Mixed Number
Fractions & DecimalsA mixed number combines a whole number and a proper fraction to represent a quantity greater than one.
Formula
a\,\frac{b}{c} = \frac{ac + b}{c}
Definition
A mixed number has a whole number part and a fraction part together. It shows amounts bigger than $1$. For example, $2$ and $3/4$ means $2$ whole things plus $3/4$ of another.
Example
If you bake $2$ full pans of brownies and have $1/2$ a pan left over, you have $2$ and $1/2$ pans. The $2$ is the whole number part and the $1/2$ is the fraction part.
Key Insight
Mixed numbers are great for everyday life ("I ran $3$ and a half miles") but improper fractions are easier for calculating. Knowing how to switch between them is a key math skill.
Definition
A mixed number $n$ and $a/b$ (where $0 \le a < b$) represents the value $n + a/b$, with $n$ a non-negative integer and $a/b$ a proper fraction. To convert to an improper fraction: multiply the whole number by the denominator, add the numerator, and keep the same denominator: $(nb + a)/b$.
Example
Add $1$ and $2/3 + 2$ and $3/4$. Convert: $5/3 + 11/4$. LCD $= 12$: $20/12 + 33/12 = 53/12 = 4$ and $5/12$. It is much harder (and error-prone) to add without converting first.
Key Insight
When subtracting mixed numbers, sometimes you must "borrow" from the whole number part, similar to regrouping in subtraction: $3$ and $1/4 - 1$ and $3/4$ requires rewriting $3$ and $1/4$ as $2$ and $5/4$.
Definition
A mixed number is the canonical representation of a rational number $q > 0$ as $\lfloor q \rfloor + \text{frac}(q)$, where $\lfloor q \rfloor$ is the greatest-integer function and $\text{frac}(q) = q - \lfloor q \rfloor$ in $[0,1)$. This is a consequence of the division algorithm applied to the numerator and denominator of $q$.
Example
The continued fraction $[3; 2, 1, 4] = 3 + \cfrac{1}{2 + \cfrac{1}{1 + \frac{1}{4}}} = 3 + \cfrac{1}{2 + 4/5} = 3 + 5/14 = 47/14$. The leading term $3$ is the whole-number part of the mixed number $3$ and $5/14$.
Key Insight
The mixed-number representation is unique for rationals but not for reals in general. For real numbers, the decomposition into integer and fractional parts underpins the definition of the sawtooth wave and is used in the theory of Fourier series and equidistribution (Weyl's theorem).