Binary ↘ Double: The 64 Bit Double Precision IEEE 754 Binary Floating Point Standard Representation Number 0 - 100 0001 1100 - 1110 0100 1111 0001 1000 0010 0000 0110 0000 0000 0000 0000 0100 Converted and Written as a Base Ten Decimal System Number (as a Double)
0 - 100 0001 1100 - 1110 0100 1111 0001 1000 0010 0000 0110 0000 0000 0000 0000 0100: 64 bit double precision IEEE 754 binary floating point standard representation number converted to decimal system (base ten)
1. Identify the elements that make up the binary representation of the number:
The first bit (the leftmost) indicates the sign,
1 = negative, 0 = positive.
0
The next 11 bits contain the exponent:
100 0001 1100
The last 52 bits contain the mantissa:
1110 0100 1111 0001 1000 0010 0000 0110 0000 0000 0000 0000 0100
2. Convert the exponent from binary (from base 2) to decimal (in base 10).
The exponent is allways a positive integer.
100 0001 1100(2) =
1 × 210 + 0 × 29 + 0 × 28 + 0 × 27 + 0 × 26 + 0 × 25 + 1 × 24 + 1 × 23 + 1 × 22 + 0 × 21 + 0 × 20 =
1,024 + 0 + 0 + 0 + 0 + 0 + 16 + 8 + 4 + 0 + 0 =
1,024 + 16 + 8 + 4 =
1,052(10)
3. Adjust the exponent.
Subtract the excess bits: 2(11 - 1) - 1 = 1023,
that is due to the 11 bit excess/bias notation.
The exponent, adjusted = 1,052 - 1023 = 29
4. Convert the mantissa from binary (from base 2) to decimal (in base 10).
The mantissa represents the fractional part of the number (what comes after the whole part of the number, separated from it by a comma).
1110 0100 1111 0001 1000 0010 0000 0110 0000 0000 0000 0000 0100(2) =
1 × 2-1 + 1 × 2-2 + 1 × 2-3 + 0 × 2-4 + 0 × 2-5 + 1 × 2-6 + 0 × 2-7 + 0 × 2-8 + 1 × 2-9 + 1 × 2-10 + 1 × 2-11 + 1 × 2-12 + 0 × 2-13 + 0 × 2-14 + 0 × 2-15 + 1 × 2-16 + 1 × 2-17 + 0 × 2-18 + 0 × 2-19 + 0 × 2-20 + 0 × 2-21 + 0 × 2-22 + 1 × 2-23 + 0 × 2-24 + 0 × 2-25 + 0 × 2-26 + 0 × 2-27 + 0 × 2-28 + 0 × 2-29 + 1 × 2-30 + 1 × 2-31 + 0 × 2-32 + 0 × 2-33 + 0 × 2-34 + 0 × 2-35 + 0 × 2-36 + 0 × 2-37 + 0 × 2-38 + 0 × 2-39 + 0 × 2-40 + 0 × 2-41 + 0 × 2-42 + 0 × 2-43 + 0 × 2-44 + 0 × 2-45 + 0 × 2-46 + 0 × 2-47 + 0 × 2-48 + 0 × 2-49 + 1 × 2-50 + 0 × 2-51 + 0 × 2-52 =
0.5 + 0.25 + 0.125 + 0 + 0 + 0.015 625 + 0 + 0 + 0.001 953 125 + 0.000 976 562 5 + 0.000 488 281 25 + 0.000 244 140 625 + 0 + 0 + 0 + 0.000 015 258 789 062 5 + 0.000 007 629 394 531 25 + 0 + 0 + 0 + 0 + 0 + 0.000 000 119 209 289 550 781 25 + 0 + 0 + 0 + 0 + 0 + 0 + 0.000 000 000 931 322 574 615 478 515 625 + 0.000 000 000 465 661 287 307 739 257 812 5 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0.000 000 000 000 000 888 178 419 700 125 232 338 905 334 472 656 25 + 0 + 0 =
0.5 + 0.25 + 0.125 + 0.015 625 + 0.001 953 125 + 0.000 976 562 5 + 0.000 488 281 25 + 0.000 244 140 625 + 0.000 015 258 789 062 5 + 0.000 007 629 394 531 25 + 0.000 000 119 209 289 550 781 25 + 0.000 000 000 931 322 574 615 478 515 625 + 0.000 000 000 465 661 287 307 739 257 812 5 + 0.000 000 000 000 000 888 178 419 700 125 232 338 905 334 472 656 25 =
0.894 310 118 164 868 050 882 887 473 562 732 338 905 334 472 656 25(10)
5. Put all the numbers into expression to calculate the double precision floating point decimal value:
(-1)Sign × (1 + Mantissa) × 2(Adjusted exponent) =
(-1)0 × (1 + 0.894 310 118 164 868 050 882 887 473 562 732 338 905 334 472 656 25) × 229 =
1.894 310 118 164 868 050 882 887 473 562 732 338 905 334 472 656 25 × 229 =
1 017 000 000.750 000 476 837 158 203 125
0 - 100 0001 1100 - 1110 0100 1111 0001 1000 0010 0000 0110 0000 0000 0000 0000 0100 converted from a 64 bit double precision IEEE 754 binary floating point standard representation number to a decimal system number, written in base ten (double) = 1 017 000 000.750 000 476 837 158 203 125(10)
Spaces were used to group digits: for binary, by 4, for decimal, by 3.
More operations with 64 bit double precision IEEE 754 binary floating point standard representation numbers: