Numerical errors arise from the use of approximations to represent exact mathematical operations and quantities. These include truncation errors, which result when approximations are used to represent exact mathematical procedures, and round-off errors, which result when numbers having limited significant figures are used to represent exact numbers. For both types, the relationship between the exact, or true, result and the approximation can be formulated as = and = 100% Where is used to designate the exact value of the error. The subscript t is included to designate that this is the "true" error and designates the true percent relative error. Suppose that you have the task of measuring the lengths of a bridge and a rivet and come up with 9999 and 9 cm, respectively. If the true values are 10,000 and 10 cm, respectively, compute (a) the true error and (b) the true percent relative error for each case