Fixed Point and Floating Point Number Representations

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• two major approaches to store real numbers (i.e., numbers with fractional component

• This representation has fixed number of bits for integer part and for fractional part

• f given fixed-point representation is IIII.FFF

• then you can store minimum value is 0000.0001 and maximum value is 9999.9999.

• Assume number is using 32-bit format which reserve 1 bit for the sign, 15 bits for the integer part and 16 bits for the fractional part. Then, -43.625 is represented as following:

• Where, 0 is used to represent + and 1 is used to represent. 000000000101011 is 15 bit binary value for decimal 43 and 1010000000000000 is 16 bit binary value for fractional 0.625.

• The advantage of using a fixed-point representation is performance and disadvantage is  relatively limited range of values that they can represent.

• So, it is usually inadequate for numerical analysis as it does not allow enough numbers and accuracy

• This representation does not reserve a specific number of bits for the integer part or the fractional part. Instead it reserves a certain number of bits for the number (called the mantissa or significand) and a certain number of bits to say where within that number the decimal place sits (called the exponent).

• e first part represents a signed fixed point number called mantissa. The second part of designates the position of the decimal (or binary) point and is called the exponent.

• Only the mantissa m and the exponent e are physically represented in the register

• floating-point binary number is represented in a similar manner except that is uses base 2 for the exponent. A floating-point number is said to be normalized if the most significant digit of the mantissa is 1.

• The floating point representation is more flexible