In computer systems, exponential numbers are represented using floating-point notation, a method designed to handle both very large and very small real numbers.

Floating-point notation is analogous to scientific notation, which you may be familiar with from mathematics. In scientific notation, numbers are expressed as the product of two components: a base number and an exponent. For instance, the number $300$ can be represented as $3 \times 10^2$ in scientific notation.

In the context of computer systems, floating-point notation operates similarly but utilizes binary numbers instead of decimal. A floating-point number consists of three main components: a sign bit, an exponent, and a fraction (also referred to as the significand or mantissa). The sign bit indicates whether the number is positive or negative, the exponent specifies the power to which the base number is raised, and the fraction represents the base number itself.

The IEEE 754 standard governs floating-point arithmetic. According to this standard, single-precision floating-point numbers are encoded using $32$ bits, which include $1$ bit for the sign, $8$ bits for the exponent, and $23$ bits for the fraction. In contrast, double-precision floating-point numbers are represented with $64$ bits, comprising $1$ bit for the sign, $11$ bits for the exponent, and $52$ bits for the fraction.

This method of representing exponential numbers allows for a vast range of values with varying degrees of precision. However, it is crucial to recognize that not all real numbers can be represented exactly, which can lead to rounding errors. This limitation is a fundamental consideration in numerical computation that computer scientists must be mindful of.

In summary, exponential numbers in computer systems are represented using floating-point notation, a binary version of scientific notation. While this approach facilitates the representation of a wide array of values, it can also result in rounding errors.