these are two equations for two unknowns and we can easily eliminate by dividing the equations by each other then solving for gives ri ln ei =ei ln ni =ni ( we have introduced subscript in since the estimated value for varies with hopefullyri approaches the correct convergence rate as the number of intervals increases and finite precision of floating-point numbers the test procedures above lead to comparison of numbers for checking that calculations were correct such comparison is more complicated than what newcomer might think suppose we have calculation and want to check that the result is what we expect we start with expected =expected true then we proceed with expected =expected false so why is : : $ : the reason is that real numbers cannot in general be exactly represented on computer they must instead be approximated by floating-point number that can only store finite amount of informationusually about digits of real number let us print and with decimalsprint ' \ \ \ ( we see that all of the numbers have an inaccurate digit in the th decimal place because evaluates to and is represented as these two numbers are not equal in generalreal numbers in python have (at most correct decimals when we compute with real numbersthese numbers are inaccurately represented on the computerand arithmetic operations with inaccurate numbers lead to small rounding errors in the final results depending on the type of numerical algorithmthe rounding errors may or may not accumulate