64 bit double precision IEEE 754 binary floating point number 0 - 011 1010 1101 - 1011 0100 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0001 converted to decimal base ten (double)

How to convert 64 bit double precision IEEE 754 binary floating point:
0 - 011 1010 1101 - 1011 0100 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0001
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 1010 1101


The last 52 bits contain the mantissa:
1011 0100 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0001

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

The exponent is allways a positive integer.

011 1010 1101(2) =


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


0 + 512 + 256 + 128 + 0 + 32 + 0 + 8 + 4 + 0 + 1 =


512 + 256 + 128 + 32 + 8 + 4 + 1 =


941(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 = 941 - 1023 = -82


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 0100 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0001(2) =

1 × 2-1 + 0 × 2-2 + 1 × 2-3 + 1 × 2-4 + 0 × 2-5 + 1 × 2-6 + 0 × 2-7 + 0 × 2-8 + 0 × 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 + 1 × 2-52 =


0.5 + 0 + 0.125 + 0.062 5 + 0 + 0.015 625 + 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 + 0 + 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 + 0.015 625 + 0.000 000 000 000 000 222 044 604 925 031 308 084 726 333 618 164 062 5 =


0.703 125 000 000 000 222 044 604 925 031 308 084 726 333 618 164 062 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 + 0.703 125 000 000 000 222 044 604 925 031 308 084 726 333 618 164 062 5) × 2-82 =


1.703 125 000 000 000 222 044 604 925 031 308 084 726 333 618 164 062 5 × 2-82 =


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

Conclusion:

0 - 011 1010 1101 - 1011 0100 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0001
converted from
64 bit double precision IEEE 754 binary floating point
to
base ten decimal system (double) =

0.000 000 000 000 000 000 000 000 352 197 995 188 593 860 609 090 061 166 019 651 273 589 423 088 486 636 593 381 123 909 668 808 935 613 526 500 674 197 450 280 189 514 160 156 25(10)

More operations of this kind:

0 - 011 1010 1101 - 1011 0100 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = ?

0 - 011 1010 1101 - 1011 0100 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 = ?


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 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 Nov 29 22:06 UTC (GMT)
1 - 101 1100 1011 - 1111 1010 0101 0100 0011 1111 1101 0011 1100 0110 1110 1111 0010 = -5 888 313 541 400 969 261 334 440 588 458 420 370 499 169 642 722 017 378 660 695 026 763 208 464 851 677 153 871 817 481 294 454 644 482 703 975 895 014 986 236 485 102 986 495 787 008 Nov 29 22:05 UTC (GMT)
0 - 000 0000 0100 - 1100 1110 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 0.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 1 Nov 29 22:04 UTC (GMT)
1 - 011 1111 1100 - 1100 1010 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 = -0.223 632 812 500 000 055 511 151 231 257 827 021 181 583 404 541 015 625 Nov 29 22:03 UTC (GMT)
0 - 100 0001 0101 - 1101 0101 1010 0001 0001 0000 0100 1111 0111 0010 1111 0011 0101 = 7 694 404.077 586 938 627 064 228 057 861 328 125 Nov 29 22:01 UTC (GMT)
0 - 111 1111 1001 - 1010 1000 0001 1100 0010 0000 0000 0000 0000 0000 0000 0000 0000 = 4 653 438 657 520 132 554 137 100 919 186 945 963 059 966 535 683 786 320 763 351 905 397 772 024 664 718 212 333 873 841 383 003 306 824 458 991 093 547 798 049 902 875 454 053 865 275 233 088 218 049 482 003 738 537 537 076 753 038 312 040 358 620 629 569 649 463 904 835 973 324 050 907 325 900 106 709 665 118 665 332 025 888 537 574 753 061 499 771 868 179 996 580 943 670 766 022 949 793 777 347 198 976 Nov 29 22:01 UTC (GMT)
1 - 001 1101 1011 - 0110 1010 0100 1100 1010 0001 0000 1011 0000 0100 0000 0000 0001 = -0 Nov 29 21:57 UTC (GMT)
0 - 100 1010 1101 - 0000 0100 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 24 319 387 245 186 224 558 909 315 616 398 949 447 074 717 515 120 640 Nov 29 21:55 UTC (GMT)
0 - 100 0001 1011 - 0101 0001 1000 1111 1011 0101 1111 0111 1011 1001 0010 1110 1100 = 353 958 751.482 710 599 899 291 992 187 5 Nov 29 21:53 UTC (GMT)
0 - 100 0001 0100 - 0000 0000 0001 0101 0000 0000 0000 0000 0000 0000 0000 0000 0001 = 2 097 824.000 000 000 465 661 287 307 739 257 812 5 Nov 29 21:52 UTC (GMT)
0 - 000 0000 0000 - 0100 0100 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 0 Nov 29 21:51 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 Nov 29 21:50 UTC (GMT)
0 - 000 0000 0000 - 1111 1000 0011 0001 0100 0010 0001 0111 0011 1111 0000 0000 0000 = 0 Nov 29 21:49 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)