Real numbers in binary are represented using a method known as floating-point representation.

In computer science, the IEEE 754 floating-point standard is commonly employed to represent real numbers in binary format. This standard breaks down real numbers into three main components: the sign, the exponent, and the mantissa (also referred to as the significand).

The sign component consists of a single bit that indicates whether the number is positive or negative. A value of ‘0’ signifies a positive number, while a value of ‘1’ denotes a negative number.

The exponent serves to represent the magnitude of the number. It is a binary number that specifies the power to which $2$ should be raised. In the IEEE 754 standard, the exponent is biased, meaning that a specific value (for single-precision floating-point numbers, this value is $127$) is subtracted from the actual exponent to derive the stored exponent. This biasing technique enables the representation of both very large and very small numbers.

The mantissa captures the precision bits of the number. It is expressed as a binary fraction and contains the significant digits of the number. The mantissa is interpreted as a binary fraction within the range $[1.0, 2.0)$ or $[0.0, 1.0)$, depending on the specific standard used. In normalized numbers, the leading ‘1’ before the binary point is typically not stored explicitly, as it is considered implicit.

For instance, the number $-12.375$ in binary can be represented as follows: the sign bit is ‘1’ (indicating that the number is negative), the exponent is ‘10000010’ (which corresponds to $130$ in decimal or $3$ after subtracting the bias of $127$), and the mantissa is ‘10001100000000000000000’ (representing the fraction $1.5$).

This floating-point representation system allows for a wide range of real numbers to be represented, including both very large and very small values, with a high degree of precision. However, it also comes with certain limitations, such as the inability to represent specific numbers exactly (due to the finite precision of the mantissa) and the potential for rounding errors.