64 bit double precision IEEE 754 binary floating point number 0 - 010 0001 1000 - 1111 0100 0011 1000 1101 1010 1010 0000 0110 0000 0000 0000 1000 converted to decimal base ten (double)

How to convert 64 bit double precision IEEE 754 binary floating point:
0 - 010 0001 1000 - 1111 0100 0011 1000 1101 1010 1010 0000 0110 0000 0000 0000 1000.

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:
010 0001 1000


The last 52 bits contain the mantissa:
1111 0100 0011 1000 1101 1010 1010 0000 0110 0000 0000 0000 1000

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

010 0001 1000(2) =


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


0 + 512 + 0 + 0 + 0 + 0 + 16 + 8 + 0 + 0 + 0 =


512 + 16 + 8 =


536(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 = 536 - 1023 = -487

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):

1111 0100 0011 1000 1101 1010 1010 0000 0110 0000 0000 0000 1000(2) =

1 × 2-1 + 1 × 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 + 1 × 2-11 + 1 × 2-12 + 1 × 2-13 + 0 × 2-14 + 0 × 2-15 + 0 × 2-16 + 1 × 2-17 + 1 × 2-18 + 0 × 2-19 + 1 × 2-20 + 1 × 2-21 + 0 × 2-22 + 1 × 2-23 + 0 × 2-24 + 1 × 2-25 + 0 × 2-26 + 1 × 2-27 + 0 × 2-28 + 0 × 2-29 + 0 × 2-30 + 0 × 2-31 + 0 × 2-32 + 0 × 2-33 + 1 × 2-34 + 1 × 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 + 1 × 2-49 + 0 × 2-50 + 0 × 2-51 + 0 × 2-52 =


0.5 + 0.25 + 0.125 + 0.062 5 + 0 + 0.015 625 + 0 + 0 + 0 + 0 + 0.000 488 281 25 + 0.000 244 140 625 + 0.000 122 070 312 5 + 0 + 0 + 0 + 0.000 007 629 394 531 25 + 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 + 0.000 000 119 209 289 550 781 25 + 0 + 0.000 000 029 802 322 387 695 312 5 + 0 + 0.000 000 007 450 580 596 923 828 125 + 0 + 0 + 0 + 0 + 0 + 0 + 0.000 000 000 058 207 660 913 467 407 226 562 5 + 0.000 000 000 029 103 830 456 733 703 613 281 25 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0.000 000 000 000 001 776 356 839 400 250 464 677 810 668 945 312 5 + 0 + 0 + 0 =


0.5 + 0.25 + 0.125 + 0.062 5 + 0.015 625 + 0.000 488 281 25 + 0.000 244 140 625 + 0.000 122 070 312 5 + 0.000 007 629 394 531 25 + 0.000 003 814 697 265 625 + 0.000 000 953 674 316 406 25 + 0.000 000 476 837 158 203 125 + 0.000 000 119 209 289 550 781 25 + 0.000 000 029 802 322 387 695 312 5 + 0.000 000 007 450 580 596 923 828 125 + 0.000 000 000 058 207 660 913 467 407 226 562 5 + 0.000 000 000 029 103 830 456 733 703 613 281 25 + 0.000 000 000 000 001 776 356 839 400 250 464 677 810 668 945 312 5 =


0.953 992 523 340 277 287 502 431 136 090 308 427 810 668 945 312 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 + 0.953 992 523 340 277 287 502 431 136 090 308 427 810 668 945 312 5) × 2-487 =


1.953 992 523 340 277 287 502 431 136 090 308 427 810 668 945 312 5 × 2-487 =


0.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 004 890 069 248 867 174 909 091 468 481 550 199 098 634 382 576 342 990 970 713 022 212 346 673 306 879 677 380 149 470 239 614 710 521 622 033 813 005 222 721 761 678 014 243 155 292 474 784 181 438 597 413 090 940 6

0 - 010 0001 1000 - 1111 0100 0011 1000 1101 1010 1010 0000 0110 0000 0000 0000 1000
converted from
64 bit double precision IEEE 754 binary floating point
to
base ten decimal system (double) =


0.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 004 890 069 248 867 174 909 091 468 481 550 199 098 634 382 576 342 990 970 713 022 212 346 673 306 879 677 380 149 470 239 614 710 521 622 033 813 005 222 721 761 678 014 243 155 292 474 784 181 438 597 413 090 940 6(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 - 010 0001 1000 - 1111 0100 0011 1000 1101 1010 1010 0000 0110 0000 0000 0000 1000 = 0.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 004 890 069 248 867 174 909 091 468 481 550 199 098 634 382 576 342 990 970 713 022 212 346 673 306 879 677 380 149 470 239 614 710 521 622 033 813 005 222 721 761 678 014 243 155 292 474 784 181 438 597 413 090 940 6 Jun 19 19:18 UTC (GMT)
1 - 100 0000 0100 - 0100 0111 1111 1111 1111 1001 0000 0000 0000 0000 0000 0000 0000 = -40.999 986 648 559 570 312 5 Jun 19 19:18 UTC (GMT)
1 - 101 1111 1110 - 0000 0000 0000 0000 0000 0000 0000 0000 1011 0000 0100 0000 1101 = -6 703 903 966 045 943 256 596 664 455 459 930 392 200 504 982 615 228 739 437 299 624 569 509 602 037 250 085 004 983 900 787 816 941 498 011 968 109 645 906 984 947 337 584 590 496 995 080 585 296 740 352 Jun 19 19:16 UTC (GMT)
0 - 100 0001 1011 - 0000 1100 0111 1010 1100 1101 1111 1001 0111 1000 1111 0000 0011 = 281 521 375.592 025 935 649 871 826 171 875 Jun 19 19:12 UTC (GMT)
0 - 110 0110 0101 - 1100 1010 0000 1110 1010 1001 0101 0101 1100 0100 1010 1110 1010 = 121 645 809 849 328 817 838 453 027 721 477 665 430 371 818 032 316 022 618 992 546 564 605 743 267 254 531 046 833 805 967 248 094 130 521 624 124 275 271 003 622 589 874 418 714 440 228 137 384 492 468 931 423 112 757 283 680 325 852 425 355 264 Jun 19 19:11 UTC (GMT)
1 - 100 0111 1110 - 1111 1111 1111 1111 1111 1110 0000 0000 0000 0000 0000 0000 0000 = -340 282 346 638 528 859 811 704 183 484 516 925 440 Jun 19 19:10 UTC (GMT)
0 - 011 1111 1011 - 1001 1001 1001 1001 1001 1001 1001 1001 1001 1001 1001 1001 1010 = 0.100 000 000 000 000 005 551 115 123 125 782 702 118 158 340 454 101 562 5 Jun 19 19:05 UTC (GMT)
1 - 000 0111 1100 - 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = -0.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 236 610 437 233 354 942 194 550 319 503 605 306 3 Jun 19 19:05 UTC (GMT)
0 - 011 1111 1100 - 1001 1001 1001 1001 1001 1001 1001 1001 1001 1001 1001 1001 1001 = 0.199 999 999 999 999 983 346 654 630 622 651 893 645 524 978 637 695 312 5 Jun 19 19:03 UTC (GMT)
0 - 100 0000 0010 - 1000 1000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 12.25 Jun 19 19:01 UTC (GMT)
0 - 011 1111 1000 - 1011 0011 0011 0011 0011 0010 0000 0000 0000 0000 0000 0000 0000 = 0.013 281 249 441 206 455 230 712 890 625 Jun 19 18:58 UTC (GMT)
1 - 100 0000 0100 - 1011 0001 1100 0010 1000 1111 0100 1110 0001 0100 0111 1010 0111 = -54.219 999 895 095 774 888 886 808 184 906 840 324 401 855 468 75 Jun 19 18:56 UTC (GMT)
0 - 000 0000 0001 - 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0001 = 0 Jun 19 18:53 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)