 4 can 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
3015-621: 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 2 3 {\displaystyle {\tfrac {2}{3}}} ? 1 2 {\displaystyle {\tfrac {1}{2}}} gives 4 6 > 3 6 {\displaystyle {\tfrac {4}{6}}>{\tfrac {3}{6}}} . For
3082-434: 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
3149-408: The fraction 3/6. 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
3216-399: The fraction becomes cd / ce , which can be reduced by dividing both the numerator and denominator by c to give the reduced fraction d / e . If one takes for c 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
3283-504: 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, 3 9 {\displaystyle {\tfrac {3}{9}}} is not in lowest terms because both 3 and 9 can be exactly divided by 3. In contrast, 3 8 {\displaystyle {\tfrac {3}{8}}}
3350-405: The fractions using distinct separators or by adding explicit parentheses, in this instance ( 5 / 10 ) / 20 {\displaystyle (5/10){\big /}20} or 5 / ( 10 / 20 ) . {\displaystyle 5{\big /}(10/20).} A compound fraction is a fraction of a fraction, or any number of fractions connected with
3417-445: 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 ,
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#17328012317493484-400: 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
3551-831: The main source of learning arithmetic and algebra in medieval India. The work was also translated into Persian and was influential in West Asia. The Leelavati Prize was handed out for the first time at the closing ceremony of the International Congress of Mathematicians (ICM) 2010 in Hyderabad, India. Established by the Executive Organising Committee (EOC) of the ICM with the endorsement of the IMU Executive Committee (EC),
3618-616: The more laborious question 5 18 {\displaystyle {\tfrac {5}{18}}} ? 4 17 , {\displaystyle {\tfrac {4}{17}},} multiply top and bottom of each fraction by the denominator of the other fraction, to get a common denominator, yielding 5 × 17 18 × 17 {\displaystyle {\tfrac {5\times 17}{18\times 17}}} ? 18 × 4 18 × 17 {\displaystyle {\tfrac {18\times 4}{18\times 17}}} . It
3685-452: The normal ordinal fashion (e.g., 6 / 1000000 as "six-millionths", "six millionths", or "six one-millionths"). 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 a b {\displaystyle {\tfrac {a}{b}}} , where a and b are both integers . As with other fractions,
3752-440: 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, 3 + 75 / 100 . Decimal fractions can also be expressed using scientific notation with negative exponents, such as 6.023 × 10 , which represents 0.0000006023. The 10 represents a denominator of 10 . Dividing by 10 moves
3819-402: The numerator and denominator by 21: The Euclidean algorithm gives a method for finding the greatest common divisor of any two integers. Comparing fractions with the same positive denominator yields the same result as comparing the numerators: If the equal denominators are negative, then the opposite result of comparing the numerators holds for the fractions: If two positive fractions have
3886-427: 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 {\displaystyle n} , the fraction n n {\displaystyle {\tfrac {n}{n}}} equals 1. Therefore, multiplying by n n {\displaystyle {\tfrac {n}{n}}}
3953-425: The numerator read out as a whole number. For example, 3 / 1 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"
4020-410: The opposite of a positive fraction. For example, if 1 / 2 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
4087-591: The remainder divided by the divisor. For example, since 4 goes into 11 twice, with 3 left over, 11 4 = 2 + 3 4 . {\displaystyle {\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 + 1 / 2 hours or 5 3/16 inches , and remain widespread in daily life and in trades, especially in regions that do not use
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#17328012317494154-419: The rules of adding unlike quantities . For example, 2 + 3 4 = 8 4 + 3 4 = 11 4 . {\displaystyle 2+{\tfrac {3}{4}}={\tfrac {8}{4}}+{\tfrac {3}{4}}={\tfrac {11}{4}}.} Conversely, an improper fraction can be converted to a mixed number using division with remainder , with the proper fraction consisting of
4221-425: 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 a / b , where a and b are integers and b is not zero; the set of all rational numbers is commonly represented by
4288-458: 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
4355-567: The shadow of the gnomon, the Kuṭṭaka - a method to solve indeterminate equations , and combinations. Bhaskara II gives the value of pi as 22/7 in the book but suggest a more accurate ratio of 3927/1250 for use in astronomical calculations. Also according to the book, the largest number is the parardha equal to one hundred thousand billion. Lilavati includes a number of methods of computing numbers such as multiplications, squares, and progressions, with examples using kings and elephants, objects which
4422-444: The symbol Q or Q {\displaystyle \mathbb {Q} } , which stands for quotient . The term fraction and the notation a / b can also be used for mathematical expressions that do not represent a rational number (for example 2 2 {\displaystyle \textstyle {\frac {\sqrt {2}}{2}}} ), and even do not represent any number (for example
4489-661: The word of , corresponding to multiplication of fractions. To reduce a compound fraction to a simple fraction, just carry out the multiplication (see § Multiplication ). For example, 3 4 {\displaystyle {\tfrac {3}{4}}} of 5 7 {\displaystyle {\tfrac {5}{7}}} is a compound fraction, corresponding to 3 4 × 5 7 = 15 28 {\displaystyle {\tfrac {3}{4}}\times {\tfrac {5}{7}}={\tfrac {15}{28}}} . The terms compound fraction and complex fraction are closely related and sometimes one
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