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 Arithmetic or arithmetics (from the Greek word αριθμός = number) in common usage is a branch of (or the forerunner of) mathematics which records elementary properties of certain operations on numerals, though professional mathematicians often treat arithmetic as a synonym for number theory. ... The integers consist of the positive natural numbers (1, 2, 3, …), their negatives (−1, −2, −3, ...) and the number zero. ... In mathematics, especially in elementary arithmetic, division is an arithmetic operation which is the inverse of multiplication, and sometimes it can be interpreted as repeated subtraction. ...

      or             or       3/4       or       ³⁄₄

In this article, we will use the last of these notations, though it should be noted that for most serious applications, 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.) 0 (zero), alternatively called naught, nil, ought, or nought, is both a number and a numeral. ... In mathematics, a division is called a division by zero if the divisor is zero. ... In mathematics, a rational number (or informally fraction) is a ratio or quotient of two integers, usually written as the vulgar fraction a/b, where b is not zero. ...

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).

The oldest use of vulgar fractions is documented in the Egyptian mathematical texts. They appeared as remainders in a remainder arithmetic statement, with the vulgar fraction being converted to an Egyptian fraction series.

Consider this article a gentle introduction to fractions: for a more theoretical treatment see rational numbers. There is also a fraction article which can serve as a guide to different kinds of fractions. In mathematics, a rational number (or informally fraction) is a ratio or quotient of two integers, usually written as the vulgar fraction a/b, where b is not zero. ... In mathematics, a fraction is a way of expressing a quantity based on an amount that is divided into a number of equal-sized parts. ...

Introduction

To understand the meaning of any vulgar fraction 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. The word unit means any of several things: Unit of measurement or physical unit, a fundamental quantity of measurement in science or engineering. ...

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, RMP 24 asked to find 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 5/8 being the vulgar

fraction. Ahmes converted 5/8 to an Egyptian fraction series by (4 + 1)/8 = 1/2 + 8, making his answer 16 + 1/2 + 1/8.

The RMP includes 15 problems of this type, with #24 being the easiest. Each of the following 14 problems produced more difficult vulgar fractions, all easily converted to 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, http://reisnerpapyri.blogspot.com . 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; ⁵⁄₅ = 1.

Adding/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: In arithmetic and number theory the least common multiple or lowest common multiple (lcm) or smallest common multiple of two integers a and b is the smallest positive integer that is a multiple of both a and b. ... In mathematics, the lowest common denominator or least common denominator (abbreviated LCD) is the least common multiple of the denominators of a set of vulgar fractions. ...

²⁄₃ + ¹⁄₄   =   ⁸⁄₁₂ + ³⁄₁₂   =   ¹¹⁄₁₂.

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:
• sum the fractional parts:
• put it all back together:

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. A unit fraction is a rational number written as a fraction where the numerator is one and the denominator is a positive integer. ...

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. In mathematics, the absolute value (or modulus1) of a real number is its numerical value without regard to its sign. ... In algebra, an improper fraction is a fraction where the absolute value of the numerator is greater than the absolute value of the denominator. ...

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:

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׳⁄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 that cannot be expressed as integers (i.e. are not of the form ^a⁄₁) can also be written as decimals. For example

2.75 = ¹¹⁄₄.

The mark after the integer is a decimal point, though in many countries it is represented by a comma. If a number is a decimal fraction (i.e., its denominator only has 2 and/or 5 as its prime factors) then it can be written as a finite decimal. If it is rational then it can be written as a recurring decimal, and if it is irrational, then as a non-recurring decimal with an infinite number of digits after the decimal point. The decimal separator is used to mark the boundary between the integer and the fractional parts of a decimal numeral. ... Decimal, or denary, notation is the most common way of writing the base 10 numeral system, which uses various symbols for ten distinct quantities (0, 1, 2, 3, 4, 5, 6, 7, 8 and 9, called digits) together with the decimal point and the sign symbols + (plus) and − (minus) to... In number theory, the prime factors of a positive integer are the prime numbers that divide into that integer exactly, without leaving a remainder. ... A recurring decimal is an expression representing a real number in the decimal numeral system, in which after some point the same sequence of digits repeats infinitely. ...

See also

• algebra
• elementary algebra
• abstract algebra
• ordinal fraction
• Egyptian fraction