Binary ↘ Double: The 64 Bit Double Precision IEEE 754 Binary Floating Point Standard Representation Number 1 - 011 0111 1001 - 0111 1000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 0000 Converted and Written as a Base Ten Decimal System Number (as a Double)
1 - 011 0111 1001 - 0111 1000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 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:
011 0111 1001
The last 52 bits contain the mantissa:
0111 1000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 0000
2. Convert the exponent from binary (from base 2) to decimal (in base 10).
The exponent is allways a positive integer.
011 0111 1001(2) =
0 × 210 + 1 × 29 + 1 × 28 + 0 × 27 + 1 × 26 + 1 × 25 + 1 × 24 + 1 × 23 + 0 × 22 + 0 × 21 + 1 × 20 =
0 + 512 + 256 + 0 + 64 + 32 + 16 + 8 + 0 + 0 + 1 =
512 + 256 + 64 + 32 + 16 + 8 + 1 =
889(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 = 889 - 1023 = -134
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).
0111 1000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 0000(2) =
0 × 2-1 + 1 × 2-2 + 1 × 2-3 + 1 × 2-4 + 1 × 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 + 1 × 2-47 + 0 × 2-48 + 0 × 2-49 + 0 × 2-50 + 0 × 2-51 + 0 × 2-52 =
0 + 0.25 + 0.125 + 0.062 5 + 0.031 25 + 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 + 0.000 000 000 000 007 105 427 357 601 001 858 711 242 675 781 25 + 0 + 0 + 0 + 0 + 0 =
0.25 + 0.125 + 0.062 5 + 0.031 25 + 0.000 000 000 000 007 105 427 357 601 001 858 711 242 675 781 25 =
0.468 750 000 000 007 105 427 357 601 001 858 711 242 675 781 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)1 × (1 + 0.468 750 000 000 007 105 427 357 601 001 858 711 242 675 781 25) × 2-134 =
-1.468 750 000 000 007 105 427 357 601 001 858 711 242 675 781 25 × 2-134 =
-0.000 000 000 000 000 000 000 000 000 000 000 000 000 067 441 692 491 025 122 254 640 657 310 803 419 080 961 909 566 010 736 121 337 085 815 375 253 159 327 872 514 511 152 603 803 548 644 806 342 693 414 080 713 409 930 467 605 590 820 312 5
1 - 011 0111 1001 - 0111 1000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 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) = -0.000 000 000 000 000 000 000 000 000 000 000 000 000 067 441 692 491 025 122 254 640 657 310 803 419 080 961 909 566 010 736 121 337 085 815 375 253 159 327 872 514 511 152 603 803 548 644 806 342 693 414 080 713 409 930 467 605 590 820 312 5(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: