64 bit double precision IEEE 754 binary floating point number 0 - 100 0001 1010 - 0011 1111 1101 0011 0101 1001 1101 0011 1100 1100 1010 1000 0010 converted to decimal base ten (double)

How to convert 64 bit double precision IEEE 754 binary floating point:
0 - 100 0001 1010 - 0011 1111 1101 0011 0101 1001 1101 0011 1100 1100 1010 1000 0010.

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 0001 1010


The last 52 bits contain the mantissa:
0011 1111 1101 0011 0101 1001 1101 0011 1100 1100 1010 1000 0010

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

100 0001 1010(2) =


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


1,024 + 0 + 0 + 0 + 0 + 0 + 16 + 8 + 0 + 2 + 0 =


1,024 + 16 + 8 + 2 =


1,050(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 = 1,050 - 1023 = 27

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):

0011 1111 1101 0011 0101 1001 1101 0011 1100 1100 1010 1000 0010(2) =

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


0 + 0 + 0.125 + 0.062 5 + 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 + 0.000 030 517 578 125 + 0.000 015 258 789 062 5 + 0 + 0.000 003 814 697 265 625 + 0 + 0.000 000 953 674 316 406 25 + 0.000 000 476 837 158 203 125 + 0 + 0 + 0.000 000 059 604 644 775 390 625 + 0.000 000 029 802 322 387 695 312 5 + 0.000 000 014 901 161 193 847 656 25 + 0 + 0.000 000 003 725 290 298 461 914 062 5 + 0 + 0 + 0.000 000 000 465 661 287 307 739 257 812 5 + 0.000 000 000 232 830 643 653 869 628 906 25 + 0.000 000 000 116 415 321 826 934 814 453 125 + 0.000 000 000 058 207 660 913 467 407 226 562 5 + 0 + 0 + 0.000 000 000 007 275 957 614 183 425 903 320 312 5 + 0.000 000 000 003 637 978 807 091 712 951 660 156 25 + 0 + 0 + 0.000 000 000 000 454 747 350 886 464 118 957 519 531 25 + 0 + 0.000 000 000 000 113 686 837 721 616 029 739 379 882 812 5 + 0 + 0.000 000 000 000 028 421 709 430 404 007 434 844 970 703 125 + 0 + 0 + 0 + 0 + 0 + 0.000 000 000 000 000 444 089 209 850 062 616 169 452 667 236 328 125 + 0 =


0.125 + 0.062 5 + 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 + 0.000 030 517 578 125 + 0.000 015 258 789 062 5 + 0.000 003 814 697 265 625 + 0.000 000 953 674 316 406 25 + 0.000 000 476 837 158 203 125 + 0.000 000 059 604 644 775 390 625 + 0.000 000 029 802 322 387 695 312 5 + 0.000 000 014 901 161 193 847 656 25 + 0.000 000 003 725 290 298 461 914 062 5 + 0.000 000 000 465 661 287 307 739 257 812 5 + 0.000 000 000 232 830 643 653 869 628 906 25 + 0.000 000 000 116 415 321 826 934 814 453 125 + 0.000 000 000 058 207 660 913 467 407 226 562 5 + 0.000 000 000 007 275 957 614 183 425 903 320 312 5 + 0.000 000 000 003 637 978 807 091 712 951 660 156 25 + 0.000 000 000 000 454 747 350 886 464 118 957 519 531 25 + 0.000 000 000 000 113 686 837 721 616 029 739 379 882 812 5 + 0.000 000 000 000 028 421 709 430 404 007 434 844 970 703 125 + 0.000 000 000 000 000 444 089 209 850 062 616 169 452 667 236 328 125 =


0.249 318 708 618 972 539 881 042 393 972 165 882 587 432 861 328 125(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 + 0.249 318 708 618 972 539 881 042 393 972 165 882 587 432 861 328 125) × 227 =


1.249 318 708 618 972 539 881 042 393 972 165 882 587 432 861 328 125 × 227 =


167 680 718.618 732 511 997 222 900 390 625

0 - 100 0001 1010 - 0011 1111 1101 0011 0101 1001 1101 0011 1100 1100 1010 1000 0010
converted from
64 bit double precision IEEE 754 binary floating point
to
base ten decimal system (double) =


167 680 718.618 732 511 997 222 900 390 625(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 - 100 0001 1010 - 0011 1111 1101 0011 0101 1001 1101 0011 1100 1100 1010 1000 0010 = 167 680 718.618 732 511 997 222 900 390 625 Aug 25 04:54 UTC (GMT)
0 - 100 0001 0111 - 0010 1110 0111 1111 1101 1000 0000 1111 0100 0000 0000 1001 0110 = 19 824 600.059 570 871 293 544 769 287 109 375 Aug 25 04:54 UTC (GMT)
0 - 000 0000 0001 - 1000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 0 Aug 25 04:53 UTC (GMT)
0 - 100 0010 1101 - 0110 1011 1100 1100 0100 0001 1110 1001 0000 0000 0000 0000 0000 = 100 000 000 000 000 Aug 25 04:51 UTC (GMT)
1 - 100 0000 0011 - 0010 0001 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = -18.062 5 Aug 25 04:49 UTC (GMT)
1 - 000 0000 0000 - 0010 0101 1010 1001 0001 0000 0010 0100 0000 0000 0000 0000 0000 = -0 Aug 25 04:49 UTC (GMT)
1 - 100 1000 1011 - 1010 0010 1111 0100 1110 1011 0010 0110 0011 1110 0000 1111 0101 = -2 281 017 311 509 344 046 224 521 182 127 760 082 993 152 Aug 25 04:49 UTC (GMT)
0 - 000 0001 0101 - 1100 0000 0000 0000 0000 1011 1000 0000 0000 0100 0000 1000 0001 = 0.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 040 830 1 Aug 25 04:44 UTC (GMT)
1 - 100 0000 0010 - 0001 0011 0011 0011 0011 0011 0011 0011 0011 0011 0011 0011 0011 = -8.599 999 999 999 999 644 728 632 119 949 907 064 437 866 210 937 5 Aug 25 04:43 UTC (GMT)
1 - 100 0011 0000 - 1111 0001 1000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = -1 094 014 069 637 120 Aug 25 04:42 UTC (GMT)
0 - 100 0000 0000 - 1010 1111 0111 1011 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 3.370 941 162 109 375 Aug 25 04:42 UTC (GMT)
0 - 011 1111 1111 - 0011 1110 1011 1000 0101 0001 1110 1011 1000 0101 0001 1110 1100 = 1.245 000 000 000 000 106 581 410 364 015 027 880 668 640 136 718 75 Aug 25 04:42 UTC (GMT)
0 - 000 0000 0001 - 1000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 0 Aug 25 04:41 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)