Floating point representation

Why fixed point is not enough

A fixed point number puts the binary point in a set position, so the range and the precision are both fixed. Floating point splits the number into a mantissa (the significant digits) and an exponent (how far to move the point), which lets the same number of bits cover a far wider range.

$\text{Giá trị}=\text{định trị}\times2^{\text{số mũ}}$

Both parts are two's complement

The mantissa and the exponent are each held in two's complement, so both can be negative. The binary point sits immediately after the first bit of the mantissa, which is the sign bit.

Normalisation — the rule to memorise

A normalised number has its first two mantissa bits different: a positive number starts $0.1\ldots$ and a negative number starts $1.0\ldots$

Two reasons it is required. First, without it the same value has many representations ($0.1100\times2^3$ and $0.0110\times2^4$ are the same number), so comparison becomes unreliable. Second, unnormalised numbers waste leading bits that carry no information, which loses precision.

The trade-off inside a fixed word length

Give more bits to the mantissa and you gain precision but lose range; give more to the exponent and the reverse happens. That trade-off is a favourite exam question, and the answer is always that the total is fixed so one gains at the other's expense.

Errors

Overflow: the number is too large for the exponent field. Underflow: too small, so it rounds towards zero. Rounding error: many decimal fractions such as $0{,}1$ have no exact binary representation, so a stored value differs slightly from the intended one. That is why financial software does not test two floating point values for exact equality.