**1. To convert the floating point into decimal, we have 3 elements in a 32-bit floating point representation:**

i) Sign

ii) Exponent

iii) Mantissa

**Sign**bit is the first bit of the binary representation. '1' implies negative number and '0' implies positive number.

**Example:**11000001110100000000000000000000 This is negative number.**Exponent**is decided by the next 8 bits of binary representation. 127 is the unique number for 32 bit floating point representation. It is known as bias. It is determined by 2^{k-1}-1 where 'k' is the number of bits in exponent field.There are 3 exponent bits in 8-bit representation and 8 exponent bits in 32-bit representation.

Thus

bias = 3 for 8 bit conversion (2^{3-1}-1 = 4-1 = 3)

bias = 127 for 32 bit conversion. (2^{8-1}-1 = 128-1 = 127)**Example:**01000001110100000000000000000000

10000011 = (131)_{2}

131-127 = 4

Hence the exponent of 2 will be 4 i.e. 2

^{4}= 16.**Mantissa**is calculated from the remaining 23 bits of the binary representation. It consists of '1' and a fractional part which is determined by:**Example:**01000001110100000000000000000000

The fractional part of mantissa is given by:

1*(1/2) + 0*(1/4) + 1*(1/8) + 0*(1/16) +……… = 0.625

Thus the mantissa will be 1 + 0.625 = 1.625

The decimal number hence given as: Sign*Exponent*Mantissa = (-1)*(16)*(1.625) = -26

**2. To convert the decimal into floating point, we have 3 elements in a 32-bit floating point representation:**

i) Sign (MSB)

ii) Exponent (8 bits after MSB)

iii) Mantissa (Remaining 23 bits)

**Sign bit**is the first bit of the binary representation. '1' implies negative number and '0' implies positive number.

Example: To convert -17 into 32-bit floating point representation Sign bit = 1**Exponent**is decided by the nearest smaller or equal to 2^{n}number. For 17, 16 is the nearest 2^{n}. Hence the exponent of 2 will be 4 since 2^{4}= 16. 127 is the unique number for 32 bit floating point representation. It is known as bias. It is determined by 2^{k-1}-1 where 'k' is the number of bits in exponent field.Thus bias = 127 for 32 bit. (2

^{8-1}-1 = 128-1 = 127)Now, 127 + 4 = 131 i.e. 10000011 in binary representation.

**Mantissa:**17 in binary = 10001.Move the binary point so that there is only one bit from the left. Adjust the exponent of 2 so that the value does not change. This is normalizing the number. 1.0001 x 2

^{4}. Now, consider the fractional part and represented as 23 bits by adding zeros.00010000000000000000000

Thus the floating point representation of -17 is 1 10000011 00010000000000000000000

Related Link:

https://www.youtube.com/watch?v=03fhijH6e2w

More questions on number representation:

http://quiz.geeksforgeeks.org/number-representation/

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