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 the section on multiplication. The terms compound fraction and complex fraction are closely related and sometimes one is used as a synonym for the other. Like whole numbers, fractions obey the commutative , associative , and distributive laws, and the rule against division by zero. 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.
Dividing the numerator and denominator of a fraction by the same non-zero number will also yield 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 equal fraction with a smaller numerator and a smaller denominator. To do this, the greatest common factor is identified, and both the numerator and the denominator are divided by this factor. If the numerator and the denominator do not share any factor greater than 1, then the fraction is said to be irreducible, in lowest terms, or in simplest terms.
The Euclidean algorithm gives a method for finding the greatest common divisor of any two positive integers. Comparing fractions with the same positive denominator yields the same result as comparing the numerators:.
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If the equal denominators are negative, then the opposite result of comparing the numerators holds for the fractions:. 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. 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.
Fraction (mathematics) - Wikipedia
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. 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:. To add fractions containing unlike quantities e. 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.
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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. 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. 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,. 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. 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. Since a whole number can be rewritten as itself divided by 1, normal fraction multiplication rules can still apply.
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When multiplying mixed numbers, it is considered preferable [ citation needed ] to convert the mixed number into an improper fraction. 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. To divide a number by a fraction, multiply that number by the reciprocal of that fraction.
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. Thus 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.
The preferred way to indicate a repeating decimal is to place a bar known as a vinculum over the digits that repeat, for example 0. For repeating patterns where the repeating pattern begins immediately after the decimal point, a simple division of the pattern by the same number of nines as numbers it has will suffice.
In case leading zeros precede the pattern, the nines are suffixed by the same number of trailing zeros :. In case a non-repeating set of decimals precede the pattern such as 0. 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. 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:.
Furthermore, the relation , specified as. 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. Formally, for addition of fractions. 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. 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. Algebraic fractions are subject to the same field properties as arithmetic fractions. The terminology used to describe algebraic fractions is similar to that used for ordinary fractions. 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 expressions are the field of fractions of polynomials. There are different integral domains of polynomials, depending on the integral domain the coefficients of the polynomials are from e. These same expressions, however, would not be considered elements of the field of fractions generated by polynomials with integer coefficients.
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The term partial fraction is used when decomposing rational expressions into sums. The goal is to write the rational expression as the sum of other rational expressions with denominators of lesser degree.
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This is useful in many areas such as integral calculus and differential equations. 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:.
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. 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.