64 bit double precision IEEE 754 binary floating point number 1 - 100 1100 0000 - 0000 1111 1101 0111 1010 1110 0001 0100 0111 1010 1110 0001 0100 converted to decimal base ten (double)

How to convert 64 bit double precision IEEE 754 binary floating point:
1 - 100 1100 0000 - 0000 1111 1101 0111 1010 1110 0001 0100 0111 1010 1110 0001 0100
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 1100 0000


The last 52 bits contain the mantissa:
0000 1111 1101 0111 1010 1110 0001 0100 0111 1010 1110 0001 0100

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

The exponent is allways a positive integer.

100 1100 0000(2) =


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


1,024 + 0 + 0 + 128 + 64 + 0 + 0 + 0 + 0 + 0 + 0 =


1,024 + 128 + 64 =


1,216(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 = 1,216 - 1023 = 193


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)

0000 1111 1101 0111 1010 1110 0001 0100 0111 1010 1110 0001 0100(2) =

0 × 2-1 + 0 × 2-2 + 0 × 2-3 + 0 × 2-4 + 1 × 2-5 + 1 × 2-6 + 1 × 2-7 + 1 × 2-8 + 1 × 2-9 + 1 × 2-10 + 0 × 2-11 + 1 × 2-12 + 0 × 2-13 + 1 × 2-14 + 1 × 2-15 + 1 × 2-16 + 1 × 2-17 + 0 × 2-18 + 1 × 2-19 + 0 × 2-20 + 1 × 2-21 + 1 × 2-22 + 1 × 2-23 + 0 × 2-24 + 0 × 2-25 + 0 × 2-26 + 0 × 2-27 + 1 × 2-28 + 0 × 2-29 + 1 × 2-30 + 0 × 2-31 + 0 × 2-32 + 0 × 2-33 + 1 × 2-34 + 1 × 2-35 + 1 × 2-36 + 1 × 2-37 + 0 × 2-38 + 1 × 2-39 + 0 × 2-40 + 1 × 2-41 + 1 × 2-42 + 1 × 2-43 + 0 × 2-44 + 0 × 2-45 + 0 × 2-46 + 0 × 2-47 + 1 × 2-48 + 0 × 2-49 + 1 × 2-50 + 0 × 2-51 + 0 × 2-52 =


0 + 0 + 0 + 0 + 0.031 25 + 0.015 625 + 0.007 812 5 + 0.003 906 25 + 0.001 953 125 + 0.000 976 562 5 + 0 + 0.000 244 140 625 + 0 + 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 + 0.000 001 907 348 632 812 5 + 0 + 0.000 000 476 837 158 203 125 + 0.000 000 238 418 579 101 562 5 + 0.000 000 119 209 289 550 781 25 + 0 + 0 + 0 + 0 + 0.000 000 003 725 290 298 461 914 062 5 + 0 + 0.000 000 000 931 322 574 615 478 515 625 + 0 + 0 + 0 + 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 + 0.000 000 000 007 275 957 614 183 425 903 320 312 5 + 0 + 0.000 000 000 001 818 989 403 545 856 475 830 078 125 + 0 + 0.000 000 000 000 454 747 350 886 464 118 957 519 531 25 + 0.000 000 000 000 227 373 675 443 232 059 478 759 765 625 + 0.000 000 000 000 113 686 837 721 616 029 739 379 882 812 5 + 0 + 0 + 0 + 0 + 0.000 000 000 000 003 552 713 678 800 500 929 355 621 337 890 625 + 0 + 0.000 000 000 000 000 888 178 419 700 125 232 338 905 334 472 656 25 + 0 + 0 =


0.031 25 + 0.015 625 + 0.007 812 5 + 0.003 906 25 + 0.001 953 125 + 0.000 976 562 5 + 0.000 244 140 625 + 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 + 0.000 001 907 348 632 812 5 + 0.000 000 476 837 158 203 125 + 0.000 000 238 418 579 101 562 5 + 0.000 000 119 209 289 550 781 25 + 0.000 000 003 725 290 298 461 914 062 5 + 0.000 000 000 931 322 574 615 478 515 625 + 0.000 000 000 058 207 660 913 467 407 226 562 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 + 0.000 000 000 007 275 957 614 183 425 903 320 312 5 + 0.000 000 000 001 818 989 403 545 856 475 830 078 125 + 0.000 000 000 000 454 747 350 886 464 118 957 519 531 25 + 0.000 000 000 000 227 373 675 443 232 059 478 759 765 625 + 0.000 000 000 000 113 686 837 721 616 029 739 379 882 812 5 + 0.000 000 000 000 003 552 713 678 800 500 929 355 621 337 890 625 + 0.000 000 000 000 000 888 178 419 700 125 232 338 905 334 472 656 25 =


0.061 884 765 624 999 893 418 589 635 984 972 119 331 359 863 281 25(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 + 0.061 884 765 624 999 893 418 589 635 984 972 119 331 359 863 281 25) × 2193 =


-1.061 884 765 624 999 893 418 589 635 984 972 119 331 359 863 281 25 × 2193 =


-13 331 117 410 170 731 205 260 214 543 462 065 083 788 970 515 408 577 626 112

Conclusion:

1 - 100 1100 0000 - 0000 1111 1101 0111 1010 1110 0001 0100 0111 1010 1110 0001 0100
converted from
64 bit double precision IEEE 754 binary floating point
to
base ten decimal system (double) =

-13 331 117 410 170 731 205 260 214 543 462 065 083 788 970 515 408 577 626 112(10)

More operations of this kind:

1 - 100 1100 0000 - 0000 1111 1101 0111 1010 1110 0001 0100 0111 1010 1110 0001 0011 = ?

1 - 100 1100 0000 - 0000 1111 1101 0111 1010 1110 0001 0100 0111 1010 1110 0001 0101 = ?


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)

1 - 100 1100 0000 - 0000 1111 1101 0111 1010 1110 0001 0100 0111 1010 1110 0001 0100 = -13 331 117 410 170 731 205 260 214 543 462 065 083 788 970 515 408 577 626 112 Nov 29 09:41 UTC (GMT)
1 - 000 0010 1111 - 1111 1100 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0001 = -0 Nov 29 09:41 UTC (GMT)
0 - 100 0000 0000 - 0110 1010 1011 1011 1001 1000 1100 0111 1101 1111 1111 1111 1101 = 2.833 849 999 995 435 187 116 754 605 085 588 991 641 998 291 015 625 Nov 29 09:38 UTC (GMT)
0 - 100 0000 0011 - 0010 0001 1001 1001 1001 1001 1000 0000 0000 0000 0000 0000 0000 = 18.099 999 904 632 568 359 375 Nov 29 09:36 UTC (GMT)
0 - 100 0000 1010 - 0000 1000 0000 0000 0010 0000 0000 0000 0000 0000 0000 0111 1111 = 2 112.003 906 250 057 752 913 562 580 943 107 604 980 468 75 Nov 29 09:35 UTC (GMT)
1 - 100 1100 0110 - 1100 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = -1 406 070 788 726 616 491 099 216 830 798 517 277 206 927 619 559 943 730 888 704 Nov 29 09:34 UTC (GMT)
0 - 100 0000 0100 - 0101 0010 0001 1100 1010 1100 0000 1000 0011 0001 0010 0111 0000 = 42.264 000 000 000 010 004 441 719 502 210 617 065 429 687 5 Nov 29 09:32 UTC (GMT)
1 - 011 1111 1101 - 1001 1001 1001 1001 1001 1001 1001 1001 1001 1001 1001 1001 1000 = -0.399 999 999 999 999 911 182 158 029 987 476 766 109 466 552 734 375 Nov 29 09:31 UTC (GMT)
1 - 011 1111 1110 - 0011 0011 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 = -0.601 562 499 999 999 888 977 697 537 484 345 957 636 833 190 917 968 75 Nov 29 09:29 UTC (GMT)
0 - 000 0000 0000 - 1010 0000 0000 0100 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 0 Nov 29 09:29 UTC (GMT)
0 - 100 0001 0110 - 1010 0100 0110 1111 1111 1111 0110 0010 0001 1001 1011 0110 1110 = 13 776 895.691 602 434 962 987 899 780 273 437 5 Nov 29 09:29 UTC (GMT)
1 - 111 1100 0011 - 1111 1111 1111 1000 0000 0000 0000 0001 1111 1111 1111 1110 1110 = -311 831 014 620 979 079 230 784 107 199 979 823 593 422 595 401 136 343 399 636 748 963 416 389 177 680 665 105 512 870 520 441 919 822 812 768 838 856 693 145 022 958 873 439 252 043 686 524 347 639 802 301 717 876 361 109 162 511 713 667 423 188 644 043 115 650 603 151 647 786 939 191 528 530 744 736 177 313 497 225 713 814 186 246 512 071 244 514 165 448 079 612 571 790 079 950 848 Nov 29 09:28 UTC (GMT)
0 - 100 0001 1011 - 0100 1000 1110 0100 1010 0101 1011 1110 0010 0001 0100 1011 0011 = 344 869 467.883 128 345 012 664 794 921 875 Nov 29 09:26 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)