Converting to IEEE 23.59375 23 12 1 6 0 3 1 1 1 0 1 . .59375 1 .1875 0 .375 0 .75 1 .5 1 .0 10111.10011 1.011110011 * 2**4 Sign field = 0 Exp field = 10000011 Frac = 01111001100000000000000 IEEE value: 01000001101111001100000000000000 4 1 B C C 0 0 0 **************************************************************** 0.1 0 . .1 0 .2 0 .4 0 .8 1 .6 1 .2 0 .4 \ 0 .8 \ Repeat section 1 .6 / 1 .2 / .... 0.0001100110011001100110011001100110011001100110011 .... 1.100110011001100110011001100110011001100110011 * 2**-4 Sign field = 0 Exp field = 01111011 Frac = 100110011001100110011001100110011001100110011 ~ 10011001100110011001101 IEEE value: 00111101110011001100110011001101 3 D C C C C C D Conversion in other direction Sign field = 0 Exp field = 01111011 Frac = 10011001100110011001101 exp base 2 is 123 - 127 fraction as integer is 5033165 1 + 5033165/(2**24) 1 + 5033165/8388608 1 + 0.60000002384185791015625 1.60000002384185791015625 1.60000002384185791015625 / 16 = 0.100000001490116119384765625 IEEE value: 00111101110011001100110011001100 3 D C C C C C C Conversion in other direction Sign field = 0 Exp field = 01111011 Frac = 10011001100110011001100 exp base 2 is 123 - 127 fraction as integer is 5033164 1 + 5033164/(2**24) 1 + 5033164/8388608 1 + 0.599999904632568359375 1.599999904632568359375 1.599999904632568359375 / 16 = 0.0999999940395355224609375 The alternatives 0.100000001490116119384765625 0.0999999940395355224609375 64-bit IEEE floating point 0.0001100110011001100110011001100110011001100110011 .... 1.100110011001100110011001100110011001100110011 * 2**-4 Sign field = 1 Exp field = 1111111011 Frac = 100110011001100110011001100110011001100110011001100110011 ... ~ 1001100110011001100110011001100110011001100110011010 1111111101110011100110011001100110011001100110011001100110011010 F F 7 3 9 9 9 9 9 9 9 9 9 9 9 A Conversion in other direction Sign field = 0 Exp field = 1111111011 Frac = 1001100110011001100110011001100110011001100110011010 exp base 2 is 1019 - 1023 or -4 fraction as integer is 2702159776422298 1 + 2702159776422298 / (2**52) 1 + 2702159776422298 / 4503599627370496 1 + 0.600000000000000088817841970012523233890533447265625 1.600000000000000088817841970012523233890533447265625 1.600000000000000088817841970012523233890533447265625 / 16 is 0.1000000000000000055511151231257827021181583404541015625 0.0001100110011001100110011001100110011001100110011 .... 1.100110011001100110011001100110011001100110011 * 2**-4 Sign field = 1 Exp field = 1111111011 Frac = 100110011001100110011001100110011001100110011001100110011 ... ~ 1001100110011001100110011001100110011001100110011001 1111111101110011100110011001100110011001100110011001100110011001 F F 7 3 9 9 9 9 9 9 9 9 9 9 9 9 Conversion in other direction Sign field = 0 Exp field = 1111111011 Frac = 1001100110011001100110011001100110011001100110011001 exp base 2 is 1019 - 1023 or -4 fraction as integer is 2702159776422298 1 + 2702159776422297 / (2**52) 1 + 2702159776422297 / 4503599627370496 1 + 0.5999999999999998667732370449812151491641998291015625 1.5999999999999998667732370449812151491641998291015625 1.5999999999999998667732370449812151491641998291015625 / 16 is 0.09999999999999999167332731531132594682276248931884765625 The alternatives 0.1000000000000000055511151231257827021181583404541015625 0.09999999999999999167332731531132594682276248931884765625