Question

Question 3: Represent the following numbers in the memory using 32-bit floating-point representation, then in Hexadecimal. 1. (-1313.3125) 2.(0.00011011) Question 4:_Represent the following data in the memory (show your work): 1.93 // Integer : Two bytes 2.-156 //Integer: Two bytes using Two's complement Question 5: Find the real number (in decimal) which is represented in the memory using 32-bit floating-point representation as follows, please show your steps follows: JOJO AOL

Answer 1

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)

a) -1313.3125

Since -1313.3125 is a negative number therefore sign bit=1

Exponent is decided by the nearest smaller or equal to 2ⁿ number. Since 2¹⁰ = 1024 which is the nearest smaller number to 1313.3125 , therefore n=10.127 is the unique number for 32 bit floating point representation. It is known as bias

Thus 127 + 10 =137 = 10001001 in binary representation.

Mantissa. Convert the decimal represenatation of number -1313.3125 to binary i.e -10100100001.0101 and move the decimal point so that there is only one number present at the left

-10100100001.0101 = -1.01001000010101 x 2¹⁰

Now, consider the fractional part and represented as 23 bits by adding zeros.

therefore mantissa = 01001000010101000000000

Thus the floating point representation of -1313.3125 is 1 10001001 01001000010101000000000

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By hexadecimal method :

_-1313.3125 = -521.5

Steps:

• Divide the number by 16.
• Get the integer quotient for the next iteration.
• Get the remainder for the hex digit.
• Repeat the steps until the quotient is equal to 0.

division by 16	Quotient	Remainder(decimal)	Remainder(hex)
1313/16	82	1	1
82/16	5	2	2

5/16

0

5

5

The fractional part of number is found by multiplying .3125 and 16 = .3125 x 16 = 5

Thus 0.3125₁₀ = 0.50000000000₁₆

Therfore hexadecimal of whole number + fractional number = -521.5

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b) (0.00011011)₂ = 0.10546875( Decimal value)

Since 0.10546875 is a positive number therefore sign bit=0

Exponent is decided by the nearest smaller or equal to 2ⁿ number. Since 2⁰ = 1 which is the nearest smaller number to 0.10546875 , therefore n=0.127 is the unique number for 32 bit floating point representation. It is known as bias

Thus 127 + 0 =127 = 1111111 in binary representation.

Mantissa.(0.00011011)₂ is binary , move the decimal point so that there is only one number present at the left. Since already there is only one number in the left, so no changes

0.00011011 = 0.00011011

Now, consider the fractional part and represented as 23 bits by adding zeros.

therefore mantissa = 00011011000000000000000

Thus the floating-point representation of -1313.3125 is 0 1111111 00011011000000000000000

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By hexadecimal method :

(0.00011011)₂ =  0.1B

Steps:

• Take 4 binary digits and convert them using binary to hexadecimal
• 0001 = 1 in hexadecimal
• 1011 = B in hexadecimal