64 bit double precision IEEE 754 binary floating point number 0 - 100 1010 1000 - 0110 1100 0011 0111 1010 1000 1010 0000 0000 0000 0000 0000 0000 converted to decimal base ten (double)

How to convert 64 bit double precision IEEE 754 binary floating point:
0 - 100 1010 1000 - 0110 1100 0011 0111 1010 1000 1010 0000 0000 0000 0000 0000 0000.

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 1010 1000


The last 52 bits contain the mantissa:
0110 1100 0011 0111 1010 1000 1010 0000 0000 0000 0000 0000 0000

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

100 1010 1000(2) =


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


1,024 + 0 + 0 + 128 + 0 + 32 + 0 + 8 + 0 + 0 + 0 =


1,024 + 128 + 32 + 8 =


1,192(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 = 1,192 - 1023 = 169

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):

0110 1100 0011 0111 1010 1000 1010 0000 0000 0000 0000 0000 0000(2) =

0 × 2-1 + 1 × 2-2 + 1 × 2-3 + 0 × 2-4 + 1 × 2-5 + 1 × 2-6 + 0 × 2-7 + 0 × 2-8 + 0 × 2-9 + 0 × 2-10 + 1 × 2-11 + 1 × 2-12 + 0 × 2-13 + 1 × 2-14 + 1 × 2-15 + 1 × 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 + 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 + 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 + 0 × 2-52 =


0 + 0.25 + 0.125 + 0 + 0.031 25 + 0.015 625 + 0 + 0 + 0 + 0 + 0.000 488 281 25 + 0.000 244 140 625 + 0 + 0.000 061 035 156 25 + 0.000 030 517 578 125 + 0.000 015 258 789 062 5 + 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 + 0.000 000 029 802 322 387 695 312 5 + 0 + 0.000 000 007 450 580 596 923 828 125 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 =


0.25 + 0.125 + 0.031 25 + 0.015 625 + 0.000 488 281 25 + 0.000 244 140 625 + 0.000 061 035 156 25 + 0.000 030 517 578 125 + 0.000 015 258 789 062 5 + 0.000 007 629 394 531 25 + 0.000 001 907 348 632 812 5 + 0.000 000 476 837 158 203 125 + 0.000 000 029 802 322 387 695 312 5 + 0.000 000 007 450 580 596 923 828 125 =


0.422 724 284 231 662 750 244 140 625(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 + 0.422 724 284 231 662 750 244 140 625) × 2169 =


1.422 724 284 231 662 750 244 140 625 × 2169 =


1 064 608 701 888 006 151 252 602 686 964 999 266 829 445 191 696 384

0 - 100 1010 1000 - 0110 1100 0011 0111 1010 1000 1010 0000 0000 0000 0000 0000 0000
converted from
64 bit double precision IEEE 754 binary floating point
to
base ten decimal system (double) =


1 064 608 701 888 006 151 252 602 686 964 999 266 829 445 191 696 384(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 1010 1000 - 0110 1100 0011 0111 1010 1000 1010 0000 0000 0000 0000 0000 0000 = 1 064 608 701 888 006 151 252 602 686 964 999 266 829 445 191 696 384 Dec 12 00:13 UTC (GMT)
1 - 100 0000 0110 - 0100 0000 0110 1010 0000 0101 1001 1010 0111 0011 1011 0100 0011 = -160.207 074 000 000 005 753 463 483 415 544 033 050 537 109 375 Dec 12 00:09 UTC (GMT)
0 - 100 0001 0001 - 1000 1011 0110 1010 1111 0100 0001 1000 0100 0010 1001 1101 0000 = 404 907.813 980 725 593 864 917 755 126 953 125 Dec 12 00:08 UTC (GMT)
0 - 000 0000 0000 - 1101 1100 1000 0100 0110 1011 0101 1011 0000 0000 0000 0000 0000 = 0 Dec 12 00:06 UTC (GMT)
0 - 000 1110 1100 - 1000 1010 0110 1011 1101 1000 0000 0000 0000 0000 0000 0000 0000 = 0.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 001 892 839 404 563 030 969 718 881 298 740 470 443 655 319 022 170 502 571 233 212 264 204 350 7 Dec 12 00:05 UTC (GMT)
0 - 100 0001 1000 - 1001 0001 0111 1011 1110 0111 0000 0011 1101 1101 1110 1010 0000 = 52 623 310.030 209 779 739 379 882 812 5 Dec 12 00:03 UTC (GMT)
0 - 111 1111 1111 - 0001 1111 1111 1111 1111 1000 0000 0000 0000 0000 0000 0000 0000 = SNaN, Signalling Not a Number Dec 12 00:03 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 Dec 12 00:02 UTC (GMT)
1 - 111 1100 0110 - 1100 1100 1100 1101 0100 0000 1100 0110 1111 1100 1001 1110 0000 = -2 245 328 971 236 279 803 757 806 293 868 626 284 142 583 072 539 824 605 203 195 507 396 828 364 822 105 444 825 803 858 528 555 408 079 091 660 699 319 927 705 388 681 351 349 252 342 020 131 169 186 922 703 498 125 638 356 245 639 596 424 468 203 794 654 578 821 226 463 947 761 939 855 555 462 077 010 927 238 560 338 434 724 803 165 416 998 323 061 121 608 355 544 801 478 973 587 456 Dec 12 00:01 UTC (GMT)
0 - 011 1111 1110 - 1000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 0.75 Dec 12 00:00 UTC (GMT)
1 - 100 0110 0010 - 1000 1101 1010 1010 0001 1110 1010 1001 1100 0110 0011 1000 0101 = -984 569 688 729 487 743 750 756 106 240 Dec 12 00:00 UTC (GMT)
1 - 100 0101 0111 - 0000 0100 0001 0011 0100 0000 0000 0010 0001 0000 0000 0000 0001 = -314 411 618 803 217 682 516 672 512 Dec 12 00:00 UTC (GMT)
0 - 100 0001 1110 - 0001 1100 0111 0100 1010 1110 0000 0000 0000 0000 0000 0000 0000 = 2 386 188 032 Dec 12 00:00 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)