A fixed-point number representation is a real data type for a number that has a fixed number of digits after (and sometimes also before) the radix point. Fixed-point number representation can be compared to the more complicated floating-point number representation.
In computing, a fixed-point number representation is a real data type for a number that has a fixed number of digits after (and sometimes also before) the radix point (after the decimal point '$.$' in English decimal notation). Fixed-point number representation can be compared to the more complicated (and more computationally demanding) floating-point number representation.
Fixed-point numbers are useful for representing fractional values, usually in base $2$ or base $10$, when the executing processor has no floating point unit (FPU) as is the case for older or low-cost embedded microprocessors and microcontrollers, if fixed-point provides improved performance or accuracy for the application at hand, or if their use is more natural for the problem (such as for the representation of angles).
A value of a fixed-point data type is essentially an integer that is scaled by an implicit specific factor determined by the type. For example, the value $1.23$ can be represented as 1230 in a fixed-point data type with scaling factor of $\frac 1 {1000}$, and the value $1,230,000$ can be represented as $1230$ with a scaling factor of $1000$. Unlike floating-point data types, the scaling factor is the same for all values of the same type, and does not change during the entire computation.
The scaling factor is usually a power of $10$ (for human convenience) or a power of $2$ (for computational efficiency). However, other scaling factors may be used occasionally, e.g. a time value in hours may be represented as a fixed-point type with a scale factor of $\frac 1 {3600}$ to obtain values with one-second accuracy.
The maximum value of a fixed-point type is simply the largest value that can be represented in the underlying integer type multiplied by the scaling factor; and similarly for the minimum value.
To convert a number from a fixed point type with scaling factor $R$ to another type with scaling factor $S$, the underlying integer must be multiplied by $R$ and divided by $S$; that is, multiplied by the ratio $\frac R S$. Thus, for example, to convert the value $1.23 = \frac {123} {100} $ from a type with scaling factor $R=\frac 1 {100}$ to one with scaling factor $S=\frac 1 {1000}$, the underlying integer $123$ must be multiplied by $\frac {\frac 1 {100}} {\frac 1 {1000}} = 10$, yielding the representation $\frac {1230} {1000}$. If $S$ does not divide $R$ (in particular, if the new scaling factor $S$ is greater than the original $R$), the new integer will have to be rounded. The rounding rules and methods are usually part of the language's specification.
To add or subtract two values of the same fixed-point type, it is sufficient to add or subtract the underlying integers, and keep their common scaling factor. The result can be exactly represented in the same type, as long as no overflow occurs (i.e. provided that the sum of the two integers fits in the underlying integer type). If the numbers have different fixed-point types, with different scaling factors, then one of them must be converted to the other before the sum.
To multiply two fixed-point numbers, it suffices to multiply the two underlying integers, and assume that the scaling factor of the result is the product of their scaling factors. This operation involves no rounding. For example, multiplying the numbers $123$ scaled by $\frac 1 {1000}$ ($0.123$) and $25$ scaled by $\frac 1 {10}$ ($2.5$) yields the integer $123×25 = 3075$ scaled by $\frac 1 {1000}\times\frac 1 {10} = \frac 1 {10000}$, that is $\frac {3075} {10000} = 0.3075$. If the two operands belong to the same fixed-point type, and the result is also to be represented in that type, then the product of the two integers must be explicitly multiplied by the common scaling factor; in this case the result may have to be rounded, and overflow may occur. For example, if the common scaling factor is $\frac 1 {100}$, multiplying $1.23$ by $0.25$ entails multiplying $123$ by $25$ to yield $3075$ with an intermediate scaling factor of $\frac 1 {10000}$. This then must be multiplied by $\frac 1 {100}$ to yield either $31$ ($0.31$) or $30$ ($0.30$), depending on the rounding method used, to result in a final scale factor of $\frac 1 {100}$.
To divide two fixed-point numbers, one takes the integer quotient of their underlying integers, and assumes that the scaling factor is the quotient of their scaling factors. The first division involves rounding in general. For example, division of $3456$ scaled by $\frac 1 {100}$ ($34.56$) and $1234$ scaled by $\frac 1 {1000}$ ($1.234$) yields the integer $3456 \div 1234 = 3$ (rounded) with scale factor $\frac {\frac 1 {100}}{\frac 1 {1000}} = 10$, that is, $30$. One can obtain a more accurate result by first converting the dividend to a more precise type: in the same example, converting $3456$ scaled by $\frac 1 {100}$ ($34.56$) to $3,456,000$ scaled by $\frac 1 {100000}$, before dividing by $1234$ scaled by $\frac 1 {1000}$ ($1.234$), would yield $3456000 \div 1234 = 2801$ (rounded) with scaling factor $\frac {\frac 1 {100000}} {\frac 1 {1000}} = \frac 1 {100}$, that is $28.01$ (instead of $30$). If both operands and the desired result all have the same scaling factor, then the quotient of the two integers must be explicitly multiplied by that common scaling factor.
Source: Wikipedia
