A type of approximation in scientific computing is introduced due to the representation of real numbers in computers. This approximation is further magnified by performing arithmetic operations on these real numbers. In this section, we will discuss this representation of real numbers, arithmetic operations on these numbers, and their possible impact on the results of the computation. These approximation errors not only arise in computerized computations, however; they may arise in non-computerized manual computation because of the rounding done to reduce the complexity. However, it is not the case that these approximations arise only in the case of computerized computations. They can also be observed in non-computerized, manual computations because of rounding done to reduce complexities in calculations.

Before advancing the discussion of the computerized representation of real numbers, let's first recall the well-known scientific notation used in mathematics. In scientific notation, to simplify the representation of a very large or very small number into a short form, we write nearly the same quantity multiplied by some powers of 10. Also, in scientific notation, numbers are represented in the form of "a multiplied by 10 to the power b" that is, a X 10b. For example, 0.000000987654 and 987,654 can represented as 9.87654 x 10^-7 and 9.87654 x 10^5 respectively. In this representation, the exponent is an integer quantity and the coefficient is a real number called mantissa.

The Institute of Electrical and Electronics Engineers (IEEE) has standardized the floating-point number representation in IEEE 754. Most modern machines use this standard as it addresses most of the problems found in various floating-point number representations. The latest version of this standard is published in 2008 and is known as IEEE 754-2008. The standard defines arithmetic formats, interchange formats, rounding rules, operations, and exception handling. It also includes recommendations for advanced exception handling, additional operations, and evaluation of expressions, and tells us how to achieve reproducible results.