Mixed Number
A 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, showing amounts bigger than $1$: for example, $2$ and $3/4$ means $2$ whole things plus $3/4$ of another. Formally, 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; converting to an improper fraction multiplies the whole number by the denominator, adds the numerator, and keeps the same denominator: $(nb + a)/b$. This is the canonical representation of a rational number $q > 0$ as $\lfloor q \rfloor + \text{frac}(q)$, the greatest-integer function plus the fractional remainder in $[0,1)$, a consequence of the division algorithm applied to the numerator and denominator of $q$.
Example
If you bake $2$ full pans of brownies and have $1/2$ a pan left over, you have $2$ and $1/2$ pans, where $2$ is the whole number part and $1/2$ is the fraction part. To add $1$ and $2/3 + 2$ and $3/4$, convert to improper fractions first: $5/3 + 11/4$; with LCD $12$, that is $20/12 + 33/12 = 53/12 = 4$ and $5/12$, much easier than trying to add without converting. The continued fraction $[3; 2, 1, 4] = 3 + \cfrac{1}{2 + \cfrac{1}{1 + \frac{1}{4}}} = 3 + \cfrac{1}{2 + 4/5} = 3 + 5/14 = 47/14$, where the leading term $3$ is the whole-number part of the mixed number $3$ and $5/14$.
Key Insight
Mixed numbers are great for everyday life ("I ran $3$ and a half miles") but improper fractions are easier for calculating, and knowing how to switch between them is a key math skill; when subtracting mixed numbers, sometimes you must "borrow" from the whole number part, similar to regrouping in subtraction, rewriting $3$ and $1/4$ as $2$ and $5/4$ before subtracting $1$ and $3/4$. 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).