Binary ↘ Double: The 64 Bit Double Precision IEEE 754 Binary Floating Point Standard Representation Number 0 - 100 0100 1011 - 0111 0110 0000 1111 0101 0100 0101 0001 1101 0111 0111 0101 1100 Converted and Written as a Base Ten Decimal System Number (as a Double)
0 - 100 0100 1011 - 0111 0110 0000 1111 0101 0100 0101 0001 1101 0111 0111 0101 1100: 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:
100 0100 1011
The last 52 bits contain the mantissa:
0111 0110 0000 1111 0101 0100 0101 0001 1101 0111 0111 0101 1100
2. Convert the exponent from binary (from base 2) to decimal (in base 10).
The exponent is allways a positive integer.
100 0100 1011(2) =
1 × 210 + 0 × 29 + 0 × 28 + 0 × 27 + 1 × 26 + 0 × 25 + 0 × 24 + 1 × 23 + 0 × 22 + 1 × 21 + 1 × 20 =
1,024 + 0 + 0 + 0 + 64 + 0 + 0 + 8 + 0 + 2 + 1 =
1,024 + 64 + 8 + 2 + 1 =
1,099(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 = 1,099 - 1023 = 76
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 0110 0000 1111 0101 0100 0101 0001 1101 0111 0111 0101 1100(2) =
0 × 2-1 + 1 × 2-2 + 1 × 2-3 + 1 × 2-4 + 0 × 2-5 + 1 × 2-6 + 1 × 2-7 + 0 × 2-8 + 0 × 2-9 + 0 × 2-10 + 0 × 2-11 + 0 × 2-12 + 1 × 2-13 + 1 × 2-14 + 1 × 2-15 + 1 × 2-16 + 0 × 2-17 + 1 × 2-18 + 0 × 2-19 + 1 × 2-20 + 0 × 2-21 + 1 × 2-22 + 0 × 2-23 + 0 × 2-24 + 0 × 2-25 + 1 × 2-26 + 0 × 2-27 + 1 × 2-28 + 0 × 2-29 + 0 × 2-30 + 0 × 2-31 + 1 × 2-32 + 1 × 2-33 + 1 × 2-34 + 0 × 2-35 + 1 × 2-36 + 0 × 2-37 + 1 × 2-38 + 1 × 2-39 + 1 × 2-40 + 0 × 2-41 + 1 × 2-42 + 1 × 2-43 + 1 × 2-44 + 0 × 2-45 + 1 × 2-46 + 0 × 2-47 + 1 × 2-48 + 1 × 2-49 + 1 × 2-50 + 0 × 2-51 + 0 × 2-52 =
0 + 0.25 + 0.125 + 0.062 5 + 0 + 0.015 625 + 0.007 812 5 + 0 + 0 + 0 + 0 + 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 + 0.000 000 953 674 316 406 25 + 0 + 0.000 000 238 418 579 101 562 5 + 0 + 0 + 0 + 0.000 000 014 901 161 193 847 656 25 + 0 + 0.000 000 003 725 290 298 461 914 062 5 + 0 + 0 + 0 + 0.000 000 000 232 830 643 653 869 628 906 25 + 0.000 000 000 116 415 321 826 934 814 453 125 + 0.000 000 000 058 207 660 913 467 407 226 562 5 + 0 + 0.000 000 000 014 551 915 228 366 851 806 640 625 + 0 + 0.000 000 000 003 637 978 807 091 712 951 660 156 25 + 0.000 000 000 001 818 989 403 545 856 475 830 078 125 + 0.000 000 000 000 909 494 701 772 928 237 915 039 062 5 + 0 + 0.000 000 000 000 227 373 675 443 232 059 478 759 765 625 + 0.000 000 000 000 113 686 837 721 616 029 739 379 882 812 5 + 0.000 000 000 000 056 843 418 860 808 014 869 689 941 406 25 + 0 + 0.000 000 000 000 014 210 854 715 202 003 717 422 485 351 562 5 + 0 + 0.000 000 000 000 003 552 713 678 800 500 929 355 621 337 890 625 + 0.000 000 000 000 001 776 356 839 400 250 464 677 810 668 945 312 5 + 0.000 000 000 000 000 888 178 419 700 125 232 338 905 334 472 656 25 + 0 + 0 =
0.25 + 0.125 + 0.062 5 + 0.015 625 + 0.007 812 5 + 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 + 0.000 000 953 674 316 406 25 + 0.000 000 238 418 579 101 562 5 + 0.000 000 014 901 161 193 847 656 25 + 0.000 000 003 725 290 298 461 914 062 5 + 0.000 000 000 232 830 643 653 869 628 906 25 + 0.000 000 000 116 415 321 826 934 814 453 125 + 0.000 000 000 058 207 660 913 467 407 226 562 5 + 0.000 000 000 014 551 915 228 366 851 806 640 625 + 0.000 000 000 003 637 978 807 091 712 951 660 156 25 + 0.000 000 000 001 818 989 403 545 856 475 830 078 125 + 0.000 000 000 000 909 494 701 772 928 237 915 039 062 5 + 0.000 000 000 000 227 373 675 443 232 059 478 759 765 625 + 0.000 000 000 000 113 686 837 721 616 029 739 379 882 812 5 + 0.000 000 000 000 056 843 418 860 808 014 869 689 941 406 25 + 0.000 000 000 000 014 210 854 715 202 003 717 422 485 351 562 5 + 0.000 000 000 000 003 552 713 678 800 500 929 355 621 337 890 625 + 0.000 000 000 000 001 776 356 839 400 250 464 677 810 668 945 312 5 + 0.000 000 000 000 000 888 178 419 700 125 232 338 905 334 472 656 25 =
0.461 171 407 681 340 461 692 798 271 542 415 022 850 036 621 093 75(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 + 0.461 171 407 681 340 461 692 798 271 542 415 022 850 036 621 093 75) × 276 =
1.461 171 407 681 340 461 692 798 271 542 415 022 850 036 621 093 75 × 276 =
110 402 990 101 789 124 067 328
0 - 100 0100 1011 - 0111 0110 0000 1111 0101 0100 0101 0001 1101 0111 0111 0101 1100 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) = 110 402 990 101 789 124 067 328(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: