A real number X can be represented as X=\\(\\sum_{i=-A}^B b_i r^{-i}\\) where bi represents the digit, ‘r’ is the radix or base, A is the number of integer digits, and B is the number of fractional digits.

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

The number (-3/8) is stored in the computer as the 2’s complement of (3/8)

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 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

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

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

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).