What decimal floating point number does this big-endian IEEE 754 single precisio

Question

What decimal floating point number does this big-endian IEEE 754 single precision number represent: n = 0 times 6C84_3175? For explanation, I want you to document the steps you perform, in this order: (1) What is n in binary; (2) What is the value of the sign bit; What does this value signify about the final number; (3) What are the binary and decimal values of the biased exponent; (4) What is the binary value of the mantissa, with the 1. part preceding the binary point? (5) What is the decimal value of the unbiased exponent; (6) What is the decimal value of the mantissa, with the leading 1. part? (7) What is the final decimal real number, written in the form [-] d.ddddddddd ddddded times 10^e where d represents a decimal digit 0-9 and there is an optional leading negative sign. Write exactly 15 digits after the decimal point (even if they are 0's) and round the final 15th digit up or down as required based on the value of the 16th digit (16th digit

Explanation / Answer

1.

So the number is -765543210.98765432101234 * 1018

we can also write this as -765543210987654321012340000

Now let's cover this into binary

100111100100111110000101

and our floating point representation will be like

1)

-765.54321098765432101234 * 1024

2)

100.111100100111110000101 * 221

3)

1.00111100100111110000101* 223

4)

Mantissa is 00111100100111110000101

5)

Biased exponent is

11011000

6)

11101100 00011110 01001111 10000101

7)

in hexadecimal it will be 0xEC1E4F85

3.

First of all it will be converted to binary number and that is

0.0000000000000000000000000000000000000000000000000000000011110100011001010110001100000011110001110001

i.e, 1.1110100011001010110001100000011110001110001 * 2-57, now exponent's value is -57.

sign is positive and mantissa is 1110100011001010110001100000011110001110001.

now we can write

and in hexa decimal the number is

0x3C6E8CAC6078E200

Now, in order to write it in double precision floating point format we need to write this number in normalized form

So the answer will be

1) 132.487359893280127146769272484e-19

2) 111.1010001100101011000110000001111000111000100000000000000000000000000000000000000000000000000000000000000000

3) 1.1110100011001010110001100000011110001110001 *2-57

4) 1110100011001010110001100000011110001110001000000000

5) Biased exponent (in binary): 01111000110

   Biased exponent (in decimal): 966

6) 0011110001101110100011001010110001100000011110001110001000000000

7) 0x3C6E8CAC6078E200

Sign Exponent Mantissa 1 11011000 00111100100111110000101

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