Binary ↘ Double: The 64 Bit Double Precision IEEE 754 Binary Floating Point Standard Representation Number 0 - 011 0101 1010 - 0000 1001 1101 1000 0111 1001 0010 1111 1011 0100 1100 0011 1001 Converted and Written as a Base Ten Decimal System Number (as a Double)
0 - 011 0101 1010 - 0000 1001 1101 1000 0111 1001 0010 1111 1011 0100 1100 0011 1001: 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 0101 1010
The last 52 bits contain the mantissa:
0000 1001 1101 1000 0111 1001 0010 1111 1011 0100 1100 0011 1001
2. Convert the exponent from binary (from base 2) to decimal (in base 10).
The exponent is allways a positive integer.
011 0101 1010(2) =
0 × 210 + 1 × 29 + 1 × 28 + 0 × 27 + 1 × 26 + 0 × 25 + 1 × 24 + 1 × 23 + 0 × 22 + 1 × 21 + 0 × 20 =
0 + 512 + 256 + 0 + 64 + 0 + 16 + 8 + 0 + 2 + 0 =
512 + 256 + 64 + 16 + 8 + 2 =
858(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 = 858 - 1023 = -165
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 1001 1101 1000 0111 1001 0010 1111 1011 0100 1100 0011 1001(2) =
0 × 2-1 + 0 × 2-2 + 0 × 2-3 + 0 × 2-4 + 1 × 2-5 + 0 × 2-6 + 0 × 2-7 + 1 × 2-8 + 1 × 2-9 + 1 × 2-10 + 0 × 2-11 + 1 × 2-12 + 1 × 2-13 + 0 × 2-14 + 0 × 2-15 + 0 × 2-16 + 0 × 2-17 + 1 × 2-18 + 1 × 2-19 + 1 × 2-20 + 1 × 2-21 + 0 × 2-22 + 0 × 2-23 + 1 × 2-24 + 0 × 2-25 + 0 × 2-26 + 1 × 2-27 + 0 × 2-28 + 1 × 2-29 + 1 × 2-30 + 1 × 2-31 + 1 × 2-32 + 1 × 2-33 + 0 × 2-34 + 1 × 2-35 + 1 × 2-36 + 0 × 2-37 + 1 × 2-38 + 0 × 2-39 + 0 × 2-40 + 1 × 2-41 + 1 × 2-42 + 0 × 2-43 + 0 × 2-44 + 0 × 2-45 + 0 × 2-46 + 1 × 2-47 + 1 × 2-48 + 1 × 2-49 + 0 × 2-50 + 0 × 2-51 + 1 × 2-52 =
0 + 0 + 0 + 0 + 0.031 25 + 0 + 0 + 0.003 906 25 + 0.001 953 125 + 0.000 976 562 5 + 0 + 0.000 244 140 625 + 0.000 122 070 312 5 + 0 + 0 + 0 + 0 + 0.000 003 814 697 265 625 + 0.000 001 907 348 632 812 5 + 0.000 000 953 674 316 406 25 + 0.000 000 476 837 158 203 125 + 0 + 0 + 0.000 000 059 604 644 775 390 625 + 0 + 0 + 0.000 000 007 450 580 596 923 828 125 + 0 + 0.000 000 001 862 645 149 230 957 031 25 + 0.000 000 000 931 322 574 615 478 515 625 + 0.000 000 000 465 661 287 307 739 257 812 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 + 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 + 0.000 000 000 003 637 978 807 091 712 951 660 156 25 + 0 + 0 + 0.000 000 000 000 454 747 350 886 464 118 957 519 531 25 + 0.000 000 000 000 227 373 675 443 232 059 478 759 765 625 + 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 + 0.000 000 000 000 001 776 356 839 400 250 464 677 810 668 945 312 5 + 0 + 0 + 0.000 000 000 000 000 222 044 604 925 031 308 084 726 333 618 164 062 5 =
0.031 25 + 0.003 906 25 + 0.001 953 125 + 0.000 976 562 5 + 0.000 244 140 625 + 0.000 122 070 312 5 + 0.000 003 814 697 265 625 + 0.000 001 907 348 632 812 5 + 0.000 000 953 674 316 406 25 + 0.000 000 476 837 158 203 125 + 0.000 000 059 604 644 775 390 625 + 0.000 000 007 450 580 596 923 828 125 + 0.000 000 001 862 645 149 230 957 031 25 + 0.000 000 000 931 322 574 615 478 515 625 + 0.000 000 000 465 661 287 307 739 257 812 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 + 0.000 000 000 029 103 830 456 733 703 613 281 25 + 0.000 000 000 014 551 915 228 366 851 806 640 625 + 0.000 000 000 003 637 978 807 091 712 951 660 156 25 + 0.000 000 000 000 454 747 350 886 464 118 957 519 531 25 + 0.000 000 000 000 227 373 675 443 232 059 478 759 765 625 + 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 + 0.000 000 000 000 001 776 356 839 400 250 464 677 810 668 945 312 5 + 0.000 000 000 000 000 222 044 604 925 031 308 084 726 333 618 164 062 5 =
0.038 459 371 706 961 897 885 435 064 381 454 139 947 891 235 351 562 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)0 × (1 + 0.038 459 371 706 961 897 885 435 064 381 454 139 947 891 235 351 562 5) × 2-165 =
1.038 459 371 706 961 897 885 435 064 381 454 139 947 891 235 351 562 5 × 2-165 =
0.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 022 204 460 492 503 053 237 984 266 794 626 450 276 957 423 969 771 299 104 079 819 203 694 089 564 992 726 335 276 214 612 798 817 529 464 833 197 123 148 829 297 243 928 709 848 503 899 593 197 274 953 126 907 348 632 812 5
0 - 011 0101 1010 - 0000 1001 1101 1000 0111 1001 0010 1111 1011 0100 1100 0011 1001 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) = 0.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 022 204 460 492 503 053 237 984 266 794 626 450 276 957 423 969 771 299 104 079 819 203 694 089 564 992 726 335 276 214 612 798 817 529 464 833 197 123 148 829 297 243 928 709 848 503 899 593 197 274 953 126 907 348 632 812 5(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: