Binary ↘ Double: The 64 Bit Double Precision IEEE 754 Binary Floating Point Standard Representation Number 0 - 110 0000 0000 - 0011 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)
0 - 110 0000 0000 - 0011 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.
0
The next 11 bits contain the exponent:
110 0000 0000
The last 52 bits contain the mantissa:
0011 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.
110 0000 0000(2) =
1 × 210 + 1 × 29 + 0 × 28 + 0 × 27 + 0 × 26 + 0 × 25 + 0 × 24 + 0 × 23 + 0 × 22 + 0 × 21 + 0 × 20 =
1,024 + 512 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 =
1,024 + 512 =
1,536(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 = 1,536 - 1023 = 513
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).
0011 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000(2) =
0 × 2-1 + 0 × 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 + 0 + 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 =
0.125 + 0.062 5 =
0.187 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)0 × (1 + 0.187 5) × 2513 =
1.187 5 × 2513 =
31 843 543 833 613 668 111 488 309 370 738 884 552 763 493 823 906 934 272 093 458 428 839 189 571 424 674 069 904 451 458 146 395 640 763 825 663 192 904 370 777 665 471 678 373 103 622 779 916 389 449 728
0 - 110 0000 0000 - 0011 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) = 31 843 543 833 613 668 111 488 309 370 738 884 552 763 493 823 906 934 272 093 458 428 839 189 571 424 674 069 904 451 458 146 395 640 763 825 663 192 904 370 777 665 471 678 373 103 622 779 916 389 449 728(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: