Time 4 Math: Fractions

A fraction is a number that names a part of a whole, for example, 1/2 and 2/3.

A fraction is made up of two numbers. The top number is called numerator. The bottom number is called the denominator. the larger the denominator, the smaller each of the equal parts are. Let us take 1/3 and 1/2. Here, 1/2 is smaller than 1/2.

Common, vulgar, or simple fractions

A common fraction (also known as a vulgar fraction or simple fraction) is a rational number written as a/b or $\tfrac{a}{b}$ , where the integers a and b are called the numerator and the denominator, respectively.The numerator represents a number of equal parts and the denominator, which cannot be zero, indicates how many of those parts make up a unit or a whole. In the examples 2/5 and 7/3, the slanting line is called a solidus or forward slash. In the examples $\tfrac{2}{5}$ and $\tfrac{7}{3}$ , the horizontal line is called a vinculum or, informally, a "fraction bar".

Writing simple fractions

In computer displays and typography, simple fractions are sometimes printed as a single character, e.g. ½ (one half).

Scientific publishing distinguishes four ways to set fractions, together with guidelines on use:

case fractions: $\tfrac{1}{2}$ , generally used for simple fractions and for showing mathematical operations;
special fractions: ½, not used in modern mathematical notation, but in other contexts;
shilling 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;
built-up fractions: $\frac{1}{2}$ , while large and legible, these can be disruptive, particularly for simple fractions or within complex fractions.

Proper and improper common 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^.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 between -1 and 1 (but not equal to -1 or 1). It is said to be an improper fraction (U.S., British or Australian) or top-heavy fraction (British, occasionally North America) 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; examples of improper fractions are 9/4, -4/3, and 8/3.

Mixed numbers

A mixed numeral (often called a mixed number, also called a mixed fraction) is the sum of a non-zero integer and a proper fraction. This sum is implied without the use of any visible operator such as "+". For example, in referring to two entire cakes and three quarters of another cake, the whole and fractional parts of the number are written next to each other: $2+\tfrac{3}{4}=2\tfrac{3}{4}$ .

This is not to be confused with the algebra rule of implied multiplication. When two algebraic expressions are written next to each other, the operation of multiplication is said to be "understood". In algebra, $a \tfrac{b}{c}$ for example is not a mixed number. Instead, multiplication is understood: $a \tfrac{b}{c} = a \times \tfrac{b}{c}$ .

An improper fraction is another way to write a whole plus a part. A mixed number can be converted to an improper fraction as follows:

Write the mixed number $2\tfrac{3}{4}$ as a sum $2+\tfrac{3}{4}$ .
Convert the whole number to an improper fraction with the same denominator as the fractional part, $\tfrac{8}{4}+\tfrac{3}{4}$ .
Add the fractions. The resulting sum is the improper fraction. In the example, $2\tfrac{3}{4}=\tfrac{11}{4}$ .

Similarly, an improper fraction can be converted to a mixed number as follows:

Divide the numerator by the denominator. In the example, $\tfrac{11}{4}$ , divide 11 by 4. 11 ÷ 4 = 2 with remainder 3.
The quotient (without the remainder) becomes the whole number part of the mixed number. The remainder becomes the numerator of the fractional part. In the example, 2 is the whole number part and 3 is the numerator of the fractional part.
The new denominator is the same as the denominator of the improper fraction. In the example, they are both 4. Thus $\tfrac{11}{4} =2\tfrac{3}{4}$ .

Mixed numbers can also be negative, as in $-2\tfrac{3}{4}$ , which equals $-(2+\tfrac{3}{4}) = -2-\tfrac{3}{4}$ .

Reciprocals and the "invisible denominator"

The reciprocal of a fraction is another fraction with the numerator and denominator reversed. The reciprocal of $\tfrac{3}{7}$ , for instance, is $\tfrac{7}{3}$ . The product of a fraction and its reciprocal is 1, hence the reciprocal is the multiplicative inverse of a fraction. Any integer can be written as a fraction with the number one as denominator. For example, 17 can be written as $\tfrac{17}{1}$ , where 1 is sometimes referred to as the invisible denominator. Therefore, every fraction or integer except for zero has a reciprocal. The reciprocal of 17 is $\tfrac{1}{17}$ .

Complex 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, $\frac{\tfrac{1}{2}}{\tfrac{1}{3}}$ and $\frac{12\tfrac{3}{4}}{26}$ are complex fractions. To reduce a complex fraction to a simple fraction, treat the longest fraction line as representing division. For example:

$\frac{\tfrac{1}{2}}{\tfrac{1}{3}}=\tfrac{1}{2}\times\tfrac{3}{1}=\tfrac{3}{2}=1\tfrac{1}{2}.$

$\frac{12\tfrac{3}{4}}{26} = 12\tfrac{3}{4} \cdot \tfrac{1}{26} = \tfrac{12 \cdot 4 + 3}{4} \cdot \tfrac{1}{26} = \tfrac{51}{4} \cdot \tfrac{1}{26} = \tfrac{51}{104}$

$\frac{\tfrac{3}{2}}5=\tfrac{3}{2}\times\tfrac{1}{5}=\tfrac{3}{10}.$

$\frac{8}{\tfrac{1}{3}}=8\times\tfrac{3}{1}=24.$

If, in a complex fraction, there is no clear way to tell which fraction line takes precedence, then the expression is improperly formed, and meaningless.

Compound fractions

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). 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.

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, a raised period (•), 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, viz. 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, $3\tfrac{75}{100}$ .

Decimal fractions can also be expressed using scientific notation with negative exponents, such as 6.023×10⁻⁷, which represents 0.0000006023. The 10⁻⁷ represents a denominator of 10⁷. Dividing by 10⁷ 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, 1/3 = 0.333... represents the infinite series 3/10 + 3/100 + 3/1000 + ... .

Another kind of fraction is the percentage (Latin per centum meaning "per hundred", represented by the symbol %), in which the implied denominator is always 100. Thus, 75% means 75/100. Related concepts are the permille, with 1000 as implied denominator, and the more general parts-per notation, as in 75 parts per million, meaning 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, 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.

Special cases

A unit fraction is a vulgar fraction with a numerator of 1, e.g. $\tfrac{1}{7}$ . 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.

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 $\tfrac{1}{3}+\tfrac{1}{4}+\tfrac{1}{6}+\tfrac{1}{68}$ .

A dyadic fraction is a vulgar fraction in which the denominator is a power of two, e.g. $\tfrac{1}{8}$ .

Reference: http://en.wikipedia.org/wiki/Fraction_%28mathematics%29

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