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.

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.

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).

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

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.

Let us consider two numbers X=M.2E and Y=N.2F

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.

Since the binary digit b-A is the first bit in the representation of the real number, it is called as the most significant bit(MSB) of the number.

(101.01)2=1*22+0*21+1*20+0*2-1+1*2-2=(5.25)10

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