64 bit double precision IEEE 754 binary floating point number 1 - 101 1000 0000 - 1100 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 converted to decimal base ten (double)

How to convert 64 bit double precision IEEE 754 binary floating point:
1 - 101 1000 0000 - 1100 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000
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:
101 1000 0000


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

2. Convert the exponent from binary (base 2) to decimal (base 10):

The exponent is allways a positive integer.

101 1000 0000(2) =


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


1,024 + 0 + 256 + 128 + 0 + 0 + 0 + 0 + 0 + 0 + 0 =


1,024 + 256 + 128 =


1,408(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 = 1,408 - 1023 = 385


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)

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

1 × 2-1 + 1 × 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 + 0 × 2-49 + 0 × 2-50 + 0 × 2-51 + 0 × 2-52 =


0.5 + 0.25 + 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 + 0 =


0.5 + 0.25 =


0.75(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 + 0.75) × 2385 =


-1.75 × 2385 =


-137 907 021 687 380 677 242 976 640 350 502 648 317 779 087 446 629 063 337 819 026 914 860 026 200 240 237 139 949 931 892 097 204 742 823 197 966 073 856

Conclusion:

1 - 101 1000 0000 - 1100 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000
converted from
64 bit double precision IEEE 754 binary floating point
to
base ten decimal system (double) =

-137 907 021 687 380 677 242 976 640 350 502 648 317 779 087 446 629 063 337 819 026 914 860 026 200 240 237 139 949 931 892 097 204 742 823 197 966 073 856(10)

More operations of this kind:

1 - 101 1000 0000 - 1011 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 = ?

1 - 101 1000 0000 - 1100 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0001 = ?


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 - 101 1000 0000 - 1100 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = -137 907 021 687 380 677 242 976 640 350 502 648 317 779 087 446 629 063 337 819 026 914 860 026 200 240 237 139 949 931 892 097 204 742 823 197 966 073 856 Nov 29 23:08 UTC (GMT)
1 - 110 1101 1010 - 1010 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = -18 356 090 731 603 413 730 169 207 474 330 081 111 783 800 427 035 253 910 442 112 550 607 509 897 347 790 249 388 180 486 468 994 980 310 836 125 738 845 977 757 801 429 628 209 541 159 572 595 075 165 149 691 800 174 594 637 654 783 722 108 471 202 572 930 204 422 836 612 424 447 377 276 928 Nov 29 23:05 UTC (GMT)
0 - 100 0001 0010 - 1111 0001 1001 0100 0000 1111 1111 1100 1100 1110 1001 1011 0110 = 1 019 040.499 610 236 613 079 905 509 948 730 468 75 Nov 29 23:05 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 Nov 29 23:04 UTC (GMT)
1 - 100 0111 0010 - 1000 0111 0110 1011 0010 1011 0001 1111 0000 0011 1111 1111 1100 = -63 511 303 310 142 696 000 182 433 523 695 616 Nov 29 23:02 UTC (GMT)
0 - 100 0001 0110 - 1010 0100 0110 1111 1111 1111 0110 0010 0001 1001 1011 0110 1110 = 13 776 895.691 602 434 962 987 899 780 273 437 5 Nov 29 23:01 UTC (GMT)
0 - 100 0001 1001 - 0110 1100 1110 1001 0010 1100 0000 0111 0110 0001 1001 0111 1010 = 95 659 184.115 331 560 373 306 274 414 062 5 Nov 29 23:00 UTC (GMT)
0 - 100 0001 1000 - 1000 0011 1101 0011 1110 1111 1111 0101 0001 0001 1110 1000 0100 = 50 833 375.914 608 985 185 623 168 945 312 5 Nov 29 23:00 UTC (GMT)
0 - 100 0000 1000 - 1111 1000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 1 008 Nov 29 23:00 UTC (GMT)
0 - 011 1110 1100 - 1111 1111 0101 0011 1000 1111 0000 0000 0000 0000 0000 0000 0000 = 0.000 003 809 678 560 173 779 260 367 155 075 073 242 187 5 Nov 29 23:00 UTC (GMT)
0 - 100 0000 0111 - 1100 1100 0010 0011 1101 0111 0000 1010 0011 1101 0111 0000 1001 = 460.139 999 999 999 929 514 160 612 598 061 561 584 472 656 25 Nov 29 22:59 UTC (GMT)
0 - 100 0000 1001 - 0010 0000 0001 0000 0000 0000 0000 0000 0000 0000 0001 0010 0000 = 1 152.250 000 000 065 483 618 527 650 833 129 882 812 5 Nov 29 22:59 UTC (GMT)
1 - 100 1101 0110 - 1000 1100 0000 0100 0011 0000 0100 0000 1110 0010 1100 0011 1001 = -81 455 840 909 427 417 337 246 493 223 197 667 724 784 370 830 030 238 870 096 838 656 Nov 29 22:59 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)