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 0100 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 0100
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 0100

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 0100(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 + 0 × 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 =


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 =


0.901 011 001 602 058 492 210 289 841 750 636 696 815 490 722 656 25(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 + 0.901 011 001 602 058 492 210 289 841 750 636 696 815 490 722 656 25) × 2-160 =


-1.901 011 001 602 058 492 210 289 841 750 636 696 815 490 722 656 25 × 2-160 =


-0.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 001 300 724 510 356 224 086 770 801 350 579 956 065 710 200 557 579 543 357 030 516 677 559 273 847 026 342 880 589 399 098 883 805 775 828 305 918 555 991 018 922 303 459 337 541 653 439 984 656 870 365 142 822 265 625

Conclusion:

1 - 011 0101 1111 - 1110 0110 1010 1000 1010 1000 0011 0001 0011 0111 1001 0000 0100
converted from
64 bit double precision IEEE 754 binary floating point
to
base ten decimal system (double) =

-0.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 001 300 724 510 356 224 086 770 801 350 579 956 065 710 200 557 579 543 357 030 516 677 559 273 847 026 342 880 589 399 098 883 805 775 828 305 918 555 991 018 922 303 459 337 541 653 439 984 656 870 365 142 822 265 625(10)

More operations of this kind:

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

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


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 0100 = -0.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 001 300 724 510 356 224 086 770 801 350 579 956 065 710 200 557 579 543 357 030 516 677 559 273 847 026 342 880 589 399 098 883 805 775 828 305 918 555 991 018 922 303 459 337 541 653 439 984 656 870 365 142 822 265 625 Nov 27 00:00 UTC (GMT)
1 - 000 0000 0000 - 1010 0011 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 = -0 Nov 26 23:56 UTC (GMT)
1 - 100 0011 1111 - 1000 0111 0101 1100 1001 0110 0000 0000 0000 0000 0000 0000 0000 = -28 200 579 893 431 369 728 Nov 26 23:55 UTC (GMT)
0 - 100 0000 0100 - 1111 1010 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 63.25 Nov 26 23:53 UTC (GMT)
1 - 010 0000 1100 - 0010 0010 1101 0010 0011 1111 1111 1111 1111 1111 1111 1111 1111 = -0 Nov 26 23:52 UTC (GMT)
0 - 100 0000 0000 - 1111 0000 0000 0000 1111 0000 1111 0000 1110 1111 1111 1111 1111 = 3.875 028 722 424 758 154 119 217 579 136 602 580 547 332 763 671 875 Nov 26 23:49 UTC (GMT)
0 - 100 0001 0000 - 0001 0101 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0001 = 141 824.000 000 000 029 103 830 456 733 703 613 281 25 Nov 26 23:47 UTC (GMT)
0 - 100 0000 0001 - 0111 0101 1110 0000 1011 1010 1011 0000 1010 0101 1111 1011 1111 = 5.841 841 385 372 332 773 329 162 591 835 483 908 653 259 277 343 75 Nov 26 23:47 UTC (GMT)
0 - 100 0001 0001 - 0000 0100 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 266 240 Nov 26 23:47 UTC (GMT)
1 - 100 0000 1001 - 0100 0000 0000 0010 0000 0000 0000 0000 0000 0000 0000 0000 0000 = -1 280.031 25 Nov 26 23:45 UTC (GMT)
0 - 101 1111 0000 - 0011 0010 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 489 090 588 875 107 134 875 017 601 186 154 951 454 244 368 207 606 360 147 405 839 178 426 241 064 370 319 235 488 597 779 163 414 105 552 404 952 653 018 422 275 625 245 625 454 235 507 094 454 272 Nov 26 23:42 UTC (GMT)
1 - 100 0000 0001 - 1100 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = -7 Nov 26 23:41 UTC (GMT)
0 - 000 0001 0000 - 0000 0010 0000 0000 0110 1010 0010 1000 0111 0111 0000 0001 1011 = 0.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 734 8 Nov 26 23:41 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)