Fraction

Fraction

Hello ScienceBee, let's discuss fraction in this lesson.
 

 

What is a fraction?

A fraction (from Latin fractus, "broken") represents a part of a whole or, more generally, any number of equal parts.

Video: Let's Learn Fractions 
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: and 17/3) consists of an integer numerator displayed above a line (or before a slash), and a non-zero integer denominator, displayed below (or after) that line. Numerators and denominators are also used in fractions that are not common, including compound fractions, complex fractions, and mixed numerals.
A common fraction is a numeral which represents a rational number. That same number can also be represented as a decimal, a percent, or with a negative exponent. For example, 0.01, 1%, and 10−2 all equal the fraction 1/100. An integer such as the number 7 can be thought of as having an implicit denominator of one: 7 equals 7/1.
Other uses for fractions are to represent ratios and division. Thus the fraction 3/4 is also used to represent the ratio 3:4 (the ratio of the part to the whole) and the division 3 ÷ 4 (three divided by four). The non-zero denominator in the case using a fraction to represent division is an example of the rule that division by zero is undefined.
We can also write negative fractions, which represent 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 require that, for example, negative divided by positive is negative, −1/2, -1/2, 1/-2, and −-1/-2 all represent the same fraction, negative one-half. Because negative divided by negative is positive, -1/-2 represents positive one-half.

 

What are the forms of fractions - Simple, common, or vulgar fractions?

Video: Vulgar fractions
A simple fraction (also known as a common fraction or vulgar fraction) is a rational number written as a/b or , where a and b are both integers. As with other fractions, the denominator (b) cannot be zero. Examples include , ,  , , and 3/17. Simple fractions can be positive or negative, proper, or improper (see below). Compound fractions, complex fractions, mixed numerals, and decimals (see below) are not simple fractions, though, unless irrational, they can be evaluated to a simple fraction.
  • A unit fraction is a common fraction with a numerator of 1, e.g. . 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.  .

 

What are proper and improper fractions?

Video: 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. 
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; examples of improper fractions are 9/4, –4/3, and 3/3.

 

What are reciprocals and the "invisible denominator"?

The reciprocal of a fraction is another fraction with the numerator and denominator exchanged. The reciprocal of , for instance, is .
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), its value is 1, and the fraction therefore is improper. Its reciprocal also has the value 1, and is improper, too.
Any integer can be written as a fraction with the number one as denominator.
17 can be written as , 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 .

 

What are 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
2 are white,
6 are red, and
4 are yellow,
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.
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 ⅓ 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

Video: Decimals, Percents, and Fractions Song
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,  .
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, ⅓ = 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, 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.

 

What are mixed numbers?

Video: What is a Mixed Number?
A mixed numeral (also called a mixed fraction or mixed number) is a traditional denotation of the sum of a non-zero integer and a proper fraction (having the same sign).
It is used primarily in measurement:   inches, for example. Scientific measurements almost invariably use decimal notation rather than mixed numbers. The sum is implied without the use of a visible operator such as the appropriate "+". For example, in referring to two entire cakes and three quarters of another cake, the numerals denoting the integer part and the fractional part of the cakes are written next to each other as instead of the unambiguous notation Negative mixed numerals, as in  , are treated like  Any such sum of a whole plus a part can be converted to an improper fraction by applying the rules of adding unlike quantities.
This tradition is, formally, in conflict with the notation in algebra where adjacent factors denote a product, without an explicit infix operator. When two algebraic expressions are written next to each other, the operation of multiplication is implied by this general rule: always means the product of   and , even if the value of is a fraction. The expression for example is not a mixed number, instead, multiplication is expressly required, where
For better readability, the multiplication is sometimes made explicit or parentheses are added. So, may be written as 
or   or  
An improper fraction can be converted to a mixed number as follows:
  1. Divide the numerator by the denominator. In the example, , divide 11 by 4. 11 ÷ 4 = 2 with remainder 3.
  2. 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.
  3. The new denominator is the same as the denominator of the improper fraction. In the example, they are both 4. Thus .

 

What are 'Complex' and 'Compound' fractions?

Both notions are outdated and nowadays used in no well defined manner, partly even taken synonymously for each other or for mixed numerals. They 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".
  • Complex fractions
Not to be confused with fractions involving complex numbers
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,   are complex fractions. To reduce a complex fraction to a simple fraction, treat the longest fraction line as representing division. For example:
 
 
 
 
If, in a complex fraction, there is no unique way to tell which fraction lines takes precedence, then this expression is improperly formed, because of ambiguity. So 5/10/20/40 is not a valid mathematical expression, because of multiple possible interpretations, e.g. as
or as
  • Compound fractions

Video: 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, of is a compound fraction, corresponding to . 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 is equivalent to the complex fraction  .)

 

Let's Review

  1. Common fractions can be classified as either proper or improper. When the numerator and the denominator are both positive, the fraction is called ____ if the numerator is less than the denominator, and ____ otherwise. 

  2. The product of a fraction and its reciprocal is 1, hence the reciprocal is the ____   ____  of a fraction.

  3. The reciprocal of a proper fraction is ____ , and the reciprocal of an improper fraction not equal to 1, that is, numerator and denominator are not equal, is a ____  fraction.

  4. When the numerator and denominator of a fraction are equal (, for example), its value is 1, and the fraction therefore is ____ . Its reciprocal also has the value 1, and is ____ , too.

  5. Decimal fractions can also be expressed using ____  ____  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.

  6. Decimal fractions with infinitely many digits to the right of the decimal separator represent an ____  ____ . For example, ⅓ = 0.333... represents the infinite series 3/10 + 3/100 + 3/1000 + ... .

  7. A ____  ____ (also called a mixed fraction or mixed number) is a traditional denotation of the sum of a non-zero integer and a proper fraction (having the same sign).

Answer

  1.  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. 

  2. The product of a fraction and its reciprocal is 1, hence the reciprocal is the multiplicative inverse of a fraction.

  3. 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.

  4. When the numerator and denominator of a fraction are equal (, for example), its value is 1, and the fraction therefore is improper. Its reciprocal also has the value 1, and is improper, too.

  5. 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.

  6. Decimal fractions with infinitely many digits to the right of the decimal separator represent an infinite series. For example, ⅓ = 0.333... represents the infinite series 3/10 + 3/100 + 3/1000 + ... .

  7. A mixed numeral (also called a mixed fraction or mixed number) is a traditional denotation of the sum of a non-zero integer and a proper fraction (having the same sign).

 

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