64 bit double precision IEEE 754 binary floating point number 1 - 011 0101 1111 - 1110 0110 1010 1000 1010 1000 0011 0001 0011 0111 1001 0000 0101 converted to decimal base ten (double)

How to convert 64 bit double precision IEEE 754 binary floating point:
1 - 011 0101 1111 - 1110 0110 1010 1000 1010 1000 0011 0001 0011 0111 1001 0000 0101
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 0101 1111


The last 52 bits contain the mantissa:
1110 0110 1010 1000 1010 1000 0011 0001 0011 0111 1001 0000 0101

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

The exponent is allways a positive integer.

011 0101 1111(2) =


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


0 + 512 + 256 + 0 + 64 + 0 + 16 + 8 + 4 + 2 + 1 =


512 + 256 + 64 + 16 + 8 + 4 + 2 + 1 =


863(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 = 863 - 1023 = -160


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)

1110 0110 1010 1000 1010 1000 0011 0001 0011 0111 1001 0000 0101(2) =

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


0.5 + 0.25 + 0.125 + 0 + 0 + 0.015 625 + 0.007 812 5 + 0 + 0.001 953 125 + 0 + 0.000 488 281 25 + 0 + 0.000 122 070 312 5 + 0 + 0 + 0 + 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 + 0 + 0 + 0 + 0 + 0.000 000 007 450 580 596 923 828 125 + 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 + 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 + 0.000 000 000 001 818 989 403 545 856 475 830 078 125 + 0.000 000 000 000 909 494 701 772 928 237 915 039 062 5 + 0.000 000 000 000 454 747 350 886 464 118 957 519 531 25 + 0 + 0 + 0.000 000 000 000 056 843 418 860 808 014 869 689 941 406 25 + 0 + 0 + 0 + 0 + 0 + 0.000 000 000 000 000 888 178 419 700 125 232 338 905 334 472 656 25 + 0 + 0.000 000 000 000 000 222 044 604 925 031 308 084 726 333 618 164 062 5 =


0.5 + 0.25 + 0.125 + 0.015 625 + 0.007 812 5 + 0.001 953 125 + 0.000 488 281 25 + 0.000 122 070 312 5 + 0.000 007 629 394 531 25 + 0.000 001 907 348 632 812 5 + 0.000 000 476 837 158 203 125 + 0.000 000 007 450 580 596 923 828 125 + 0.000 000 003 725 290 298 461 914 062 5 + 0.000 000 000 232 830 643 653 869 628 906 25 + 0.000 000 000 029 103 830 456 733 703 613 281 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 + 0.000 000 000 000 454 747 350 886 464 118 957 519 531 25 + 0.000 000 000 000 056 843 418 860 808 014 869 689 941 406 25 + 0.000 000 000 000 000 888 178 419 700 125 232 338 905 334 472 656 25 + 0.000 000 000 000 000 222 044 604 925 031 308 084 726 333 618 164 062 5 =


0.901 011 001 602 058 714 254 894 766 781 944 781 541 824 340 820 312 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 + 0.901 011 001 602 058 714 254 894 766 781 944 781 541 824 340 820 312 5) × 2-160 =


-1.901 011 001 602 058 714 254 894 766 781 944 781 541 824 340 820 312 5 × 2-160 =


-0.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 001 300 724 510 356 224 238 699 885 282 736 736 025 282 076 870 570 857 824 565 970 015 478 393 307 502 822 733 853 530 789 200 784 320 358 208 826 898 608 620 410 745 587 148 149 127 187 934 936 955 571 174 621 582 031 25

Conclusion:

1 - 011 0101 1111 - 1110 0110 1010 1000 1010 1000 0011 0001 0011 0111 1001 0000 0101
converted from
64 bit double precision IEEE 754 binary floating point
to
base ten decimal system (double) =

-0.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 001 300 724 510 356 224 238 699 885 282 736 736 025 282 076 870 570 857 824 565 970 015 478 393 307 502 822 733 853 530 789 200 784 320 358 208 826 898 608 620 410 745 587 148 149 127 187 934 936 955 571 174 621 582 031 25(10)

More operations of this kind:

1 - 011 0101 1111 - 1110 0110 1010 1000 1010 1000 0011 0001 0011 0111 1001 0000 0100 = ?

1 - 011 0101 1111 - 1110 0110 1010 1000 1010 1000 0011 0001 0011 0111 1001 0000 0110 = ?


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 - 011 0101 1111 - 1110 0110 1010 1000 1010 1000 0011 0001 0011 0111 1001 0000 0101 = -0.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 001 300 724 510 356 224 238 699 885 282 736 736 025 282 076 870 570 857 824 565 970 015 478 393 307 502 822 733 853 530 789 200 784 320 358 208 826 898 608 620 410 745 587 148 149 127 187 934 936 955 571 174 621 582 031 25 Oct 21 07:16 UTC (GMT)
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 Oct 21 07:15 UTC (GMT)
0 - 100 0001 0101 - 0001 0100 1011 0011 0011 1111 0000 0010 1010 1011 1111 1101 1111 = 4 533 455.752 609 222 196 042 537 689 208 984 375 Oct 21 07:14 UTC (GMT)
0 - 100 0001 1010 - 0111 1110 1110 1110 1110 1111 1111 0101 0011 0100 0111 1101 1011 = 200 767 359.662 657 588 720 321 655 273 437 5 Oct 21 07:13 UTC (GMT)
0 - 001 1011 1100 - 0011 0111 1000 1011 0011 1110 1111 0010 1011 1110 1000 1100 0000 = 0.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 615 052 004 544 171 247 797 952 106 146 667 686 891 797 545 880 744 216 083 194 535 561 992 456 830 260 835 561 432 024 039 243 686 434 058 396 130 827 398 415 486 825 000 165 4 Oct 21 07:12 UTC (GMT)
0 - 000 0000 0000 - 1000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 0 Oct 21 07:10 UTC (GMT)
0 - 000 0000 0000 - 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0001 0000 = 0 Oct 21 07:09 UTC (GMT)
0 - 011 1010 1101 - 1011 0100 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0001 = 0.000 000 000 000 000 000 000 000 352 197 995 188 593 860 609 090 061 166 019 651 273 589 423 088 486 636 593 381 123 909 668 808 935 613 526 500 674 197 450 280 189 514 160 156 25 Oct 21 07:08 UTC (GMT)
0 - 011 1110 1000 - 0001 0010 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 0.000 000 127 591 192 722 320 556 640 625 Oct 21 07:07 UTC (GMT)
1 - 101 1000 0000 - 1100 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = -137 907 021 687 380 677 242 976 640 350 502 648 317 779 087 446 629 063 337 819 026 914 860 026 200 240 237 139 949 931 892 097 204 742 823 197 966 073 856 Oct 21 07:06 UTC (GMT)
1 - 100 0000 0010 - 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = -8 Oct 21 07:05 UTC (GMT)
0 - 100 1010 1101 - 1110 0100 1010 1000 1001 0100 1010 0101 0011 0001 1001 0100 1100 = 45 333 069 940 373 986 391 476 432 776 396 710 987 832 706 377 711 616 Oct 21 07:05 UTC (GMT)
0 - 011 1111 1111 - 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 = 1.000 000 000 000 000 444 089 209 850 062 616 169 452 667 236 328 125 Oct 21 07:04 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)