Score originally meantwholeA part of, or more generally, any equal number of parts.In the form ofintegerThe ratio of a to an integer b (whether a is a false fraction that is a multiple of b is a fraction is controversial[1-2]）。
A fraction represents the fraction of one number to another, or the ratio of one event to all events.Divide the unit "1" into several equal shares, and the number representing such a share or shares is called fraction.The numerator is above and the denominator is below.[3]
When the denominator is 100, it can be written aspercentageIn the form of 1%[4]。
be careful:Different books have different definitions of scores
，
All are regarded as scores and belong to false scores[2]。Another view is that fractions that can be converted into integers are not fractions[1]。
Divide the unit "1" into several equal parts, and the number representing such one or several parts is called true fraction, such as:
or
, or it may become a false fraction, that is, the numerator is greater than or equal to the denominator, for example
。The denominator means to divide an object into several equal parts, and the numerator means to take several of them.
fraction
It serves asdivisionSee, divide the numerator by the denominator (because 0 cannot be divided in division, so the denominator cannot be 0). On the contrary, divide can also be expressed in fractions.
Difference between percentage and fraction:
(1) Different meanings, percentage only represents the multiple ratio relationship of two numbers, and cannot take the unit name;Fractions can represent both specific numbers and the relationship between two numbers. When representing specific numbers, the company name can be used.
Example: Can say
It can also say 70% of a meter, but not 70% of a meter.
(3) Any percentage can be written as a fraction whose denominator is 100, anddenominatorA score of 100 is not always a percentage.
Example: 61%=
, but
No 61% meaning.
Example Diagram
(4) The application range is different. In production and life, percentage is often used for investigation, statistics, analysis and comparison, while fraction is often used when integer results cannot be obtained in calculation and measurement.
history
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The earliest fraction was the reciprocal of an integer: an ancient symbol representing half, a thirdquarterwait.Egyptians use the Egyptian fraction c.1000 bc。About 4000 years ago, Egyptians separated them with slightly different scores.They useLeast common multipleAnd unit fraction.Their method gives the same answer as modern methods.Egyptians also have different representations of Akhmim wood chips and second-generation mathematical papyrus.
The Greeks use unit scores and (post) sustained scores.Followers of the Greek philosopher Pythagoras (c. 530bc) found that two square roots cannot be expressed as part of an integer.(Usually this may be wrongly attributed to Hippasus of Metafontum, who is said to have been executed to reveal this fact).Among 150 Indians in India, Jain mathematicians wrote "Sthananga Sutra", which includes number theory, arithmetic operations and operations.
The modern score called bhinnarasi seems to originate from India's work in Aryabhatta (c. ad 500), [citation needs] Brahmagupta (c. 628) and Bhaskara (c. 1150).Their works form fractions by placing molecules (Sanskrit: amsa) on the denominator (cheda), but without stripes between them.In Sanskrit literature, fractions are always expressed as the addition and subtraction of an integer.Integers are written on one line and their fractions are written on the next line of two lines.If the fraction is marked with a small circle ⟨ 0was or cross ⟨+was, it is subtracted from the integer;If no such sign appears, it is understood as being added.
significance
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An object, a figure and a unit of measurement can all be regarded as the unit "1".Divide the unit "1" into several equal portions, and the number representing such portions or portions is called fraction.In fractions, those that mean how many shares the unit "1" is divided into are called denominators, and those that mean how many shares there are are called molecules;One of them is called fractional unit.
To understand the meaning of decimals, we can start with the meaning of fractions. The meaning of fractions can be explained from the activities of segmentation and composition. When a whole (referring to the benchmark quantity) is divided equally, the quantity that gathers a part of it is called "component", and "fraction" is used to represent or record this "component".For example, 2/5 means that an integer is divided into five equal parts to form a binary "component".[5]When the whole is divided into ten equal parts, one hundred equal parts, one thousand equal parts, etc., another recording method is used for the components at this time - decimal.for example
Record as 0.1
Recorded as 0.02
Record as 0.005...... Etc.The "." is calleddecimal point, to separateInteger partAnd cannot form the decimal part of an integer.Integers other than 0 are calledwhole number with a decimal, if it is 0, it is calledpure decimal。It can be seen that the meaning of decimal is a part of the meaning of fraction.
The numerator and denominator are multiplied or divided by the same number (except 0), and the size of the fraction remains the same. This isBasic Properties of Fraction。
In a fraction, the number of equal parts described is the numerator, and the type or kind of part is the denominator.In informal texts, numerators and denominators may be distinguished only by their placement, but in formal texts they are always separated by a fractional line.The fraction line can be horizontal (e.g.), oblique (e.g.) or diagonal (e.g.).These markers are called horizontal, diagonal (US), or diagonal (UK), division, and fraction slashes, respectively.In typesetting, the horizontal fraction is also called "en fraction" or "nut fraction", and the diagonal fraction is called "em fraction", which is the width of the line they occupy.
The denominator of an English fraction is usually expressed as an ordinal number. If the numerator is not 1, read the plural of the denominator.(For example, both the denominator and the denominator are read as "fifth".) In addition, when the denominator is 2, it is always read as "half" or "halves", and when the denominator is 4, it is always read as "quarter/quarters" or "fourth/four".The denominator is 100, and it is always read "hunderedth/underths" or "percent".If the denominator of a fraction is 1, it is often omitted and only the numerator needs to be read (for example, 3).The denominator is 1, which can be omitted.
In English, if there is a hyphen between two numbers of a fraction, it represents a whole. Otherwise, it represents several fractions with 1 numerator (for example, "two five" represents two).It is worth noting that if fractions are used as adjectives, hyphens must be used.In addition, fractions can be read as the numerator "over" and denominator, and the denominator can be expressed as the cardinal number (for example, in English, 3/1 can also be read as "three over one")"Over" can also be used in molecules with diagonal fractional lines (for example, in English, it can be read as "one half", "one half", or "one over two").Fractions with large denominators not to the tenth power are usually presented in the following way (for example, 1/117 is "one over one hundred seven"), while those with denominators divisible by ten are usually read in the normal ordinal way (for example, 6/1000000 is read as "six millionths", "six millionths", "six one millionths").[6]
nature
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nature
Reading: two-thirds
Writing:
The horizontal line in the middle of the fraction is calledFractional lineThe number above the fraction line is calledmoleculeThe number below the fractional line is calleddenominator。Read as fractions.
A fraction can be expressed as adivisionFormula: For example, half is equal to 1 divided by 2.Where 1 molecule is equal toDividend, - Fractional line equalsdivision sign, 2 denominator equalsDivisor, while 0.5Fractional valueIs equal to quotient.
Score can also be expressed as a ratio, for example;Half is equal to 1:2, where the numerator of 1 is equal to the preceding term, - the fractional line is equal to the ratio sign, the denominator of 2 is equal to the following term, and the value of 0.5 fraction is equal toratio。Basic properties of fraction: the numerator and denominator of a fraction are multiplied by or divided by the same nonzero number, and the resulting fraction is equal to the original fraction.
(b and c are not equal to zero)
Another interesting property of fractions is that a fraction is notterminating decimal , isInfinite recurring decimal, like pi, etcInfinite acyclic decimalIt is impossible to replace it with a fraction.
Another property of fractions is that when the numerator and denominator are multiplied or divided by the same number (except 0), the value of fractions will not change.Therefore, each fraction has an infinite number of equal fractions.With this propertyApproximationAndGeneral distribution。
The result of the power operation on a fraction cannot be an integer, and if it is the simplest fraction before the operation, the result will also be the simplest, such as
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integerIt can be regarded as a fraction with a denominator of 1, and the unit is
① The denominator must not be 0, becausedenominatorIt is equivalent to a divisor.Otherwise, the equation cannot be established, and the numerator can be equal to 0, because the numerator is equal to the dividend.It is equal to 0 divided by any number. No matter what the denominator is, the answer is 0.
② The numerator or denominator of a fraction passes throughApproximationNo irrational number (such as 2'ssquare root）, otherwise it is not a score.
③ In the denominator of a simplest fraction, there are only 2 and 5 primefactorCan becometerminating decimal ；If the denominator of the simplest fraction only contains prime factors other than 2 and 5, then it can be converted into pure cyclic decimal;If the denominator of the simplest fraction contains both 2 or 5 prime factors and prime factors other than 2 and 5, then it can be transformed intoMixed circular decimal。(Note: if it is not a simplest fraction, it must be converted to the simplest fraction before judgment; the simplest fraction with a denominator of 2 or 5 must be converted to a finite fraction, and the denominator is otherPrime numberThe simplest fraction ofPure recurring decimal）
Fractionate Decimal
The simplest fraction is to look at the prime of the denominator firstfactorIf there are only 2 and 5, they can be converted into finite decimals. If not, they cannot be converted into finite decimals.If it is not the simplest fraction, it must be reduced.
There are the following methods:
The denominator is a special number (such as 2, 4, 8, 10, 100, 1000, etc.)
1. The denominator is 2, 4, 8, etc. By using the basic properties of fractions, the denominator and numerator are multiplied by 5, 25, 125, etc. at the same time, the denominator is converted into 10, 100, 1000 numbers and directly converted into decimal.
2. Use the relationship between fraction and division: numerator/denominator=decimal
The denominator is not a special number
1. Use the relationship between fraction and division: numerator/denominator=decimal (i.e
）
2. If the result is a circular decimal, only a few decimal places should be reserved according to the actual situation.(i.e
）
Decimal fraction
terminating decimal To convert fractions, there are several zeros in the decimal part and several denominators.Example: 0.45=
If yesMixed circular decimalThe method of mixed circular decimation is to write the decimal part as a fraction, and the numerator is the difference between the number formed by the decimal part number with a circular section and the number formed by the decimal part number without a circular section.The denominator is 99 -. 0.. Where the number of 9 is the same as the number of circular sections, and the number of 0 is the number of non circular decimal parts.Example: 0.12 (2 cycles)=(12-1)/90=11/90
The value of a true fraction is less than 1.The numerator is smaller than the denominator,
Example:
、
、
etc.
The value of the false fraction is greater than or equal to 1.The numerator is greater than or equal to the denominator
Example:
、
、
etc.
With fractionA form of false fraction[2], whose value is greater than 1, the subsequent fraction must be a true fraction.
Example:
、
、
etc.
computing method
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Addition and subtraction
1. Add and subtract fractions with the same denominator, and the denominator remains unchanged, that is, the fractional unit remains unchanged, and the numerator is added and subtracted, and the points that can be reduced shouldApproximation。
Example 1:
Example 2:
Example 3:
Example 4:
2. Add and subtract fractions with different denominatorsGeneral distribution, that is, use the basic properties of fractions to divide the denominator into fractionsconversionIt refers to the offer score with the same denominator, which is calculated by adding and subtracting the scores with the same denominator after changing the score unit without changing the size.
Example 1:
Example 2:
Example 3:
Example 4:
Multiplication and division
1. Score multiplied by integer, denominator unchanged, numerator multiplied by integer, and the offer score that can be reduced finally.
Example:
2. Fraction multiplied by fraction, numerator multiplied by numerator, denominator multiplied by denominator, and the offer score that can be reduced finally.
Example:
3. Divide a fraction by an integer, and the denominator remains unchanged. If the numerator is an integermultiple, then divide the numerator by the integer, and the offer score that can be reduced finally.
Example:
4. Divide a fraction by an integer, and the denominator remains unchanged. If the numerator is not a multiple of an integer, multiply the fraction by thereciprocal, the final offer points that can be reduced.
Example:
5. The fraction divided by the fraction is equal to the reciprocal of the divisor multiplied by the dividend, and the offer score that can be reduced finally.
Example:
history
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In history, fractions are almost as old as natural numbers.As early asHuman cultural inventionAt the beginning of the 20th century, because of the need to measure and average, people introduced and used scores.
foreign country
There are records about fractions and different fractional systems in the ancient literature of many nationalities.As early as 2100 BC, the ancient Babylonians (nowIraqA band) uses a fraction whose denominator is 60.
Fractions were also used in Egyptian mathematical literature around 1850 BC, but at that time, fractions in ancient Egypt were only fractional units.
China
our countrySpring and Autumn Period(770-476 BC)《Master Zuo's Spring and Autumn Annals》The size of the capital city of the feudal lords is stipulated in: the maximum size should not exceed one third of the capital of the Kingdom of Zhou Wen, the medium size should not exceed one fifth, and the small size should not exceed one ninth.First Emperor of QinThe calendar of the times stipulates that the number of days in a year is 365 and 1/4.This shows that scores have appeared very early in China and are used for social production and life.
The earliest number in human history is natural number (non negativeinteger）In the future, when measuring and averaging scores, it is often impossible to get exactly the integer result, which results in scores.
Use a standard quantity (unit of measurement) to measure another quantity. Only when the equivalent is exactly measured several times can an integer be used to express the measurement result.If the quantity cannot be exactly measured for several times, there are two cases:
For example, use b as the standard to measure a:
One case is to divide b into n equal parts, and use one of them as a new unit of measurement to measure a, which means that a contains m equal parts after b is divided into n equal parts.For example, divide b into four equal parts, measure a with one of them, and measure it exactly nine times. In this case, it is not possible to use an integer to express the result of measuring a with b, so it is necessary to introduce a new number - fraction to express the result of measurement.
Another situation is that no matter how many equal parts of b are divided into, one of them can not be used as a new measure of a (for example, the diameter of a circle can be used to measure the diameter of the same circlePerimeter）。In this case, it is necessary to introduce a new number-Irrational number。In integer division, when two numbers are divided, sometimes the integer quotient cannot be obtained.In order to make division always possible, it is also necessary to introduce a new number fraction.
The history of fractions can be traced back to Egypt more than 3000 years ago.
More than 3000 years ago, ancient Egypt used special symbols to represent fractions whose numerator was 1 in order to express numbers without being able to divide integers.More than 2000 years ago, China had scores. However, the scores in the Qin and Han Dynasties were different in form.India has developed a fraction representation similar to that of China.Later, the Arabs invented the fraction line, and today's fraction representation is derived from it.
More than 200 years ago, Euler, a Swiss mathematician, wrote in General Motorsarithmetic》It is said in the book that it is impossible to divide a 7 meter long rope into three equal parts, because a suitable number cannot be found to express it. If we divide it into three equal parts, each part is
Mi. Image
It is a new number, which we call fraction.
name
fraction
Why is it called fraction?The name fraction visually and vividly represents the characteristics of this number.For example, if four people divide a watermelon equally, won't they divide it into four equal pieces?From this example, we can see that fractions are generated from the needs of measurement and mathematics itself - the needs of division.
Why are the ones on the line of fraction called numerators and those below the line of fraction called denominators?The so-called fraction means to divide the number. Therefore, the number above the fractional line becomes an equal number, while the number below represents a whole number.Now let's look at why the above question is called "molecule", which involves "fractional unit". When you divide a number into several equal parts, one of them is one of many parts, which is called fractional unit.The value of the fraction will be equal to 1 only when the numbers on and off the line of the fraction are equal. In any other case, it will be less than 1.Since the number above the fractional line is usually smaller than the number below, it is easy to understand that the small number above is called "sub" and the large number below is called "parent".
English Pronunciation
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ScoredecimalandpercentagePronunciation of;In fractions, numerator is expressed by cardinal number, and denominator is expressed by ordinal number.Read the numerator first, then the denominator.When the numerator is greater than 1, add "s" to the denominator.
For example:
—two thirds
Formula: numerator cardinal number word, denominator ordinal number word, numerator greater than 1, denominator plus s[8]
