Binary ↘ Double: The 64 Bit Double Precision IEEE 754 Binary Floating Point Standard Representation Number 0 - 111 1111 1110 - 1111 1111 1111 1111 1100 0101 0111 1100 1010 1000 0010 1010 0110 Converted and Written as a Base Ten Decimal System Number (as a Double)
0 - 111 1111 1110 - 1111 1111 1111 1111 1100 0101 0111 1100 1010 1000 0010 1010 0110: 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:
111 1111 1110
The last 52 bits contain the mantissa:
1111 1111 1111 1111 1100 0101 0111 1100 1010 1000 0010 1010 0110
2. Convert the exponent from binary (from base 2) to decimal (in base 10).
The exponent is allways a positive integer.
111 1111 1110(2) =
1 × 210 + 1 × 29 + 1 × 28 + 1 × 27 + 1 × 26 + 1 × 25 + 1 × 24 + 1 × 23 + 1 × 22 + 1 × 21 + 0 × 20 =
1,024 + 512 + 256 + 128 + 64 + 32 + 16 + 8 + 4 + 2 + 0 =
1,024 + 512 + 256 + 128 + 64 + 32 + 16 + 8 + 4 + 2 =
2,046(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 = 2,046 - 1023 = 1023
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 1111 1111 1111 1100 0101 0111 1100 1010 1000 0010 1010 0110(2) =
1 × 2-1 + 1 × 2-2 + 1 × 2-3 + 1 × 2-4 + 1 × 2-5 + 1 × 2-6 + 1 × 2-7 + 1 × 2-8 + 1 × 2-9 + 1 × 2-10 + 1 × 2-11 + 1 × 2-12 + 1 × 2-13 + 1 × 2-14 + 1 × 2-15 + 1 × 2-16 + 1 × 2-17 + 1 × 2-18 + 0 × 2-19 + 0 × 2-20 + 0 × 2-21 + 1 × 2-22 + 0 × 2-23 + 1 × 2-24 + 0 × 2-25 + 1 × 2-26 + 1 × 2-27 + 1 × 2-28 + 1 × 2-29 + 1 × 2-30 + 0 × 2-31 + 0 × 2-32 + 1 × 2-33 + 0 × 2-34 + 1 × 2-35 + 0 × 2-36 + 1 × 2-37 + 0 × 2-38 + 0 × 2-39 + 0 × 2-40 + 0 × 2-41 + 0 × 2-42 + 1 × 2-43 + 0 × 2-44 + 1 × 2-45 + 0 × 2-46 + 1 × 2-47 + 0 × 2-48 + 0 × 2-49 + 1 × 2-50 + 1 × 2-51 + 0 × 2-52 =
0.5 + 0.25 + 0.125 + 0.062 5 + 0.031 25 + 0.015 625 + 0.007 812 5 + 0.003 906 25 + 0.001 953 125 + 0.000 976 562 5 + 0.000 488 281 25 + 0.000 244 140 625 + 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 + 0.000 007 629 394 531 25 + 0.000 003 814 697 265 625 + 0 + 0 + 0 + 0.000 000 238 418 579 101 562 5 + 0 + 0.000 000 059 604 644 775 390 625 + 0 + 0.000 000 014 901 161 193 847 656 25 + 0.000 000 007 450 580 596 923 828 125 + 0.000 000 003 725 290 298 461 914 062 5 + 0.000 000 001 862 645 149 230 957 031 25 + 0.000 000 000 931 322 574 615 478 515 625 + 0 + 0 + 0.000 000 000 116 415 321 826 934 814 453 125 + 0 + 0.000 000 000 029 103 830 456 733 703 613 281 25 + 0 + 0.000 000 000 007 275 957 614 183 425 903 320 312 5 + 0 + 0 + 0 + 0 + 0 + 0.000 000 000 000 113 686 837 721 616 029 739 379 882 812 5 + 0 + 0.000 000 000 000 028 421 709 430 404 007 434 844 970 703 125 + 0 + 0.000 000 000 000 007 105 427 357 601 001 858 711 242 675 781 25 + 0 + 0 + 0.000 000 000 000 000 888 178 419 700 125 232 338 905 334 472 656 25 + 0.000 000 000 000 000 444 089 209 850 062 616 169 452 667 236 328 125 + 0 =
0.5 + 0.25 + 0.125 + 0.062 5 + 0.031 25 + 0.015 625 + 0.007 812 5 + 0.003 906 25 + 0.001 953 125 + 0.000 976 562 5 + 0.000 488 281 25 + 0.000 244 140 625 + 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 + 0.000 007 629 394 531 25 + 0.000 003 814 697 265 625 + 0.000 000 238 418 579 101 562 5 + 0.000 000 059 604 644 775 390 625 + 0.000 000 014 901 161 193 847 656 25 + 0.000 000 007 450 580 596 923 828 125 + 0.000 000 003 725 290 298 461 914 062 5 + 0.000 000 001 862 645 149 230 957 031 25 + 0.000 000 000 931 322 574 615 478 515 625 + 0.000 000 000 116 415 321 826 934 814 453 125 + 0.000 000 000 029 103 830 456 733 703 613 281 25 + 0.000 000 000 007 275 957 614 183 425 903 320 312 5 + 0.000 000 000 000 113 686 837 721 616 029 739 379 882 812 5 + 0.000 000 000 000 028 421 709 430 404 007 434 844 970 703 125 + 0.000 000 000 000 007 105 427 357 601 001 858 711 242 675 781 25 + 0.000 000 000 000 000 888 178 419 700 125 232 338 905 334 472 656 25 + 0.000 000 000 000 000 444 089 209 850 062 616 169 452 667 236 328 125 =
0.999 996 512 349 903 721 172 950 099 571 608 006 954 193 115 234 375(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 + 0.999 996 512 349 903 721 172 950 099 571 608 006 954 193 115 234 375) × 21023 =
1.999 996 512 349 903 721 172 950 099 571 608 006 954 193 115 234 375 × 21023 =
179 768 999 999 999 846 655 805 729 361 357 710 983 215 993 486 600 050 819 310 148 452 523 113 945 126 894 963 663 950 115 071 346 648 896 576 909 456 687 286 089 112 364 646 871 623 256 612 020 185 241 611 839 016 078 596 068 679 202 359 046 666 205 141 570 619 467 393 497 008 024 615 905 047 862 949 116 171 908 995 883 575 770 750 919 440 496 560 352 660 477 351 346 255 096 262 442 236 680 216 630 001 664
0 - 111 1111 1110 - 1111 1111 1111 1111 1100 0101 0111 1100 1010 1000 0010 1010 0110 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) = 179 768 999 999 999 846 655 805 729 361 357 710 983 215 993 486 600 050 819 310 148 452 523 113 945 126 894 963 663 950 115 071 346 648 896 576 909 456 687 286 089 112 364 646 871 623 256 612 020 185 241 611 839 016 078 596 068 679 202 359 046 666 205 141 570 619 467 393 497 008 024 615 905 047 862 949 116 171 908 995 883 575 770 750 919 440 496 560 352 660 477 351 346 255 096 262 442 236 680 216 630 001 664(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: