Vulgar fraction

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In arithmetic, a vulgar fraction (or common fraction) consists of one integer divided by a non-zero integer. The fraction "three divided by four" or "three over four" or "three fourths" or "three quarters" can be written as

\frac{3}{4}

3 \div 4

or 3/4 or ³⁄₄

In this article, we will use the last of these notations, though the first is often preferred.

The top number of the fraction is called the numerator, and the bottom number is called the denominator.

The word "numerator" is related to the word "enumerate." To enumerate means to "tell how many"; thus the numerator tells us how many fractional parts we have in the indicated fraction. To denominate means to "give a name" or "tell what kind"; thus the denominator tells us what kind of parts we have (halves, thirds, fourths, etc.).

The denominator can never be zero because division by zero is not defined. All vulgar fractions are rational numbers and, by definition, all rational numbers can be expressed as vulgar fractions, although the representation is not unique. (For example, ³⁄₄ = ⁶⁄₈.)

Naturally, a fraction which is not a vulgar fraction is one in which either the numerator or the denominator is something other than a simple integer (for instance, a square root expression).

Introduction

To understand the meaning of any vulgar fraction, from any time period, consider some unit (e.g., a cake) divided into an equal number of parts (or slices). The number of slices into which the cake is divided is the denominator. The number of slices in consideration is the numerator. So were I to eat 2 slices of a cake divided into 8 equal slices then I would have eaten ²⁄₈ (or two eighths) of the cake. Note that had the cake been divided into 4 slices and had I eaten one of those then I would have eaten the same amount of cake as before. Hence, ²⁄₈ = ¹⁄₄. Had I eaten 1 and a half full cakes I would have eaten 12 of the one eighth of a cake slices or ¹²⁄₈. If the cakes been divided into quarters I would have eaten ⁶⁄₄ cakes.

¹²⁄₈ cakes = ⁶⁄₄ cakes = ³⁄₂ cakes = (1 + ¹⁄₂) cakes.

Egyptians loved working problems of this nature. Scribes always looked for exact answers, which were always possible when vulgar fractions appeared as remainders. For example, considering simple algebra Rhind Mathematical Papyrus (or RMP) 24 asked to find unknown x and 1/7th of x to equal a fixed number, in this case 19. Ahmes, the Egyptian scribe, worked the problem this way:

(8/7)x = 19, or x = 133/8 = 16 + 5/8,

with 133/8 being the initial vulgar fraction and 5/8 being the remainder vulgar fraction term. Ahmes converted 5/8 to an Egyptian fraction series by (4 + 1)/8 = 1/2 + 1/8, making his final quotient plus remainder based answer x = 16 + 1/2 + 1/8.

The RMP includes 15 algebra problems of this type, with #24 being the easiest. Each of the RMP's other 14 algebra problems produced increasingly difficult vulgar fractions. Yet, all were easily converted to an optimal (short and small last term) Egyptian fraction series.

Arithmetic

Several rules for calculation with fractions are useful:

One follows a remainder arithmetic structure as found in the Reisner Papyus, 1800 BCE Egypt^*. This form of arithmetic creates a quotient and an exact remainder, and is often discussed within finite arithmetic. The Reisner Papyrus rated workers in units of 10, using remainder arithmetic, with their digging rates being calculated.

Cancelling

If both the numerator and the denominator of a fraction are multiplied or divided by the same non-zero number, then the fraction does not change its value. For instance, ⁴⁄₆ = ²⁄₃* ²⁄₂.

Adding and subtracting fractions

To add or subtract two fractions, you first need to change the two fractions so that they have a common denominator, for example the lowest common multiple of the denominators which is called the lowest common denominator; then you can add or subtract the numerators. For instance:

²⁄₃ + ¹⁄₄ = ⁸⁄₁₂ + ³⁄₁₂ = ¹¹⁄₁₂.

and, for subtraction, as Egyptians practiced in 2,000 BCE

2/71 - 1/40 = (80 - 71)/(40*71) = 9/2840 (modern version)

as Egyptians easily converted to their unit fraction system

by (5 + 4)/2840 = 1/568 + 1/710 (ancient version)

If the fractions are improper but you want a mixed-number result, you may first change them into mixed numbers. After that you find a common denominator. Then you change the fractions so that both the fractions share the same denominator. After that like two normal fractions you either add or subtract the numerator. Remember the denominator stays the same. When you are done adding or subtracting the numerators then you write it as a fraction. The answer you just got as the numerator and the denominator is the same from before you added or subtracted. If it turns out to be an improper fraction then you change it to a mixed fraction and add your first whole number(s) that you got when you started. There is your answer. Example:

change to mixed fractions: $\begin{matrix} {12 \over 7} + {8 \over 5} = 1 {5 \over 7} + 1 {3 \over 5} \end{matrix}$
sum the fractional parts: $\begin{matrix} {5 \over 7} + {3 \over 5} = {25 \over 35} + {21 \over 35} = {46 \over 35} = 1 {11 \over 35} \end{matrix}$
put it all back together: $\begin{matrix} 1 + 1 + 1 {11 \over 35} = \mathbf{3 {11 \over 35}} \end{matrix}$

Multiplying fractions

To multiply two fractions, multiply the numerators to get the new numerator, and multiply the denominators to get the new denominator. For instance:

²⁄₃ × ¹⁄₄ = (2×1) / (3×4) = ²⁄₁₂ = ¹⁄₆.

It is helpful to read "²⁄₃ × ¹⁄₄" as "one quarter of two thirds". If one took ²⁄₃ of a cake and gave ¹⁄₄ of that part away, the part one gave away would be equivalent to ¹⁄₆ of a full cake.

Reciprocal of fractions. To take the reciprocal of fractions, simply swap the numerator and the denominator, so the reciprocal of ²⁄₃ is ³⁄₂. If the numerator is 1, i.e. the fraction is a unit fraction, then the reciprocal is an integer, namely the denominator, so the reciprocal of ¹⁄₃ is ³⁄₁ or 3.

Dividing fractions. As dividing is the same as multiplying by the reciprocal, to divide one fraction by another one, flip numerator and denominator of the second one, and then multiply the two fractions. For instance:

(²⁄₃) / (⁴⁄₅) = ²⁄₃ × ⁵⁄₄ = (2×5) / (3×4) = ¹⁰⁄₁₂ = ⁵⁄₆.

Other ways of writing fractions

Improper fraction

Any rational number can be written as a vulgar fraction. If the absolute value of a fraction is greater than or equal to 1, i.e. the absolute value of the numerator is greater than or equal to the absolute value of the denominator—then it is also known as an improper fraction. An example is ¹¹⁄₄, which is a little less than 3.

Mixed number

A fraction greater than 1 can also be written as a mixed number, i.e. as the sum of a positive integer and a fraction between 0 and 1 (sometimes called a proper fraction). For example

2³⁄₄ = ¹¹⁄₄.

In general:

A\ \frac{b}{c} = \frac{Ac+b}{c}.

This notation has the advantage that one can readily tell the approximate size of the fraction; it is rather dangerous however, because 2³⁄₄ risks being understood as 2×³⁄₄, which would equal ³⁄₂, (or even as ²³⁄₄), rather than 2+³⁄₄ . To indicate multiplication between an integer and a fraction, the fraction is instead put inside parentheses: 2 (³⁄₄) = 2 × ³⁄₄.

Decimal notation

Fractions can also be written as decimals. For example

⁵⁄₄ = 1.25

The mark after the integer part is a decimal point (though in many countries it is represented by a comma). If the denominator of a fraction has only 2 and/or 5 for prime factors (which includes all decimal fractions), then it can be written as a finite decimal, one with a finite number of non-zero digits. Any rational number can be written as a recurring decimal if not a simple finite decimal. An irrational number would be expressed as a non-recurring decimal with an infinite number of digits after the decimal point.

History

See also Egyptian fractions.

The oldest formal use of vulgar fractions is documented in the Egyptian Middle Kingdom mathematical texts. They appeared as quotients and remainders in a remainder arithmetic statement, with the remainder vulgar fraction term always converted to an exact unit (Egyptian) fraction series. For example, vulgar fractions appeared whenever Egyptian scribes like Ahmes (1650 BCE) divided a hekat unity, 64/64, by any number up to 64. One of Ahmes' problems solved 6400/70 using a related technique, extending the divisors of a hekat unity beyond 64 to n, by increasing the size of the numerator.

The older Akhmim (Cairo) Wooden Tablet (2000 BCE) reported the method that Ahmes had followed in structure and technique, divided (64/64) by 13. The 2,000 BCE method created two types of vulgar fractions in the same statement. The first type of vulgar fraction was a quotient, 4/64, stated as a binary series. The second type of vulgar fraction was a remainder, (12/(13*64)) = 12/832. Taken together the two-part statement, one of the earliest forms of mixed number, was followed by the word ro.

The scribe first simplified the first vulgar fraction 4/64 to 1/4.

The second vulgar fraction 12/832 was written 60/13 times 1/320, since 1/64 = 5/320. Thus the scribal answer became 1/4 + 60/13 * ro (with ro = 1/320, a common divisor) further simplied the data. Ahmes' last step converted the remainder term's vulgar fraction 60/13 times 1/320 to (4 + 8/13)*1/320 = (4 + 1/2 + 3/(26))*1/320 = (4 + 1/2 + 1/13 + 1/26)*1/320.

The awkward looking second vulgar fraction (remainder) was then added to the first vulgar fraction (quotient), creating the final exact, two-part (mixed) statement. The 2,000 BCE answer was written as:

1/4 + (4 + 1/2 + 1/13 + 1/26) ro.

The scribe was then asked to prove his work, clearing up any error that may have crept in. Proof was obtained by multiplying the (64/64)/n answer by n, obtaining his initial (64/64) starting unity vulgar fraction. The proof step was not fully seen in 1906, though suggested, when Georges Daressy first translated the document. However, by 2002 the proof statement was generally found to equal the original (unity) vulgar fraction, 64/64. The proof step was documented by Hana Vymazalova (Charles U., Prague), for all five divisors, 3, 7, 10, 11 and 13, contained in the wooden tablets, altering a 100 year old point of view on the document and its connections to Ahmes' arithmetic.