Mixed numbers
Definitions
- A mixed number is one in the form \(a\frac{b}{c}\), where all \(a,b\) and \(c\) are integers. For example \(1\frac{2}{3}\) is a mixed number.
- The whole part is \(a\), and the fractional part is \(\frac{b}{c}\). The fractional part of a mixed number must be a proper fraction.
- The notation of a mixed number is a shorthand for \(\left(a+\frac{b}{c}\right)\). The parentheses are important because they affect the order of operations.
Negative mixed numbers
A negative sign before a mixed number means that the entire mixed number is negative, not that only the whole part is negative; i.e.,
\[-2\frac{3}{4}=-\left(2+\frac{3}{4}\right)=-2-\frac{3}{4}\]Interconversion between rational and mixed numbers
A mixed number \(a\frac{b}{c}\) is equivalent to the rational number \(\frac{a\times c+b}{c}\).
\begin{align} a\tfrac{b}{c} &= \phantom{-}\left(a\tfrac{b}{c}\right) = \phantom{-}\tfrac{a\times c}{c}+\tfrac{b}{c} = \tfrac{a\times c+b}{c} \\ -a\tfrac{b}{c} &= -\left(a\tfrac{b}{c}\right) = -\left(\tfrac{a\times c}{c}+\tfrac{b}{c}\right) = \tfrac{-(a\times c+b)}{c} \end{align}The whole part of a proper fraction is zero; we do not write it as a mixed number.
We can convert an improper fraction to a mixed number by dividing the numerator by the denominator. The quotient of division is the whole part, and the remainder makes the numerator of the fractional part. For example, to convert \(\frac{30}{7}\) to a mixed number, we divide \(30\) by \(7\). The quotient is \(4\), and the remainder is \(2\). Therefore \(\frac{30}{7}=4\frac{2}{7}\).
Multiplication and division
The easiest and most efficient method is to convert them to rational numbers first. Afterwards, we multiply or divide them as rational numbers, which is quite straightforward. For example, to multiply \(1\frac{2}{3}\times 2\frac{3}{4}\), we do it as follows.
\[1\tfrac{2}{3}\times 2\tfrac{3}{4}=\frac{5}{3}\times\frac{11}{4}=\frac{55}{12}=4\tfrac{7}{12}\]Addition and subtraction
The easiest way is to do the whole parts together and the fractional parts together. Converting to rational numbers before adding or subtracting may actually complicate the process a whole lot.
\begin{align} 2\tfrac{3}{7}+1\tfrac{1}{3} &= (2+1)+\left(\tfrac{3}{7}+\tfrac{1}{3}\right) \\ &= 3+\left(\tfrac{9}{21}+\tfrac{7}{21}\right) \\ &=3+\tfrac{16}{21}=3\tfrac{16}{21} \end{align}If one (or both) of the mixed numbers is (are) negative, the order of operations may not be very obvious. All we need to do is to understand the notation properly and ‘expand’ the shorthand.
\begin{align} -2\tfrac{1}{3}-1\tfrac{1}{2} &= -\left(2+\tfrac{1}{3}\right)-\left(1+\tfrac{1}{2}\right) \\ &= -2-\tfrac{1}{3}-1-\tfrac{1}{2}=-3-\tfrac{2}{6}-\tfrac{3}{6} \\ &= -3-\tfrac{5}{6}=-\left(3+\tfrac{5}{6}\right)=-3\tfrac{5}{6} \end{align}Raising to a power
The easiest way is to first convert a mixed number to a rational number and do the operation on the rational number. If needed, we can then convert the result to a different form.