How do you represent real numbers in binary?

Real numbers in binary are represented using a system called floating-point representation.

In computer science, real numbers are typically represented using a standard called IEEE 754 floating-point standard. This standard represents real numbers in binary form using three components: the sign, the exponent, and the mantissa (also known as the significand).

The sign is a single bit that represents whether the number is positive or negative. A '0' indicates a positive number, while a '1' indicates a negative number.

The exponent is used to represent the magnitude of the number. It is a binary number that indicates the power to which 2 should be raised. The exponent is biased in the IEEE 754 standard, meaning that a certain value (for single-precision floating-point numbers, this is 127) is subtracted from the actual exponent to get the stored exponent. This allows for the representation of both very large and very small numbers.

The mantissa represents the precision bits of the number. It is a fraction in binary, and it represents the actual digits of the number. The mantissa is interpreted as a binary fraction in the range [1.0, 2.0) or [0.0, 1.0) depending on the standard used. The leading '1' before the binary point is often not stored as it is implicit in normalised numbers.

For example, the number -12.375 in binary would be represented as follows: the sign bit is '1' (as the number is negative), the exponent is '10000010' (which represents 130 in decimal, or 3 after subtracting the bias of 127), and the mantissa is '10001100000000000000000' (which represents the fraction 1.5).

This system allows for the representation of a wide range of real numbers, including very large and very small numbers, with a high degree of precision. However, it also has some limitations, such as the inability to represent certain numbers exactly (due to the finite precision of the mantissa), and the possibility of rounding errors.