Question

Convert the (base 10)decimal numbers into 14-bit, excess-16 floating point representation.

a) 44₁₀

b) 22.487₁₀

c) -44₁₀

d) -78.8125₁₀

Answer 1

a)
44
Converting 44.0 to binary
   Convert decimal part first, then the fractional part
   > First convert 44 to binary
   Divide 44 successively by 2 until the quotient is 0
       > 44/2 = 22, remainder is 0
       > 22/2 = 11, remainder is 0
       > 11/2 = 5, remainder is 1
       > 5/2 = 2, remainder is 1
       > 2/2 = 1, remainder is 0
       > 1/2 = 0, remainder is 1
   Read remainders from the bottom to top as 101100
   So, 44 of decimal is 101100 in binary
   > Now, Convert 0.00000000 to binary
       > Multiply 0.00000000 with 2.    Since 0.00000000 is < 1. then add 0 to result
       > This is equal to 1, so, stop calculating
   0.0 of decimal is .0 in binary
   so, 44.0 in binary is 101100.0
44.0 in simple binary => 101100.0
so, 44.0 in normal binary is 101100.0 => 1.011 * 2^5

14-bit format:
--------------------
sign bit is 0(+ve)
exponent bits are (16+5=21) => 10101
   Divide 21 successively by 2 until the quotient is 0
       > 21/2 = 10, remainder is 1
       > 10/2 = 5, remainder is 0
       > 5/2 = 2, remainder is 1
       > 2/2 = 1, remainder is 0
       > 1/2 = 0, remainder is 1
   Read remainders from the bottom to top as 10101
   So, 21 of decimal is 10101 in binary
frac/significant bits are 01100000

so, 44.0 in 14-bit format is 0 10101 01100000

b)
22.487
Converting 22.487 to binary
   Convert decimal part first, then the fractional part
   > First convert 22 to binary
   Divide 22 successively by 2 until the quotient is 0
       > 22/2 = 11, remainder is 0
       > 11/2 = 5, remainder is 1
       > 5/2 = 2, remainder is 1
       > 2/2 = 1, remainder is 0
       > 1/2 = 0, remainder is 1
   Read remainders from the bottom to top as 10110
   So, 22 of decimal is 10110 in binary
   > Now, Convert 0.48700000 to binary
       > Multiply 0.48700000 with 2.    Since 0.97400000 is < 1. then add 0 to result
       > Multiply 0.97400000 with 2.    Since 1.94800000 is >= 1. then add 1 to result
       > Multiply 0.94800000 with 2.    Since 1.89600000 is >= 1. then add 1 to result
       > Multiply 0.89600000 with 2.    Since 1.79200000 is >= 1. then add 1 to result
       > Multiply 0.79200000 with 2.    Since 1.58400000 is >= 1. then add 1 to result
       > Multiply 0.58400000 with 2.    Since 1.16800000 is >= 1. then add 1 to result
       > Multiply 0.16800000 with 2.    Since 0.33600000 is < 1. then add 0 to result
       > Multiply 0.33600000 with 2.    Since 0.67200000 is < 1. then add 0 to result
       > Multiply 0.67200000 with 2.    Since 1.34400000 is >= 1. then add 1 to result
       > Multiply 0.34400000 with 2.    Since 0.68800000 is < 1. then add 0 to result
       > Multiply 0.68800000 with 2.    Since 1.37600000 is >= 1. then add 1 to result
       > Multiply 0.37600000 with 2.    Since 0.75200000 is < 1. then add 0 to result
       > Multiply 0.75200000 with 2.    Since 1.50400000 is >= 1. then add 1 to result
       > Multiply 0.50400000 with 2.    Since 1.00800000 is >= 1. then add 1 to result
       > Multiply 0.00800000 with 2.    Since 0.01600000 is < 1. then add 0 to result
       > Multiply 0.01600000 with 2.    Since 0.03200000 is < 1. then add 0 to result
       > Multiply 0.03200000 with 2.    Since 0.06400000 is < 1. then add 0 to result
       > Multiply 0.06400000 with 2.    Since 0.12800000 is < 1. then add 0 to result
       > Multiply 0.12800000 with 2.    Since 0.25600000 is < 1. then add 0 to result
       > Multiply 0.25600000 with 2.    Since 0.51200000 is < 1. then add 0 to result
       > Multiply 0.51200000 with 2.    Since 1.02400000 is >= 1. then add 1 to result
       > Multiply 0.02400000 with 2.    Since 0.04799999 is < 1. then add 0 to result
       > Multiply 0.04799999 with 2.    Since 0.09599999 is < 1. then add 0 to result
       > Multiply 0.09599999 with 2.    Since 0.19199997 is < 1. then add 0 to result
       > Multiply 0.19199997 with 2.    Since 0.38399994 is < 1. then add 0 to result
       > Multiply 0.38399994 with 2.    Since 0.76799989 is < 1. then add 0 to result
       > Multiply 0.76799989 with 2.    Since 1.53599977 is >= 1. then add 1 to result
       > Multiply 0.53599977 with 2.    Since 1.07199955 is >= 1. then add 1 to result
       > Multiply 0.07199955 with 2.    Since 0.14399910 is < 1. then add 0 to result
       > Multiply 0.14399910 with 2.    Since 0.28799820 is < 1. then add 0 to result
       > Multiply 0.28799820 with 2.    Since 0.57599640 is < 1. then add 0 to result
       > Multiply 0.57599640 with 2.    Since 1.15199280 is >= 1. then add 1 to result
       > Multiply 0.15199280 with 2.    Since 0.30398560 is < 1. then add 0 to result
       > Multiply 0.30398560 with 2.    Since 0.60797119 is < 1. then add 0 to result
       > Multiply 0.60797119 with 2.    Since 1.21594238 is >= 1. then add 1 to result
       > Multiply 0.21594238 with 2.    Since 0.43188477 is < 1. then add 0 to result
       > Multiply 0.43188477 with 2.    Since 0.86376953 is < 1. then add 0 to result
       > Multiply 0.86376953 with 2.    Since 1.72753906 is >= 1. then add 1 to result
       > Multiply 0.72753906 with 2.    Since 1.45507812 is >= 1. then add 1 to result
       > Multiply 0.45507812 with 2.    Since 0.91015625 is < 1. then add 0 to result
       > Multiply 0.91015625 with 2.    Since 1.82031250 is >= 1. then add 1 to result
       > Multiply 0.82031250 with 2.    Since 1.64062500 is >= 1. then add 1 to result
       > Multiply 0.64062500 with 2.    Since 1.28125000 is >= 1. then add 1 to result
       > Multiply 0.28125000 with 2.    Since 0.56250000 is < 1. then add 0 to result
       > Multiply 0.56250000 with 2.    Since 1.12500000 is >= 1. then add 1 to result
       > Multiply 0.12500000 with 2.    Since 0.25000000 is < 1. then add 0 to result
       > Multiply 0.25000000 with 2.    Since 0.50000000 is < 1. then add 0 to result
       > Multiply 0.50000000 with 2.    Since 1.00000000 is >= 1. then add 1 to result
       > This is equal to 1, so, stop calculating
   0.4869999999999983 of decimal is .011111001010110000001000001100010010011011101001 in binary
   so, 22.487 in binary is 10110.011111001010110000001000001100010010011011101001
22.487 in simple binary => 10110.011111001010110000001000001100010010011011101001
so, 22.487 in normal binary is 10110.011111001010110000001000001100010010011011101001 => 1.01100111 * 2^4

14-bit format:
--------------------
sign bit is 0(+ve)
exponent bits are (16+4=20) => 10100
   Divide 20 successively by 2 until the quotient is 0
       > 20/2 = 10, remainder is 0
       > 10/2 = 5, remainder is 0
       > 5/2 = 2, remainder is 1
       > 2/2 = 1, remainder is 0
       > 1/2 = 0, remainder is 1
   Read remainders from the bottom to top as 10100
   So, 20 of decimal is 10100 in binary
frac/significant bits are 01100111

so, 22.487 in 14-bit format is 0 10100 01100111

c)
-44
Converting 44.0 to binary
   Convert decimal part first, then the fractional part
   > First convert 44 to binary
   Divide 44 successively by 2 until the quotient is 0
       > 44/2 = 22, remainder is 0
       > 22/2 = 11, remainder is 0
       > 11/2 = 5, remainder is 1
       > 5/2 = 2, remainder is 1
       > 2/2 = 1, remainder is 0
       > 1/2 = 0, remainder is 1
   Read remainders from the bottom to top as 101100
   So, 44 of decimal is 101100 in binary
   > Now, Convert 0.00000000 to binary
       > Multiply 0.00000000 with 2.    Since 0.00000000 is < 1. then add 0 to result
       > This is equal to 1, so, stop calculating
   0.0 of decimal is .0 in binary
   so, 44.0 in binary is 101100.0
-44.0 in simple binary => 101100.0
so, -44.0 in normal binary is 101100.0 => 1.011 * 2^5

14-bit format:
--------------------
sign bit is 1(-ve)
exponent bits are (16+5=21) => 10101
   Divide 21 successively by 2 until the quotient is 0
       > 21/2 = 10, remainder is 1
       > 10/2 = 5, remainder is 0
       > 5/2 = 2, remainder is 1
       > 2/2 = 1, remainder is 0
       > 1/2 = 0, remainder is 1
   Read remainders from the bottom to top as 10101
   So, 21 of decimal is 10101 in binary
frac/significant bits are 01100000

so, -44.0 in 14-bit format is 1 10101 01100000

d)
-78.8125
Converting 78.8125 to binary
   Convert decimal part first, then the fractional part
   > First convert 78 to binary
   Divide 78 successively by 2 until the quotient is 0
       > 78/2 = 39, remainder is 0
       > 39/2 = 19, remainder is 1
       > 19/2 = 9, remainder is 1
       > 9/2 = 4, remainder is 1
       > 4/2 = 2, remainder is 0
       > 2/2 = 1, remainder is 0
       > 1/2 = 0, remainder is 1
   Read remainders from the bottom to top as 1001110
   So, 78 of decimal is 1001110 in binary
   > Now, Convert 0.81250000 to binary
       > Multiply 0.81250000 with 2.    Since 1.62500000 is >= 1. then add 1 to result
       > Multiply 0.62500000 with 2.    Since 1.25000000 is >= 1. then add 1 to result
       > Multiply 0.25000000 with 2.    Since 0.50000000 is < 1. then add 0 to result
       > Multiply 0.50000000 with 2.    Since 1.00000000 is >= 1. then add 1 to result
       > This is equal to 1, so, stop calculating
   0.8125 of decimal is .1101 in binary
   so, 78.8125 in binary is 1001110.1101
-78.8125 in simple binary => 1001110.1101
so, -78.8125 in normal binary is 1001110.1101 => 1.00111011 * 2^6

14-bit format:
--------------------
sign bit is 1(-ve)
exponent bits are (16+6=22) => 10110
   Divide 22 successively by 2 until the quotient is 0
       > 22/2 = 11, remainder is 0
       > 11/2 = 5, remainder is 1
       > 5/2 = 2, remainder is 1
       > 2/2 = 1, remainder is 0
       > 1/2 = 0, remainder is 1
   Read remainders from the bottom to top as 10110
   So, 22 of decimal is 10110 in binary
frac/significant bits are 00111011

so, -78.8125 in 14-bit format is 1 10110 00111011