In this chapter, we briefly discuss floating point number systems according to the recent IEEE 754 standard, considering the effect of errors due to floating point representation on algebraic systems. We look at solving systems of equations using Gaussian elimination with a view towards the effect of floating point error. We give particular attention to catastrophic cancellation, which arises with the subtraction of two nearly equal numbers and also to the need for pivoting to improve the accuracy of the solution. We show how Gaussian elimination can be implemented using a series of Gauss transforms. These Gauss transforms enable us to develop the intimate connection between the LU decomposition and the Gaussian elimination procedure. This discussion presents a natural segway into the topic of the Cholesky decomposition on symmetric, positive definite matrices. We then introduce the important idea of the condition number of a matrix and its relation to the error in the solution of a system of linear equations. Several examples are presented throughout to enhance the presentation of the material.