64 bit double precision IEEE 754 binary floating point number 0 - 000 0000 0011 - 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1100 converted to decimal base ten (double)

How to convert 64 bit double precision IEEE 754 binary floating point:
0 - 000 0000 0011 - 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1100.

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:
000 0000 0011


The last 52 bits contain the mantissa:
0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1100

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

000 0000 0011(2) =


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


0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 2 + 1 =


2 + 1 =


3(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 = 3 - 1023 = -1020

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):

0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1100(2) =

0 × 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 + 0 × 2-11 + 0 × 2-12 + 0 × 2-13 + 0 × 2-14 + 0 × 2-15 + 0 × 2-16 + 0 × 2-17 + 0 × 2-18 + 0 × 2-19 + 0 × 2-20 + 0 × 2-21 + 0 × 2-22 + 0 × 2-23 + 0 × 2-24 + 0 × 2-25 + 0 × 2-26 + 0 × 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 + 1 × 2-49 + 1 × 2-50 + 0 × 2-51 + 0 × 2-52 =


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 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 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 + 0.000 000 000 000 000 888 178 419 700 125 232 338 905 334 472 656 25 + 0 + 0 =


0.000 000 000 000 001 776 356 839 400 250 464 677 810 668 945 312 5 + 0.000 000 000 000 000 888 178 419 700 125 232 338 905 334 472 656 25 =


0.000 000 000 000 002 664 535 259 100 375 697 016 716 003 417 968 75(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 + 0.000 000 000 000 002 664 535 259 100 375 697 016 716 003 417 968 75) × 2-1020 =


1.000 000 000 000 002 664 535 259 100 375 697 016 716 003 417 968 75 × 2-1020 =


0

0 - 000 0000 0011 - 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1100
converted from
64 bit double precision IEEE 754 binary floating point
to
base ten decimal system (double) =


0(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 - 000 0000 0011 - 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1100 = 0 Mar 25 07:44 UTC (GMT)
1 - 001 1101 0111 - 1000 0010 0000 0011 0100 0000 0000 0010 0001 0000 0000 0000 0001 = -0.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 102 283 131 622 246 595 645 548 124 409 543 463 611 879 232 936 954 243 306 834 404 755 764 975 713 566 553 948 135 668 983 702 360 025 871 015 765 040 067 021 928 887 307 355 652 288 838 8 Mar 25 07:37 UTC (GMT)
0 - 011 1111 1110 - 0010 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 0.562 5 Mar 25 07:35 UTC (GMT)
0 - 100 0100 1000 - 1111 0111 0001 0100 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 18 560 306 841 913 326 043 136 Mar 25 07:32 UTC (GMT)
0 - 100 0000 1001 - 0101 0110 0101 0110 1010 1100 1000 0000 0000 0000 0000 0000 0000 = 1 369.354 278 564 453 125 Mar 25 07:30 UTC (GMT)
1 - 100 0000 0011 - 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = -16 Mar 25 07:29 UTC (GMT)
1 - 011 1111 1110 - 0111 1111 1000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = -0.749 023 437 5 Mar 25 07:29 UTC (GMT)
0 - 100 0000 0000 - 0001 1011 0110 1011 0011 1010 0101 0110 0001 1000 1001 1111 0111 = 2.214 209 835 089 828 271 037 504 237 028 770 148 754 119 873 046 875 Mar 25 07:29 UTC (GMT)
1 - 011 1000 0001 - 1110 0000 1001 0100 0000 0000 0000 0000 0000 0100 0000 0000 0000 = -0.000 000 000 000 000 000 000 000 000 000 000 000 022 067 065 276 068 824 344 366 864 336 481 199 213 765 376 181 764 592 893 150 941 173 457 722 816 921 101 812 457 153 834 579 060 003 306 949 511 170 387 268 066 406 25 Mar 25 07:29 UTC (GMT)
0 - 100 0001 0111 - 1110 0001 1101 1001 1110 1011 0011 1010 0011 1100 1010 0000 0000 = 31 578 603.227 487 564 086 914 062 5 Mar 25 07:26 UTC (GMT)
0 - 100 0001 0110 - 1100 1111 1101 1000 1111 1100 1110 0100 0000 1110 0001 0110 1111 = 15 199 358.445 419 995 114 207 267 761 230 468 75 Mar 25 07:26 UTC (GMT)
0 - 000 0000 0000 - 0111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1000 = 0 Mar 25 07:23 UTC (GMT)
1 - 100 0001 0100 - 0111 1000 1011 0000 0011 1001 1111 0111 0110 0110 0110 1001 0100 = -3 085 831.245 800 802 484 154 701 232 910 156 25 Mar 25 07:20 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)