64 bit double precision IEEE 754 binary floating point number 0 - 100 0001 1000 - 1111 0000 0100 0101 0101 1000 1011 0111 0000 1010 0110 0010 0110 converted to decimal base ten (double)

How to convert 64 bit double precision IEEE 754 binary floating point:
0 - 100 0001 1000 - 1111 0000 0100 0101 0101 1000 1011 0111 0000 1010 0110 0010 0110
to decimal system (base ten)

1. Identify the elements that make up the binary representation of the number:

First bit (the leftmost) indicates the sign,
1 = negative, 0 = positive.


The next 11 bits contain the exponent:
100 0001 1000


The last 52 bits contain the mantissa:
1111 0000 0100 0101 0101 1000 1011 0111 0000 1010 0110 0010 0110

2. Convert the exponent from binary (base 2) to decimal (base 10):

The exponent is allways a positive integer.

100 0001 1000(2) =


1 × 210 + 0 × 29 + 0 × 28 + 0 × 27 + 0 × 26 + 0 × 25 + 1 × 24 + 1 × 23 + 0 × 22 + 0 × 21 + 0 × 20 =


1,024 + 0 + 0 + 0 + 0 + 0 + 16 + 8 + 0 + 0 + 0 =


1,024 + 16 + 8 =


1,048(10)

3. Adjust the exponent.

Subtract the excess bits: 2(11 - 1) - 1 = 1023, that is due to the 11 bit excess/bias notation:

Exponent adjusted = 1,048 - 1023 = 25


4. Convert the mantissa from binary (base 2) to decimal (base 10):

Mantissa represents the number's fractional part (the excess beyond the number's integer part, comma delimited)

1111 0000 0100 0101 0101 1000 1011 0111 0000 1010 0110 0010 0110(2) =

1 × 2-1 + 1 × 2-2 + 1 × 2-3 + 1 × 2-4 + 0 × 2-5 + 0 × 2-6 + 0 × 2-7 + 0 × 2-8 + 0 × 2-9 + 1 × 2-10 + 0 × 2-11 + 0 × 2-12 + 0 × 2-13 + 1 × 2-14 + 0 × 2-15 + 1 × 2-16 + 0 × 2-17 + 1 × 2-18 + 0 × 2-19 + 1 × 2-20 + 1 × 2-21 + 0 × 2-22 + 0 × 2-23 + 0 × 2-24 + 1 × 2-25 + 0 × 2-26 + 1 × 2-27 + 1 × 2-28 + 0 × 2-29 + 1 × 2-30 + 1 × 2-31 + 1 × 2-32 + 0 × 2-33 + 0 × 2-34 + 0 × 2-35 + 0 × 2-36 + 1 × 2-37 + 0 × 2-38 + 1 × 2-39 + 0 × 2-40 + 0 × 2-41 + 1 × 2-42 + 1 × 2-43 + 0 × 2-44 + 0 × 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 + 0 + 0 + 0 + 0 + 0.000 976 562 5 + 0 + 0 + 0 + 0.000 061 035 156 25 + 0 + 0.000 015 258 789 062 5 + 0 + 0.000 003 814 697 265 625 + 0 + 0.000 000 953 674 316 406 25 + 0.000 000 476 837 158 203 125 + 0 + 0 + 0 + 0.000 000 029 802 322 387 695 312 5 + 0 + 0.000 000 007 450 580 596 923 828 125 + 0.000 000 003 725 290 298 461 914 062 5 + 0 + 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 + 0 + 0 + 0 + 0.000 000 000 007 275 957 614 183 425 903 320 312 5 + 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 + 0.000 000 000 000 113 686 837 721 616 029 739 379 882 812 5 + 0 + 0 + 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 + 0.000 976 562 5 + 0.000 061 035 156 25 + 0.000 015 258 789 062 5 + 0.000 003 814 697 265 625 + 0.000 000 953 674 316 406 25 + 0.000 000 476 837 158 203 125 + 0.000 000 029 802 322 387 695 312 5 + 0.000 000 007 450 580 596 923 828 125 + 0.000 000 003 725 290 298 461 914 062 5 + 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 + 0.000 000 000 007 275 957 614 183 425 903 320 312 5 + 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 + 0.000 000 000 000 113 686 837 721 616 029 739 379 882 812 5 + 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 =


0.938 558 144 271 504 968 259 023 371 501 825 749 874 114 990 234 375(10)

5. Put all the numbers into expression to calculate the double precision floating point decimal value:

(-1)Sign × (1 + Mantissa) × 2(Exponent adjusted) =


(-1)0 × (1 + 0.938 558 144 271 504 968 259 023 371 501 825 749 874 114 990 234 375) × 225 =


1.938 558 144 271 504 968 259 023 371 501 825 749 874 114 990 234 375 × 225 =


65 047 217.430 004 402 995 109 558 105 468 75

Conclusion:

0 - 100 0001 1000 - 1111 0000 0100 0101 0101 1000 1011 0111 0000 1010 0110 0010 0110
converted from
64 bit double precision IEEE 754 binary floating point
to
base ten decimal system (double) =

65 047 217.430 004 402 995 109 558 105 468 75(10)

More operations of this kind:

0 - 100 0001 1000 - 1111 0000 0100 0101 0101 1000 1011 0111 0000 1010 0110 0010 0101 = ?

0 - 100 0001 1000 - 1111 0000 0100 0101 0101 1000 1011 0111 0000 1010 0110 0010 0111 = ?


Convert 64 bit double precision IEEE 754 floating point standard binary numbers to base ten decimal system (double)

64 bit double precision IEEE 754 binary floating point standard representation of numbers requires three building blocks: sign (it takes 1 bit and it's either 0 for positive or 1 for negative numbers), exponent (11 bits), mantissa (52 bits)

Latest 64 bit double precision IEEE 754 floating point binary standard numbers converted to decimal base ten (double)

0 - 100 0001 1000 - 1111 0000 0100 0101 0101 1000 1011 0111 0000 1010 0110 0010 0110 = 65 047 217.430 004 402 995 109 558 105 468 75 Oct 25 17:22 UTC (GMT)
0 - 000 0000 0000 - 0000 0000 0000 1111 1111 1111 1111 1110 0010 1110 1000 0001 0100 = 0 Oct 25 17:21 UTC (GMT)
0 - 100 0000 1000 - 0100 0111 1010 0000 1000 0011 0001 0010 0110 1110 1001 0111 1000 = 655.253 999 999 999 905 412 551 015 615 463 256 835 937 5 Oct 25 17:20 UTC (GMT)
0 - 100 0001 1010 - 1111 0011 0101 0111 0001 1100 0111 1100 0000 1000 0010 0010 1000 = 261 798 115.875 993 013 381 958 007 812 5 Oct 25 17:20 UTC (GMT)
0 - 010 0110 0110 - 0100 1100 1100 1100 1100 1100 0000 0000 0000 0000 0000 0000 0010 = 0.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 983 275 254 813 387 327 875 978 075 247 727 072 342 149 797 646 347 440 240 612 140 358 885 996 100 083 319 745 068 344 954 964 821 389 668 633 231 364 322 158 177 276 377 239 624 577 163 758 921 190 764 591 233 751 806 195 299 817 465 401 557 Oct 25 17:14 UTC (GMT)
0 - 011 1111 1110 - 1100 1001 0000 1111 1101 1010 1010 0010 0010 0001 0110 1000 1011 = 0.892 699 081 698 724 028 477 442 971 052 369 102 835 655 212 402 343 75 Oct 25 17:14 UTC (GMT)
0 - 000 0001 0101 - 1100 0000 0000 0000 0000 1011 1000 0000 0000 0100 0000 1000 0000 = 0.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 040 830 1 Oct 25 17:12 UTC (GMT)
0 - 100 1000 0100 - 1000 1001 1101 0101 1111 1101 1111 1010 0111 0010 0000 1010 0000 = 16 751 926 811 721 714 852 883 337 970 407 499 104 256 Oct 25 17:12 UTC (GMT)
0 - 100 1000 0100 - 1000 1001 1101 0101 1111 1101 1111 1010 0111 0010 0000 1010 0000 = 16 751 926 811 721 714 852 883 337 970 407 499 104 256 Oct 25 17:12 UTC (GMT)
1 - 101 1000 0000 - 0011 1100 0001 0111 0010 0100 0111 0100 0101 0011 1000 1111 0000 = -97 301 530 464 225 088 042 674 953 821 686 378 464 682 583 414 175 862 037 408 670 678 775 490 544 317 284 577 432 882 459 287 509 321 353 140 816 052 224 Oct 25 17:11 UTC (GMT)
1 - 011 1111 1110 - 0111 1111 0111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 = -0.749 023 437 499 999 888 977 697 537 484 345 957 636 833 190 917 968 75 Oct 25 17:11 UTC (GMT)
1 - 100 0000 0001 - 0011 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 = -4.999 999 999 999 999 111 821 580 299 874 767 661 094 665 527 343 75 Oct 25 17:11 UTC (GMT)
0 - 111 1110 1000 - 0011 0000 1100 1001 0010 0011 1111 1001 0111 0011 1111 1100 0000 = 25 514 102 065 907 131 312 476 200 111 852 647 644 000 236 888 508 095 119 066 618 113 818 249 622 993 755 679 630 070 146 150 186 829 636 181 429 662 721 109 489 546 295 680 571 540 320 102 733 017 002 607 005 278 449 781 070 767 662 032 185 224 619 733 483 499 793 168 611 503 751 426 300 216 609 989 399 701 320 763 022 658 232 090 636 983 763 763 138 028 611 984 430 810 643 737 147 922 366 595 072 Oct 25 17:11 UTC (GMT)
All base ten decimal numbers converted to 64 bit double precision IEEE 754 binary floating point

How to convert numbers from 64 bit double precision IEEE 754 binary floating point standard to decimal system in base 10

Follow the steps below to convert a number from 64 bit double precision IEEE 754 binary floating point representation to base 10 decimal system:

  • 1. Identify the elements that make up the binary representation of the number:
    First bit (leftmost) indicates the sign, 1 = negative, 0 = pozitive.
    The next 11 bits contain the exponent.
    The last 52 bits contain the mantissa.
  • 2. Convert the exponent, that is allways a positive integer, from binary (base 2) to decimal (base 10).
  • 3. Adjust the exponent, subtract the excess bits, 2(11 - 1) - 1 = 1,023, that is due to the 11 bit excess/bias notation.
  • 4. Convert the mantissa, that represents the number's fractional part (the excess beyond the number's integer part, comma delimited), from binary (base 2) to decimal (base 10).
  • 5. Put all the numbers into expression to calculate the double precision floating point decimal value:
    (-1)Sign × (1 + Mantissa) × 2(Exponent adjusted)

Example: convert the number 1 - 100 0011 1101 - 1000 0000 0010 0001 0100 0000 0100 1110 0000 0100 0000 1010 1000 from 64 bit double precision IEEE 754 binary floating point system to base ten decimal (double):

  • 1. Identify the elements that make up the binary representation of the number:
    First bit (leftmost) indicates the sign, 1 = negative, 0 = pozitive.
    The next 11 bits contain the exponent: 100 0011 1101
    The last 52 bits contain the mantissa:
    1000 0000 0010 0001 0100 0000 0100 1110 0000 0100 0000 1010 1000
  • 2. Convert the exponent, that is allways a positive integer, from binary (base 2) to decimal (base 10):
    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 = 1,023, that is due to the 11 bit excess/bias notation:
    Exponent adjusted = 1,085 - 1,023 = 62
  • 4. Convert the mantissa, that represents the number's fractional part (the excess beyond the number's integer part, comma delimited), from binary (base 2) to decimal (base 10):
    1000 0000 0010 0001 0100 0000 0100 1110 0000 0100 0000 1010 1000(2) =
    1 × 2-1 + 0 × 2-2 + 0 × 2-3 + 0 × 2-4 + 0 × 2-5 + 0 × 2-6 + 0 × 2-7 + 0 × 2-8 + 0 × 2-9 + 0 × 2-10 + 1 × 2-11 + 0 × 2-12 + 0 × 2-13 + 0 × 2-14 + 0 × 2-15 + 1 × 2-16 + 0 × 2-17 + 1 × 2-18 + 0 × 2-19 + 0 × 2-20 + 0 × 2-21 + 0 × 2-22 + 0 × 2-23 + 0 × 2-24 + 0 × 2-25 + 1 × 2-26 + 0 × 2-27 + 0 × 2-28 + 1 × 2-29 + 1 × 2-30 + 1 × 2-31 + 0 × 2-32 + 0 × 2-33 + 0 × 2-34 + 0 × 2-35 + 0 × 2-36 + 0 × 2-37 + 1 × 2-38 + 0 × 2-39 + 0 × 2-40 + 0 × 2-41 + 0 × 2-42 + 0 × 2-43 + 0 × 2-44 + 1 × 2-45 + 0 × 2-46 + 1 × 2-47 + 0 × 2-48 + 1 × 2-49 + 0 × 2-50 + 0 × 2-51 + 0 × 2-52 =
    0.5 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0.000 488 281 25 + 0 + 0 + 0 + 0 + 0.000 015 258 789 062 5 + 0 + 0.000 003 814 697 265 625 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0.000 000 014 901 161 193 847 656 25 + 0 + 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 + 0 + 0 + 0 + 0 + 0 + 0.000 000 000 003 637 978 807 091 712 951 660 156 25 + 0 + 0 + 0 + 0 + 0 + 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 + 0.000 000 000 000 001 776 356 839 400 250 464 677 810 668 945 312 5 + 0 + 0 + 0 =
    0.5 + 0.000 488 281 25 + 0.000 015 258 789 062 5 + 0.000 003 814 697 265 625 + 0.000 000 014 901 161 193 847 656 25 + 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 + 0.000 000 000 003 637 978 807 091 712 951 660 156 25 + 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 + 0.000 000 000 000 001 776 356 839 400 250 464 677 810 668 945 312 5 =
    0.500 507 372 900 793 612 302 550 172 898 918 390 274 047 851 562 5(10)
  • 5. Put all the numbers into expression to calculate the double precision floating point decimal value:
    (-1)Sign × (1 + Mantissa) × 2(Exponent adjusted) =
    (-1)1 × (1 + 0.500 507 372 900 793 612 302 550 172 898 918 390 274 047 851 562 5) × 262 =
    -1.500 507 372 900 793 612 302 550 172 898 918 390 274 047 851 562 5 × 262 =
    -6 919 868 872 153 800 704(10)
  • 1 - 100 0011 1101 - 1000 0000 0010 0001 0100 0000 0100 1110 0000 0100 0000 1010 1000 converted from 64 bit double precision IEEE 754 binary floating point representation to a decimal number (float) in decimal system (in base 10) = -6 919 868 872 153 800 704(10)