Binary ↘ Double: The 64 Bit Double Precision IEEE 754 Binary Floating Point Standard Representation Number 1 - 111 1011 0000 - 1111 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 Converted and Written as a Base Ten Decimal System Number (as a Double)
1 - 111 1011 0000 - 1111 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000: 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.
1
The next 11 bits contain the exponent:
111 1011 0000
The last 52 bits contain the mantissa:
1111 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000
2. Convert the exponent from binary (from base 2) to decimal (in base 10).
The exponent is allways a positive integer.
111 1011 0000(2) =
1 × 210 + 1 × 29 + 1 × 28 + 1 × 27 + 0 × 26 + 1 × 25 + 1 × 24 + 0 × 23 + 0 × 22 + 0 × 21 + 0 × 20 =
1,024 + 512 + 256 + 128 + 0 + 32 + 16 + 0 + 0 + 0 + 0 =
1,024 + 512 + 256 + 128 + 32 + 16 =
1,968(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 = 1,968 - 1023 = 945
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).
1111 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000(2) =
1 × 2-1 + 1 × 2-2 + 1 × 2-3 + 1 × 2-4 + 0 × 2-5 + 0 × 2-6 + 0 × 2-7 + 0 × 2-8 + 0 × 2-9 + 0 × 2-10 + 0 × 2-11 + 0 × 2-12 + 0 × 2-13 + 0 × 2-14 + 0 × 2-15 + 0 × 2-16 + 0 × 2-17 + 0 × 2-18 + 0 × 2-19 + 0 × 2-20 + 0 × 2-21 + 0 × 2-22 + 0 × 2-23 + 0 × 2-24 + 0 × 2-25 + 0 × 2-26 + 0 × 2-27 + 0 × 2-28 + 0 × 2-29 + 0 × 2-30 + 0 × 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 + 0 × 2-50 + 0 × 2-51 + 0 × 2-52 =
0.5 + 0.25 + 0.125 + 0.062 5 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 =
0.5 + 0.25 + 0.125 + 0.062 5 =
0.937 5(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)1 × (1 + 0.937 5) × 2945 =
-1.937 5 × 2945 =
-576 219 052 035 140 936 834 249 218 033 086 467 050 863 328 143 015 352 884 111 595 484 927 004 649 656 694 748 752 721 867 260 585 853 547 005 468 699 777 057 051 151 782 648 650 961 768 946 850 177 177 917 490 901 200 673 390 701 152 790 983 810 790 373 037 217 688 365 661 181 917 351 252 623 915 178 534 160 875 839 385 105 007 167 167 478 363 480 766 219 209 072 665 690 112
1 - 111 1011 0000 - 1111 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 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) = -576 219 052 035 140 936 834 249 218 033 086 467 050 863 328 143 015 352 884 111 595 484 927 004 649 656 694 748 752 721 867 260 585 853 547 005 468 699 777 057 051 151 782 648 650 961 768 946 850 177 177 917 490 901 200 673 390 701 152 790 983 810 790 373 037 217 688 365 661 181 917 351 252 623 915 178 534 160 875 839 385 105 007 167 167 478 363 480 766 219 209 072 665 690 112(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: