64 bit double precision IEEE 754 binary floating point number 0 - 011 1111 0011 - 1011 1111 1011 1101 1111 0000 1001 0000 1111 0111 0011 0011 1011 converted to decimal base ten (double)

How to convert 64 bit double precision IEEE 754 binary floating point:
0 - 011 1111 0011 - 1011 1111 1011 1101 1111 0000 1001 0000 1111 0111 0011 0011 1011
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:
011 1111 0011


The last 52 bits contain the mantissa:
1011 1111 1011 1101 1111 0000 1001 0000 1111 0111 0011 0011 1011

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

The exponent is allways a positive integer.

011 1111 0011(2) =


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


0 + 512 + 256 + 128 + 64 + 32 + 16 + 0 + 0 + 2 + 1 =


512 + 256 + 128 + 64 + 32 + 16 + 2 + 1 =


1,011(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 = 1,011 - 1023 = -12


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)

1011 1111 1011 1101 1111 0000 1001 0000 1111 0111 0011 0011 1011(2) =

1 × 2-1 + 0 × 2-2 + 1 × 2-3 + 1 × 2-4 + 1 × 2-5 + 1 × 2-6 + 1 × 2-7 + 1 × 2-8 + 1 × 2-9 + 0 × 2-10 + 1 × 2-11 + 1 × 2-12 + 1 × 2-13 + 1 × 2-14 + 0 × 2-15 + 1 × 2-16 + 1 × 2-17 + 1 × 2-18 + 1 × 2-19 + 1 × 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 + 0 × 2-31 + 0 × 2-32 + 1 × 2-33 + 1 × 2-34 + 1 × 2-35 + 1 × 2-36 + 0 × 2-37 + 1 × 2-38 + 1 × 2-39 + 1 × 2-40 + 0 × 2-41 + 0 × 2-42 + 1 × 2-43 + 1 × 2-44 + 0 × 2-45 + 0 × 2-46 + 1 × 2-47 + 1 × 2-48 + 1 × 2-49 + 0 × 2-50 + 1 × 2-51 + 1 × 2-52 =


0.5 + 0 + 0.125 + 0.062 5 + 0.031 25 + 0.015 625 + 0.007 812 5 + 0.003 906 25 + 0.001 953 125 + 0 + 0.000 488 281 25 + 0.000 244 140 625 + 0.000 122 070 312 5 + 0.000 061 035 156 25 + 0 + 0.000 015 258 789 062 5 + 0.000 007 629 394 531 25 + 0.000 003 814 697 265 625 + 0.000 001 907 348 632 812 5 + 0.000 000 953 674 316 406 25 + 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 + 0 + 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 + 0.000 000 000 029 103 830 456 733 703 613 281 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 + 0.000 000 000 001 818 989 403 545 856 475 830 078 125 + 0.000 000 000 000 909 494 701 772 928 237 915 039 062 5 + 0 + 0 + 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 + 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 + 0.000 000 000 000 000 444 089 209 850 062 616 169 452 667 236 328 125 + 0.000 000 000 000 000 222 044 604 925 031 308 084 726 333 618 164 062 5 =


0.5 + 0.125 + 0.062 5 + 0.031 25 + 0.015 625 + 0.007 812 5 + 0.003 906 25 + 0.001 953 125 + 0.000 488 281 25 + 0.000 244 140 625 + 0.000 122 070 312 5 + 0.000 061 035 156 25 + 0.000 015 258 789 062 5 + 0.000 007 629 394 531 25 + 0.000 003 814 697 265 625 + 0.000 001 907 348 632 812 5 + 0.000 000 953 674 316 406 25 + 0.000 000 029 802 322 387 695 312 5 + 0.000 000 003 725 290 298 461 914 062 5 + 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 + 0.000 000 000 029 103 830 456 733 703 613 281 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 + 0.000 000 000 001 818 989 403 545 856 475 830 078 125 + 0.000 000 000 000 909 494 701 772 928 237 915 039 062 5 + 0.000 000 000 000 113 686 837 721 616 029 739 379 882 812 5 + 0.000 000 000 000 056 843 418 860 808 014 869 689 941 406 25 + 0.000 000 000 000 007 105 427 357 601 001 858 711 242 675 781 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 + 0.000 000 000 000 000 444 089 209 850 062 616 169 452 667 236 328 125 + 0.000 000 000 000 000 222 044 604 925 031 308 084 726 333 618 164 062 5 =


0.748 992 000 000 000 102 133 412 838 156 800 717 115 402 221 679 687 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)0 × (1 + 0.748 992 000 000 000 102 133 412 838 156 800 717 115 402 221 679 687 5) × 2-12 =


1.748 992 000 000 000 102 133 412 838 156 800 717 115 402 221 679 687 5 × 2-12 =


0.000 427 000 000 000 000 024 934 915 243 690 625 175 077 002 495 527 267 456 054 687 5

Conclusion:

0 - 011 1111 0011 - 1011 1111 1011 1101 1111 0000 1001 0000 1111 0111 0011 0011 1011
converted from
64 bit double precision IEEE 754 binary floating point
to
base ten decimal system (double) =

0.000 427 000 000 000 000 024 934 915 243 690 625 175 077 002 495 527 267 456 054 687 5(10)

More operations of this kind:

0 - 011 1111 0011 - 1011 1111 1011 1101 1111 0000 1001 0000 1111 0111 0011 0011 1010 = ?

0 - 011 1111 0011 - 1011 1111 1011 1101 1111 0000 1001 0000 1111 0111 0011 0011 1100 = ?


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 - 011 1111 0011 - 1011 1111 1011 1101 1111 0000 1001 0000 1111 0111 0011 0011 1011 = 0.000 427 000 000 000 000 024 934 915 243 690 625 175 077 002 495 527 267 456 054 687 5 Oct 30 04:55 UTC (GMT)
0 - 100 0001 1010 - 1100 1110 1110 0010 1100 0110 1111 0111 1010 0101 1010 0010 0001 = 242 685 495.738 968 878 984 451 293 945 312 5 Oct 30 04:55 UTC (GMT)
0 - 011 1111 0010 - 0000 0001 1111 0011 0001 1111 0100 0110 1110 1101 0010 0100 0110 = 0.000 123 000 000 000 000 008 198 303 147 466 390 328 190 755 099 058 151 245 117 187 5 Oct 30 04:55 UTC (GMT)
0 - 011 1011 0000 - 0001 0000 0001 0010 1110 1111 0001 0011 1010 1111 1111 1111 1111 = 0.000 000 000 000 000 000 000 001 758 236 760 359 483 378 312 103 841 136 228 680 632 237 424 405 240 276 609 369 720 227 024 624 925 604 712 188 942 357 897 758 483 886 718 75 Oct 30 04:54 UTC (GMT)
0 - 000 0000 0011 - 0110 1101 1011 0110 1101 1011 0110 1101 1011 0110 1101 1011 0101 = 0 Oct 30 04:54 UTC (GMT)
1 - 100 0101 0111 - 0000 0100 0001 0011 0100 0000 0000 0010 0001 0000 0000 0000 0001 = -314 411 618 803 217 682 516 672 512 Oct 30 04:53 UTC (GMT)
0 - 100 0001 1001 - 0000 0101 1111 0000 0110 1111 0011 0001 0011 0100 1011 0110 0100 = 68 665 788.768 842 279 911 041 259 765 625 Oct 30 04:53 UTC (GMT)
0 - 100 0000 0100 - 1110 0010 1100 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 60.343 75 Oct 30 04:53 UTC (GMT)
1 - 000 0000 0000 - 1111 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0001 = -0 Oct 30 04:53 UTC (GMT)
0 - 100 0001 0110 - 1001 0011 0100 0101 1001 0011 0001 0010 0001 1100 1000 1001 0001 = 13 214 409.535 373 957 827 687 263 488 769 531 25 Oct 30 04:53 UTC (GMT)
0 - 100 0000 0110 - 1111 1010 1001 1001 1001 1001 1001 1001 1001 1001 1001 0111 1110 = 253.299 999 999 999 215 560 819 720 849 394 798 278 808 593 75 Oct 30 04:53 UTC (GMT)
1 - 110 0000 0000 - 1010 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0001 = -43 575 375 772 313 446 527 878 410 673 780 647 294 368 573 135 458 423 902 631 601 442 605 282 923 706 739 362 403 139 693 998 224 028 149 690 986 257 208 739 041 277 117 671 355 983 841 365 900 881 035 264 Oct 30 04:53 UTC (GMT)
0 - 100 0000 0011 - 1001 0111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 = 25.499 999 999 999 996 447 286 321 199 499 070 644 378 662 109 375 Oct 30 04:53 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)