Binary ↘ Double: The 64 Bit Double Precision IEEE 754 Binary Floating Point Standard Representation Number 1 - 101 1000 1000 - 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1010 Converted and Written as a Base Ten Decimal System Number (as a Double)
1 - 101 1000 1000 - 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1010: 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:
101 1000 1000
The last 52 bits contain the mantissa:
0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1010
2. Convert the exponent from binary (from base 2) to decimal (in base 10).
The exponent is allways a positive integer.
101 1000 1000(2) =
1 × 210 + 0 × 29 + 1 × 28 + 1 × 27 + 0 × 26 + 0 × 25 + 0 × 24 + 1 × 23 + 0 × 22 + 0 × 21 + 0 × 20 =
1,024 + 0 + 256 + 128 + 0 + 0 + 0 + 8 + 0 + 0 + 0 =
1,024 + 256 + 128 + 8 =
1,416(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 = 1,416 - 1023 = 393
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).
0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1010(2) =
0 × 2-1 + 0 × 2-2 + 0 × 2-3 + 0 × 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 + 1 × 2-49 + 0 × 2-50 + 1 × 2-51 + 0 × 2-52 =
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 + 0.000 000 000 000 001 776 356 839 400 250 464 677 810 668 945 312 5 + 0 + 0.000 000 000 000 000 444 089 209 850 062 616 169 452 667 236 328 125 + 0 =
0.000 000 000 000 001 776 356 839 400 250 464 677 810 668 945 312 5 + 0.000 000 000 000 000 444 089 209 850 062 616 169 452 667 236 328 125 =
0.000 000 000 000 002 220 446 049 250 313 080 847 263 336 181 640 625(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 + 0.000 000 000 000 002 220 446 049 250 313 080 847 263 336 181 640 625) × 2393 =
-1.000 000 000 000 002 220 446 049 250 313 080 847 263 336 181 640 625 × 2393 =
-20 173 827 172 554 018 151 581 712 087 357 741 417 046 437 875 363 871 126 899 470 915 964 507 546 788 591 108 881 527 925 719 984 425 642 858 963 179 405 312
1 - 101 1000 1000 - 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1010 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) = -20 173 827 172 554 018 151 581 712 087 357 741 417 046 437 875 363 871 126 899 470 915 964 507 546 788 591 108 881 527 925 719 984 425 642 858 963 179 405 312(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: