Let the mantissa be represented by 23 bits plus a sign bit and let the exponent be represented by 7 bits plus a sign bit.

A fixed point representation of numbers allows us to cover a range of numbers, say, xmax-xmin with a resolution

For a two’s complement representation, the truncation error is always negative and falls in the range

We can represent the numbers in binary float point as

In floating point representation, the mantissa is either rounded or truncated. Due to non-uniform resolution, the corresponding error in a floating point representation is proportional to the number being quantized.

The round-off error is independent of the type of fixed point representation. The maximum error that can be introduced through rounding is 0.5(2-b+2-bm) and this can be either positive or negative, depending on the value of x. Therefore, the round-off error is symmetric about zero and falls in the range

The binary point between the digits b0 and b1 does not exist physically in the computer. Simply, the logic circuits of the computer are designed such that the computations result in numbers that correspond to the assumed location of this point.

According to the IEEE 754 standard, for a 32-bit machine, single precision floating point number is represented as X=(-1)s.2E-127(M).

The binary floating point representation commonly used in practice, consists of a mantissa M, which is the fractional part of the number and falls in the range 1/2<M<1, multiplied by the exponential factor 2E, where the exponent E is either a negative or positive integer. Hence a number X is represented as X= M.2E(1/2<M<1).

The truncation error for the sign magnitude representation is symmetric about zero and falls in the range