information:extraction, linguistic:voice_change
Mixed numbers

A mixed number (also called a mixed fraction or mixed numeral) is the sum of a non-zero integer and a proper fraction, conventionally written by juxtaposition (or concatenation) of the two parts, without the use of an intermediate plus (+) or minus (−) sign. When the fraction is written horizontally, a space is added between the integer and fraction to separate them.

As a basic example, two entire cakes and three quarters of another cake might be written as 2 3 4 {\displaystyle 2{\tfrac {3}{4}}} cakes or 2 3 / 4 {\displaystyle 2\ \,3/4} cakes, with the numeral 2 {\displaystyle 2} representing the whole cakes and the fraction 3 4 {\displaystyle {\tfrac {3}{4}}} representing the additional partial cake juxtaposed; this is more concise than the more explicit notation 2 + 3 4 {\displaystyle 2+{\tfrac {3}{4}}} cakes. The mixed number 2 + 3 / 4 is pronounced "two and three quarters", with the integer and fraction portions connected by the word and. Subtraction or negation is applied to the entire mixed numeral, so − 2 3 4 {\displaystyle -2{\tfrac {3}{4}}} means − (2 + 3 4). {\displaystyle -{\bigl (}2+{\tfrac {3}{4}}{\bigr)}.}

Any mixed number can be converted to an improper fraction by applying the rules of adding unlike quantities. For example, 2 + 3 4 = 8 4 + 3 4 = 11 4. {\displaystyle 2+{\tfrac {3}{4}}={\tfrac {8}{4}}+{\tfrac {3}{4}}={\tfrac {11}{4}}.} Conversely, an improper fraction can be converted to a mixed number using division with remainder, with the proper fraction consisting of the remainder divided by the divisor. For example, since 4 goes into 11 twice, with 3 left over, 11 4 = 2 + 3 4. {\displaystyle {\tfrac {11}{4}}=2+{\tfrac {3}{4}}.}

In primary school, teachers often insist that every fractional result should be expressed as a mixed number. Outside school, mixed numbers are commonly used for describing measurements, for instance 2 + 1 / 2 hours or 5 3/16 inches, and remain widespread in daily life and in trades, especially in regions that do not use the decimalized metric system. However, scientific measurements typically use the metric system, which is based on decimal fractions, and starting from the secondary school level, mathematics pedagogy treats every fraction uniformly as a rational number, the quotient p / q of integers, leaving behind the concepts of "improper fraction" and "mixed number". College students with years of mathematical training are sometimes confused when re-encountering mixed numbers because they are used to the convention that juxtaposition in algebraic expressions means multiplication.