Question

convert

5378.27 and 21.78 to Floating Point Standard (FPS) modified (16 bits) by changing 23 bit fractional part to a 7 bit fractional part. then multiply

Answer 1

Answer: 0 10001111 1100100

Explanation:
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Converting 5378.27 to binary
   Convert decimal part first, then the fractional part
   > First convert 5378 to binary
   Divide 5378 successively by 2 until the quotient is 0
      > 5378/2 = 2689, remainder is 0
      > 2689/2 = 1344, remainder is 1
      > 1344/2 = 672, remainder is 0
      > 672/2 = 336, remainder is 0
      > 336/2 = 168, remainder is 0
      > 168/2 = 84, remainder is 0
      > 84/2 = 42, remainder is 0
      > 42/2 = 21, remainder 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 1010100000010
   So, 5378 of decimal is 1010100000010 in binary
   > Now, Convert 0.27000000 to binary
      > Multiply 0.27000000 with 2.  Since 0.54000000 is < 1. then add 0 to result
      > Multiply 0.54000000 with 2.  Since 1.08000000 is >= 1. then add 1 to result
      > Multiply 0.08000000 with 2.  Since 0.16000000 is < 1. then add 0 to result
      > Multiply 0.16000000 with 2.  Since 0.32000000 is < 1. then add 0 to result
      > Multiply 0.32000000 with 2.  Since 0.64000000 is < 1. then add 0 to result
      > Multiply 0.64000000 with 2.  Since 1.28000000 is >= 1. then add 1 to result
      > This is equal to 1, so, stop calculating
   0.27000000000043656 of decimal is .010001 in binary
   so, 5378.27 in binary is 1010100000010.010001
5378.27 in simple binary => 1010100000010.010001
so, 5378.27 in normal binary is 1010100000010.010001 => 1.0101 * 2^12

16-bit format:
--------------------
sign bit is 0(+ve)
exp bits are (127+12=139) => 10001011
   Divide 139 successively by 2 until the quotient is 0
      > 139/2 = 69, remainder is 1
      > 69/2 = 34, remainder is 1
      > 34/2 = 17, remainder is 0
      > 17/2 = 8, remainder is 1
      > 8/2 = 4, remainder is 0
      > 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 10001011
   So, 139 of decimal is 10001011 in binary
frac bits are 0101000

so, 5378.27 in 16-bit format is 0 10001011 0101000

Converting 21.78 to binary
   Convert decimal part first, then the fractional part
   > First convert 21 to binary
   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
   > Now, Convert 0.78000000 to binary
      > Multiply 0.78000000 with 2.  Since 1.56000000 is >= 1. then add 1 to result
      > Multiply 0.56000000 with 2.  Since 1.12000000 is >= 1. then add 1 to result
      > Multiply 0.12000000 with 2.  Since 0.24000000 is < 1. then add 0 to result
      > Multiply 0.24000000 with 2.  Since 0.48000000 is < 1. then add 0 to result
      > Multiply 0.48000000 with 2.  Since 0.96000000 is < 1. then add 0 to result
      > Multiply 0.96000000 with 2.  Since 1.92000000 is >= 1. then add 1 to result
      > This is equal to 1, so, stop calculating
   0.7800000000000011 of decimal is .110001 in binary
   so, 21.78 in binary is 10101.110001
21.78 in simple binary => 10101.110001
so, 21.78 in normal binary is 10101.110001 => 1.010111 * 2^4

16-bit format:
--------------------
sign bit is 0(+ve)
exp bits are (127+4=131) => 10000011
   Divide 131 successively by 2 until the quotient is 0
      > 131/2 = 65, remainder is 1
      > 65/2 = 32, remainder is 1
      > 32/2 = 16, remainder is 0
      > 16/2 = 8, remainder is 0
      > 8/2 = 4, remainder is 0
      > 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 10000011
   So, 131 of decimal is 10000011 in binary
frac bits are 0101110

so, 21.78 in 16-bit format is 0 10000011 0101110

5378.27 x 21.78 = 1.0101 * 2^12 x 1.010111 * 2^4 = 1.1100100011 * 2^16

1.1100100011 * 2^16

16-bit format:
--------------------
sign bit is 0(+ve)
exp bits are (127+16=143) => 10001111
   Divide 143 successively by 2 until the quotient is 0
      > 143/2 = 71, remainder is 1
      > 71/2 = 35, remainder is 1
      > 35/2 = 17, remainder is 1
      > 17/2 = 8, remainder is 1
      > 8/2 = 4, remainder is 0
      > 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 10001111
   So, 143 of decimal is 10001111 in binary
frac bits are 1100100

so, 5378.27 x 21.78 in 16-bit format is 0 10001111 1100100

Answer: 0 10001111 1100100