Implementation of complex optimization-based real-time decision making is often not possible due to the limited computational capabilities of embedded computing platforms. Compared to widespread floating-point arithmetic, fundamentally more efficient fixed-point arithmetic can enable implementation in low cost devices and can result in significant performance improvements for meeting tight real-time deadlines. However, fixed-point arithmetic presents additional challenges, such as having to bound the peak value of each variable to prevent overflow errors. First, we show how the linearized KKT system, the solution of which forms the computational bottleneck in interior-point and active-set methods, can be altered to allow for reliable overflow-free fixed-point implementation. We then focus on first-order methods and present an analysis that enables one to predict a priori the numerical error introduced by a given word-length fixed-point implementation. For instance, this approach can allow for the implementation of online optimization-based controllers at megahertz sampling rates on a low cost device.