64 bit double precision IEEE 754 binary floating point number 0 - 101 1111 0000 - 0011 0010 1000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 converted to decimal base ten (double)

How to convert 64 bit double precision IEEE 754 binary floating point:
0 - 101 1111 0000 - 0011 0010 1000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000
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:
101 1111 0000


The last 52 bits contain the mantissa:
0011 0010 1000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000

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

The exponent is allways a positive integer.

101 1111 0000(2) =


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


1,024 + 0 + 256 + 128 + 64 + 32 + 16 + 0 + 0 + 0 + 0 =


1,024 + 256 + 128 + 64 + 32 + 16 =


1,520(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 = 1,520 - 1023 = 497


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)

0011 0010 1000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000(2) =

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


0 + 0 + 0.125 + 0.062 5 + 0 + 0 + 0.007 812 5 + 0 + 0.001 953 125 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 =


0.125 + 0.062 5 + 0.007 812 5 + 0.001 953 125 =


0.197 265 625(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 + 0.197 265 625) × 2497 =


1.197 265 625 × 2497 =


489 889 756 503 988 028 886 251 290 076 982 002 028 516 009 332 128 592 762 025 783 360 090 336 229 508 179 234 239 396 141 547 668 050 169 320 647 020 098 517 736 859 927 399 352 036 545 504 739 328

Conclusion:

0 - 101 1111 0000 - 0011 0010 1000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000
converted from
64 bit double precision IEEE 754 binary floating point
to
base ten decimal system (double) =

489 889 756 503 988 028 886 251 290 076 982 002 028 516 009 332 128 592 762 025 783 360 090 336 229 508 179 234 239 396 141 547 668 050 169 320 647 020 098 517 736 859 927 399 352 036 545 504 739 328(10)

More operations of this kind:

0 - 101 1111 0000 - 0011 0010 0111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 = ?

0 - 101 1111 0000 - 0011 0010 1000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0001 = ?


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 - 101 1111 0000 - 0011 0010 1000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 489 889 756 503 988 028 886 251 290 076 982 002 028 516 009 332 128 592 762 025 783 360 090 336 229 508 179 234 239 396 141 547 668 050 169 320 647 020 098 517 736 859 927 399 352 036 545 504 739 328 Oct 21 07:35 UTC (GMT)
0 - 000 0000 0000 - 0000 0000 0000 0000 0000 0001 1010 0001 1101 0000 0000 1000 1000 = 0 Oct 21 07:34 UTC (GMT)
0 - 111 1111 1011 - 1001 1001 1001 1001 0110 1010 1100 1010 0010 0000 0010 0010 0101 = 17 976 900 000 000 001 131 263 126 597 574 217 263 396 481 902 578 175 138 403 913 437 777 028 790 527 439 603 364 502 930 460 677 390 792 873 005 190 202 715 684 097 701 052 772 925 098 274 606 265 671 862 282 068 943 709 466 085 560 282 706 513 077 345 426 748 148 655 344 345 059 351 783 647 273 667 856 596 377 154 335 538 285 408 147 908 746 020 334 699 795 216 859 612 853 175 701 809 318 359 744 905 216 Oct 21 07:32 UTC (GMT)
0 - 000 0001 1000 - 0000 0011 1011 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 = 0.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 189 386 8 Oct 21 07:30 UTC (GMT)
1 - 010 0001 0010 - 0101 1110 1111 1000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = -0 Oct 21 07:29 UTC (GMT)
0 - 100 0000 0100 - 1001 0001 1001 1001 1001 1001 1001 1001 1001 1001 1001 1001 1010 = 50.200 000 000 000 002 842 170 943 040 400 743 484 497 070 312 5 Oct 21 07:28 UTC (GMT)
0 - 100 0000 0011 - 0111 1010 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0001 = 23.625 000 000 000 003 552 713 678 800 500 929 355 621 337 890 625 Oct 21 07:28 UTC (GMT)
0 - 100 0000 0100 - 0001 1110 0010 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 35.765 625 Oct 21 07:25 UTC (GMT)
0 - 011 1100 1101 - 1001 1000 0110 1001 0001 1010 0011 0000 0010 1100 1011 1000 0111 = 0.000 000 000 000 001 416 958 758 139 293 351 562 612 952 461 978 951 177 907 506 096 820 267 060 820 697 224 698 960 781 097 412 109 375 Oct 21 07:22 UTC (GMT)
0 - 100 0001 0100 - 1000 0010 1110 1110 1011 0100 0101 1101 0010 0101 1111 1001 0110 = 3 169 750.545 482 586 137 950 420 379 638 671 875 Oct 21 07:21 UTC (GMT)
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)
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)