64 bit double precision IEEE 754 binary floating point number 0 - 100 0001 1000 - 1000 1010 0110 1011 0100 0010 1000 0100 0011 0101 1001 0100 0110 converted to decimal base ten (double)

How to convert 64 bit double precision IEEE 754 binary floating point:
0 - 100 0001 1000 - 1000 1010 0110 1011 0100 0010 1000 0100 0011 0101 1001 0100 0110.

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 1000


The last 52 bits contain the mantissa:
1000 1010 0110 1011 0100 0010 1000 0100 0011 0101 1001 0100 0110

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

100 0001 1000(2) =


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


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


1,024 + 16 + 8 =


1,048(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 = 1,048 - 1023 = 25

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 1010 0110 1011 0100 0010 1000 0100 0011 0101 1001 0100 0110(2) =

1 × 2-1 + 0 × 2-2 + 0 × 2-3 + 0 × 2-4 + 1 × 2-5 + 0 × 2-6 + 1 × 2-7 + 0 × 2-8 + 0 × 2-9 + 1 × 2-10 + 1 × 2-11 + 0 × 2-12 + 1 × 2-13 + 0 × 2-14 + 1 × 2-15 + 1 × 2-16 + 0 × 2-17 + 1 × 2-18 + 0 × 2-19 + 0 × 2-20 + 0 × 2-21 + 0 × 2-22 + 1 × 2-23 + 0 × 2-24 + 1 × 2-25 + 0 × 2-26 + 0 × 2-27 + 0 × 2-28 + 0 × 2-29 + 1 × 2-30 + 0 × 2-31 + 0 × 2-32 + 0 × 2-33 + 0 × 2-34 + 1 × 2-35 + 1 × 2-36 + 0 × 2-37 + 1 × 2-38 + 0 × 2-39 + 1 × 2-40 + 1 × 2-41 + 0 × 2-42 + 0 × 2-43 + 1 × 2-44 + 0 × 2-45 + 1 × 2-46 + 0 × 2-47 + 0 × 2-48 + 0 × 2-49 + 1 × 2-50 + 1 × 2-51 + 0 × 2-52 =


0.5 + 0 + 0 + 0 + 0.031 25 + 0 + 0.007 812 5 + 0 + 0 + 0.000 976 562 5 + 0.000 488 281 25 + 0 + 0.000 122 070 312 5 + 0 + 0.000 030 517 578 125 + 0.000 015 258 789 062 5 + 0 + 0.000 003 814 697 265 625 + 0 + 0 + 0 + 0 + 0.000 000 119 209 289 550 781 25 + 0 + 0.000 000 029 802 322 387 695 312 5 + 0 + 0 + 0 + 0 + 0.000 000 000 931 322 574 615 478 515 625 + 0 + 0 + 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 + 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 + 0.000 000 000 000 014 210 854 715 202 003 717 422 485 351 562 5 + 0 + 0 + 0 + 0.000 000 000 000 000 888 178 419 700 125 232 338 905 334 472 656 25 + 0.000 000 000 000 000 444 089 209 850 062 616 169 452 667 236 328 125 + 0 =


0.5 + 0.031 25 + 0.007 812 5 + 0.000 976 562 5 + 0.000 488 281 25 + 0.000 122 070 312 5 + 0.000 030 517 578 125 + 0.000 015 258 789 062 5 + 0.000 003 814 697 265 625 + 0.000 000 119 209 289 550 781 25 + 0.000 000 029 802 322 387 695 312 5 + 0.000 000 000 931 322 574 615 478 515 625 + 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 + 0.000 000 000 000 909 494 701 772 928 237 915 039 062 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 + 0.000 000 000 000 014 210 854 715 202 003 717 422 485 351 562 5 + 0.000 000 000 000 000 888 178 419 700 125 232 338 905 334 472 656 25 + 0.000 000 000 000 000 444 089 209 850 062 616 169 452 667 236 328 125 =


0.540 699 155 118 617 991 178 098 236 559 890 210 628 509 521 484 375(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 + 0.540 699 155 118 617 991 178 098 236 559 890 210 628 509 521 484 375) × 225 =


1.540 699 155 118 617 991 178 098 236 559 890 210 628 509 521 484 375 × 225 =


51 697 285.032 885 119 318 962 097 167 968 75

0 - 100 0001 1000 - 1000 1010 0110 1011 0100 0010 1000 0100 0011 0101 1001 0100 0110
converted from
64 bit double precision IEEE 754 binary floating point
to
base ten decimal system (double) =


51 697 285.032 885 119 318 962 097 167 968 75(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 1000 - 1000 1010 0110 1011 0100 0010 1000 0100 0011 0101 1001 0100 0110 = 51 697 285.032 885 119 318 962 097 167 968 75 Jul 19 16:28 UTC (GMT)
0 - 110 0001 0010 - 0000 0100 1010 1110 0110 1100 0110 0100 0010 1101 1000 0100 0000 = 7 158 098 650 486 605 062 682 343 968 676 472 237 252 701 633 419 710 363 299 580 437 065 553 747 172 154 642 059 767 957 845 356 450 045 175 467 179 225 693 921 233 514 564 618 935 305 306 183 745 279 048 548 352 Jul 19 16:25 UTC (GMT)
1 - 111 1111 1111 - 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = -∞ (Infinity, negative) Jul 19 16:22 UTC (GMT)
0 - 100 0000 0101 - 0101 1110 0000 0000 0000 0001 0000 1100 0110 1111 0111 1010 0001 = 87.500 004 000 000 004 111 825 546 715 408 563 613 891 601 562 5 Jul 19 16:21 UTC (GMT)
0 - 011 1111 1111 - 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0001 0000 = 1.000 000 000 000 003 552 713 678 800 500 929 355 621 337 890 625 Jul 19 16:19 UTC (GMT)
1 - 100 0100 1111 - 1001 0001 0101 1001 0101 0001 0101 0010 1010 1010 0001 1011 0101 = -1 895 316 579 786 387 326 238 720 Jul 19 16:17 UTC (GMT)
0 - 010 0000 0100 - 1011 0100 1000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 0.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 004 069 457 161 461 112 804 308 132 950 249 463 275 018 537 281 066 347 011 686 897 190 555 675 643 853 883 300 130 800 424 574 733 420 209 124 768 526 465 572 855 518 319 766 957 305 764 127 763 685 146 181 2 Jul 19 16:13 UTC (GMT)
0 - 100 0001 1100 - 1111 1111 1111 1111 1111 1111 1100 0000 0000 0000 0000 0000 0000 = 1 073 741 816 Jul 19 16:08 UTC (GMT)
0 - 100 0001 0011 - 0000 1101 1111 1010 1110 0111 0110 0010 1101 1111 0100 1010 1100 = 1 105 838.461 638 729 088 008 403 778 076 171 875 Jul 19 16:04 UTC (GMT)
1 - 000 0000 0000 - 1110 0011 1101 0111 0000 1010 0011 1101 0111 0000 1010 0011 1101 = -0 Jul 19 16:04 UTC (GMT)
0 - 000 0000 0010 - 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 0 Jul 19 16:01 UTC (GMT)
0 - 100 0000 0010 - 0101 0111 0000 1010 0011 0000 0000 0000 0000 0000 0000 0000 0000 = 10.719 993 591 308 593 75 Jul 19 15:55 UTC (GMT)
0 - 100 0010 0011 - 0001 1011 0110 1101 1101 0000 0000 0000 0000 0000 0000 0000 0000 = 76 082 380 800 Jul 19 15:54 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)