How do you represent a floating-point number using IEEE 754 format?

A floating-point number in IEEE 754 format is represented using three components: sign, exponent, and mantissa.

The IEEE 754 standard is a technical standard for floating-point computation established in 1985 by the Institute of Electrical and Electronics Engineers (IEEE). The standard defines the representation and mathematical behaviour of floating-point numbers, which are used in computer systems to represent real numbers.

The first component, the sign, is a single bit that indicates whether the number is positive or negative. If the sign bit is 0, the number is positive; if it's 1, the number is negative.

The second component, the exponent, is an 8-bit field for single-precision numbers (or 11 bits for double-precision). The exponent is biased, meaning that a value is added to the actual exponent to get the stored exponent. For single-precision, the bias is 127, and for double-precision, it's 1023. This bias allows the representation of both very small and very large numbers.

The third component, the mantissa (also known as the significand), is the remaining bits. For single-precision, it's 23 bits, and for double-precision, it's 52 bits. The mantissa represents the precision bits of the number. It's interpreted as a binary fraction in the range [1.0, 2.0) for normalised numbers, or [0.0, 1.0) for denormalised numbers.

To calculate the value of the floating-point number, the formula is (-1)^sign * 2^(exponent - bias) * 1.mantissa. The '1.' before the mantissa is implicit for normalised numbers, meaning it's always assumed to be there and isn't actually stored in the number's representation.

In summary, the IEEE 754 format allows for a wide range of real numbers to be represented in a standardised way, facilitating accurate and consistent computations across different computer systems. It's important to understand this format as it's widely used in computer science and related fields.