64 bit double precision IEEE 754 binary floating point number 0 - 011 1111 1111 - 0101 0011 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 1111 1111 - 0101 0011 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0001.

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 1111


The last 52 bits contain the mantissa:
0101 0011 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0001

2. Convert the exponent, that is allways a positive integer, from binary (base 2) to decimal (base 10):

011 1111 1111(2) =


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


0 + 512 + 256 + 128 + 64 + 32 + 16 + 8 + 4 + 2 + 1 =


512 + 256 + 128 + 64 + 32 + 16 + 8 + 4 + 2 + 1 =


1,023(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 = 1,023 - 1023 = 0

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):

0101 0011 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0001(2) =

0 × 2-1 + 1 × 2-2 + 0 × 2-3 + 1 × 2-4 + 0 × 2-5 + 0 × 2-6 + 1 × 2-7 + 1 × 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 + 0.25 + 0 + 0.062 5 + 0 + 0 + 0.007 812 5 + 0.003 906 25 + 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 =


0.25 + 0.062 5 + 0.007 812 5 + 0.003 906 25 + 0.000 000 000 000 000 222 044 604 925 031 308 084 726 333 618 164 062 5 =


0.324 218 750 000 000 222 044 604 925 031 308 084 726 333 618 164 062 5(10)

Conclusion:

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 + 0.324 218 750 000 000 222 044 604 925 031 308 084 726 333 618 164 062 5) × 20 =


1.324 218 750 000 000 222 044 604 925 031 308 084 726 333 618 164 062 5 × 20 =


1.324 218 750 000 000 222 044 604 925 031 308 084 726 333 618 164 062 5

0 - 011 1111 1111 - 0101 0011 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) =


1.324 218 750 000 000 222 044 604 925 031 308 084 726 333 618 164 062 5(10)

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 1111 - 0101 0011 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0001 = 1.324 218 750 000 000 222 044 604 925 031 308 084 726 333 618 164 062 5 Jun 18 09:15 UTC (GMT)
1 - 100 0000 0111 - 1100 0000 0111 1010 0011 1010 0000 0000 0000 0000 0000 0000 0000 = -448.477 447 509 765 625 Jun 18 09:14 UTC (GMT)
1 - 100 1100 0111 - 0111 0110 0110 0000 0110 0000 1001 0011 0101 1010 1010 0101 0110 = -2 349 999 212 290 342 694 039 218 688 336 252 332 275 839 940 449 760 595 410 944 Jun 18 09:14 UTC (GMT)
0 - 011 1111 0000 - 1101 0011 0100 1000 0010 1011 1110 1000 1011 1100 0001 0110 1001 = 0.000 055 704 345 703 124 994 858 713 298 073 737 973 936 658 818 274 736 404 418 945 312 5 Jun 18 09:13 UTC (GMT)
0 - 000 0111 1100 - 0100 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 0.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 295 763 046 541 693 677 743 187 899 379 506 632 8 Jun 18 09:11 UTC (GMT)
0 - 011 0111 0111 - 0100 1001 0100 0111 1011 1101 1011 0101 1110 0011 1111 1001 1100 = 0.000 000 000 000 000 000 000 000 000 000 000 000 000 014 765 436 561 480 878 676 891 108 523 061 052 917 694 300 423 277 484 568 837 503 572 922 127 941 354 390 218 840 310 480 130 975 566 924 987 629 757 737 295 221 886 597 573 757 171 630 859 375 Jun 18 09:10 UTC (GMT)
0 - 111 1111 1000 - 1110 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 2 633 339 553 020 970 567 962 849 400 569 860 449 635 708 465 247 519 393 653 760 954 458 974 743 244 643 014 639 284 335 777 454 140 934 376 668 162 178 613 637 579 928 254 117 430 993 547 569 784 757 921 743 814 959 376 340 803 006 977 083 294 940 128 331 827 926 019 805 932 701 691 965 097 885 875 921 115 340 524 080 187 898 981 885 017 246 356 244 603 124 227 235 426 948 969 309 111 664 984 729 845 760 Jun 18 09:07 UTC (GMT)
1 - 100 0000 0101 - 1011 0001 1100 0010 1000 1111 0100 1110 0001 0100 0111 1010 0111 = -108.439 999 790 191 549 777 773 616 369 813 680 648 803 710 937 5 Jun 18 09:06 UTC (GMT)
1 - 100 0000 0110 - 1000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = -192 Jun 18 09:04 UTC (GMT)
0 - 011 1111 1000 - 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0110 = 0.007 812 500 000 000 010 408 340 855 860 842 566 471 546 888 351 440 429 687 5 Jun 18 09:04 UTC (GMT)
0 - 000 0000 0000 - 0010 0000 0010 0000 0010 0000 0010 0000 0010 0000 0101 1000 0000 = 0 Jun 18 09:03 UTC (GMT)
1 - 011 1111 1101 - 1100 1100 1100 1101 0000 0000 0000 0000 0000 0000 0000 0000 0000 = -0.450 000 762 939 453 125 Jun 18 09:03 UTC (GMT)
0 - 011 1111 0100 - 0110 0011 1110 0100 1100 1000 1110 0001 1111 1101 1110 0110 1100 = 0.000 678 813 343 482 920 824 450 976 837 738 380 709 197 372 198 104 858 398 437 5 Jun 18 09:03 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)