Binary ↘ Double: The 64 Bit Double Precision IEEE 754 Binary Floating Point Standard Representation Number 1 - 101 1110 1010 - 1101 1011 0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 0011 Converted and Written as a Base Ten Decimal System Number (as a Double)
1 - 101 1110 1010 - 1101 1011 0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 0011: 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 1110 1010
The last 52 bits contain the mantissa:
1101 1011 0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 0011
2. Convert the exponent from binary (from base 2) to decimal (in base 10).
The exponent is allways a positive integer.
101 1110 1010(2) =
1 × 210 + 0 × 29 + 1 × 28 + 1 × 27 + 1 × 26 + 1 × 25 + 0 × 24 + 1 × 23 + 0 × 22 + 1 × 21 + 0 × 20 =
1,024 + 0 + 256 + 128 + 64 + 32 + 0 + 8 + 0 + 2 + 0 =
1,024 + 256 + 128 + 64 + 32 + 8 + 2 =
1,514(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 = 1,514 - 1023 = 491
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).
1101 1011 0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 0011(2) =
1 × 2-1 + 1 × 2-2 + 0 × 2-3 + 1 × 2-4 + 1 × 2-5 + 0 × 2-6 + 1 × 2-7 + 1 × 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 + 1 × 2-48 + 0 × 2-49 + 0 × 2-50 + 1 × 2-51 + 1 × 2-52 =
0.5 + 0.25 + 0 + 0.062 5 + 0.031 25 + 0 + 0.007 812 5 + 0.003 906 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 + 0.000 000 000 000 007 105 427 357 601 001 858 711 242 675 781 25 + 0.000 000 000 000 003 552 713 678 800 500 929 355 621 337 890 625 + 0 + 0 + 0.000 000 000 000 000 444 089 209 850 062 616 169 452 667 236 328 125 + 0.000 000 000 000 000 222 044 604 925 031 308 084 726 333 618 164 062 5 =
0.5 + 0.25 + 0.062 5 + 0.031 25 + 0.007 812 5 + 0.003 906 25 + 0.000 000 000 000 007 105 427 357 601 001 858 711 242 675 781 25 + 0.000 000 000 000 003 552 713 678 800 500 929 355 621 337 890 625 + 0.000 000 000 000 000 444 089 209 850 062 616 169 452 667 236 328 125 + 0.000 000 000 000 000 222 044 604 925 031 308 084 726 333 618 164 062 5 =
0.855 468 750 000 011 324 274 851 176 596 712 321 043 014 526 367 187 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)1 × (1 + 0.855 468 750 000 011 324 274 851 176 596 712 321 043 014 526 367 187 5) × 2491 =
-1.855 468 750 000 011 324 274 851 176 596 712 321 043 014 526 367 187 5 × 2491 =
-11 862 644 491 200 842 879 201 122 355 931 645 273 395 849 286 328 554 505 509 708 659 898 823 459 077 918 365 448 752 612 137 538 310 219 034 529 554 008 446 082 002 931 751 363 082 337 111 769 088
1 - 101 1110 1010 - 1101 1011 0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 0011 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) = -11 862 644 491 200 842 879 201 122 355 931 645 273 395 849 286 328 554 505 509 708 659 898 823 459 077 918 365 448 752 612 137 538 310 219 034 529 554 008 446 082 002 931 751 363 082 337 111 769 088(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: