Binary ↘ Double: The 64 Bit Double Precision IEEE 754 Binary Floating Point Standard Representation Number 0 - 011 0110 1110 - 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0100 Converted and Written as a Base Ten Decimal System Number (as a Double)
0 - 011 0110 1110 - 0000 0000 0000 0000 0000 0000 0000 0000 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:
011 0110 1110
The last 52 bits contain the mantissa:
0000 0000 0000 0000 0000 0000 0000 0000 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.
011 0110 1110(2) =
0 × 210 + 1 × 29 + 1 × 28 + 0 × 27 + 1 × 26 + 1 × 25 + 0 × 24 + 1 × 23 + 1 × 22 + 1 × 21 + 0 × 20 =
0 + 512 + 256 + 0 + 64 + 32 + 0 + 8 + 4 + 2 + 0 =
512 + 256 + 64 + 32 + 8 + 4 + 2 =
878(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 = 878 - 1023 = -145
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 0100(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 + 0 × 2-49 + 1 × 2-50 + 0 × 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 + 0.000 000 000 000 000 888 178 419 700 125 232 338 905 334 472 656 25 + 0 + 0 =
0.000 000 000 000 000 888 178 419 700 125 232 338 905 334 472 656 25 =
0.000 000 000 000 000 888 178 419 700 125 232 338 905 334 472 656 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 + 0.000 000 000 000 000 888 178 419 700 125 232 338 905 334 472 656 25) × 2-145 =
1.000 000 000 000 000 888 178 419 700 125 232 338 905 334 472 656 25 × 2-145 =
0.000 000 000 000 000 000 000 000 000 000 000 000 000 000 022 420 775 429 197 093 048 428 562 488 292 120 961 489 163 166 421 822 236 920 032 449 971 483 246 546 936 950 822 479 927 020 957 101 123 434 002 283 574 262 293 086 576 391 942 799 091 339 111 328 125
0 - 011 0110 1110 - 0000 0000 0000 0000 0000 0000 0000 0000 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) = 0.000 000 000 000 000 000 000 000 000 000 000 000 000 000 022 420 775 429 197 093 048 428 562 488 292 120 961 489 163 166 421 822 236 920 032 449 971 483 246 546 936 950 822 479 927 020 957 101 123 434 002 283 574 262 293 086 576 391 942 799 091 339 111 328 125(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: