Binary ↘ Double: The 64 Bit Double Precision IEEE 754 Binary Floating Point Standard Representation Number 1 - 100 0011 1101 - 1111 1010 0110 1111 0110 0000 1001 0011 0111 0010 0100 0111 0101 Converted and Written as a Base Ten Decimal System Number (as a Double)
1 - 100 0011 1101 - 1111 1010 0110 1111 0110 0000 1001 0011 0111 0010 0100 0111 0101: 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:
100 0011 1101
The last 52 bits contain the mantissa:
1111 1010 0110 1111 0110 0000 1001 0011 0111 0010 0100 0111 0101
2. Convert the exponent from binary (from base 2) to decimal (in base 10).
The exponent is allways a positive integer.
100 0011 1101(2) =
1 × 210 + 0 × 29 + 0 × 28 + 0 × 27 + 0 × 26 + 1 × 25 + 1 × 24 + 1 × 23 + 1 × 22 + 0 × 21 + 1 × 20 =
1,024 + 0 + 0 + 0 + 0 + 32 + 16 + 8 + 4 + 0 + 1 =
1,024 + 32 + 16 + 8 + 4 + 1 =
1,085(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 = 1,085 - 1023 = 62
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).
1111 1010 0110 1111 0110 0000 1001 0011 0111 0010 0100 0111 0101(2) =
1 × 2-1 + 1 × 2-2 + 1 × 2-3 + 1 × 2-4 + 1 × 2-5 + 0 × 2-6 + 1 × 2-7 + 0 × 2-8 + 0 × 2-9 + 1 × 2-10 + 1 × 2-11 + 0 × 2-12 + 1 × 2-13 + 1 × 2-14 + 1 × 2-15 + 1 × 2-16 + 0 × 2-17 + 1 × 2-18 + 1 × 2-19 + 0 × 2-20 + 0 × 2-21 + 0 × 2-22 + 0 × 2-23 + 0 × 2-24 + 1 × 2-25 + 0 × 2-26 + 0 × 2-27 + 1 × 2-28 + 0 × 2-29 + 0 × 2-30 + 1 × 2-31 + 1 × 2-32 + 0 × 2-33 + 1 × 2-34 + 1 × 2-35 + 1 × 2-36 + 0 × 2-37 + 0 × 2-38 + 1 × 2-39 + 0 × 2-40 + 0 × 2-41 + 1 × 2-42 + 0 × 2-43 + 0 × 2-44 + 0 × 2-45 + 1 × 2-46 + 1 × 2-47 + 1 × 2-48 + 0 × 2-49 + 1 × 2-50 + 0 × 2-51 + 1 × 2-52 =
0.5 + 0.25 + 0.125 + 0.062 5 + 0.031 25 + 0 + 0.007 812 5 + 0 + 0 + 0.000 976 562 5 + 0.000 488 281 25 + 0 + 0.000 122 070 312 5 + 0.000 061 035 156 25 + 0.000 030 517 578 125 + 0.000 015 258 789 062 5 + 0 + 0.000 003 814 697 265 625 + 0.000 001 907 348 632 812 5 + 0 + 0 + 0 + 0 + 0 + 0.000 000 029 802 322 387 695 312 5 + 0 + 0 + 0.000 000 003 725 290 298 461 914 062 5 + 0 + 0 + 0.000 000 000 465 661 287 307 739 257 812 5 + 0.000 000 000 232 830 643 653 869 628 906 25 + 0 + 0.000 000 000 058 207 660 913 467 407 226 562 5 + 0.000 000 000 029 103 830 456 733 703 613 281 25 + 0.000 000 000 014 551 915 228 366 851 806 640 625 + 0 + 0 + 0.000 000 000 001 818 989 403 545 856 475 830 078 125 + 0 + 0 + 0.000 000 000 000 227 373 675 443 232 059 478 759 765 625 + 0 + 0 + 0 + 0.000 000 000 000 014 210 854 715 202 003 717 422 485 351 562 5 + 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 + 0.000 000 000 000 000 888 178 419 700 125 232 338 905 334 472 656 25 + 0 + 0.000 000 000 000 000 222 044 604 925 031 308 084 726 333 618 164 062 5 =
0.5 + 0.25 + 0.125 + 0.062 5 + 0.031 25 + 0.007 812 5 + 0.000 976 562 5 + 0.000 488 281 25 + 0.000 122 070 312 5 + 0.000 061 035 156 25 + 0.000 030 517 578 125 + 0.000 015 258 789 062 5 + 0.000 003 814 697 265 625 + 0.000 001 907 348 632 812 5 + 0.000 000 029 802 322 387 695 312 5 + 0.000 000 003 725 290 298 461 914 062 5 + 0.000 000 000 465 661 287 307 739 257 812 5 + 0.000 000 000 232 830 643 653 869 628 906 25 + 0.000 000 000 058 207 660 913 467 407 226 562 5 + 0.000 000 000 029 103 830 456 733 703 613 281 25 + 0.000 000 000 014 551 915 228 366 851 806 640 625 + 0.000 000 000 001 818 989 403 545 856 475 830 078 125 + 0.000 000 000 000 227 373 675 443 232 059 478 759 765 625 + 0.000 000 000 000 014 210 854 715 202 003 717 422 485 351 562 5 + 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 + 0.000 000 000 000 000 888 178 419 700 125 232 338 905 334 472 656 25 + 0.000 000 000 000 000 222 044 604 925 031 308 084 726 333 618 164 062 5 =
0.978 261 981 961 876 303 515 168 729 063 589 125 871 658 325 195 312 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 + 0.978 261 981 961 876 303 515 168 729 063 589 125 871 658 325 195 312 5) × 262 =
-1.978 261 981 961 876 303 515 168 729 063 589 125 871 658 325 195 312 5 × 262 =
-9 123 123 123 000 038 400
1 - 100 0011 1101 - 1111 1010 0110 1111 0110 0000 1001 0011 0111 0010 0100 0111 0101 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) = -9 123 123 123 000 038 400(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: