64 bit double precision IEEE 754 binary floating point number 0 - 100 0000 0011 - 1110 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1000 converted to decimal base ten (double)

How to convert 64 bit double precision IEEE 754 binary floating point:
0 - 100 0000 0011 - 1110 0000 0000 0000 0000 0000 0000 0000 0000 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:
100 0000 0011


The last 52 bits contain the mantissa:
1110 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1000

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

100 0000 0011(2) =


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


1,024 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 2 + 1 =


1,024 + 2 + 1 =


1,027(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 = 1,027 - 1023 = 4

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):

1110 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1000(2) =

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


0.5 + 0.25 + 0.125 + 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 + 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 + 0 + 0 =


0.5 + 0.25 + 0.125 + 0.000 000 000 000 001 776 356 839 400 250 464 677 810 668 945 312 5 =


0.875 000 000 000 001 776 356 839 400 250 464 677 810 668 945 312 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 + 0.875 000 000 000 001 776 356 839 400 250 464 677 810 668 945 312 5) × 24 =


1.875 000 000 000 001 776 356 839 400 250 464 677 810 668 945 312 5 × 24 =


30.000 000 000 000 028 421 709 430 404 007 434 844 970 703 125

0 - 100 0000 0011 - 1110 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1000
converted from
64 bit double precision IEEE 754 binary floating point
to
base ten decimal system (double) =


30.000 000 000 000 028 421 709 430 404 007 434 844 970 703 125(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 0000 0011 - 1110 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1000 = 30.000 000 000 000 028 421 709 430 404 007 434 844 970 703 125 Jun 03 10:10 UTC (GMT)
0 - 100 0001 1000 - 0011 1010 1001 1101 0101 1110 1010 1000 0100 1100 1110 0001 1001 = 41 237 181.314 846 225 082 874 298 095 703 125 Jun 03 10:05 UTC (GMT)
1 - 101 1100 1010 - 0001 0010 0110 0011 1000 0010 1010 1111 0101 0010 1010 1000 0010 = -1 595 490 736 029 241 154 837 177 729 981 154 941 948 596 263 229 714 338 765 839 157 038 697 237 824 497 339 153 227 557 953 318 823 221 288 655 574 310 275 200 659 640 852 120 862 720 Jun 03 10:00 UTC (GMT)
0 - 011 1111 1110 - 1100 0000 0000 0000 1000 0000 0000 0000 0000 0000 0000 0000 0000 = 0.875 003 814 697 265 625 Jun 03 09:58 UTC (GMT)
0 - 011 1111 1110 - 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 = 0.999 999 999 999 999 888 977 697 537 484 345 957 636 833 190 917 968 75 Jun 03 09:56 UTC (GMT)
0 - 100 0000 1010 - 0110 1000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 2 880 Jun 03 09:47 UTC (GMT)
0 - 110 0001 1101 - 1111 1111 1111 1111 1000 0111 0101 0111 1111 1111 1111 1111 1111 = 28 792 944 749 983 618 636 582 128 014 993 202 816 466 701 853 267 102 750 818 674 794 443 833 617 058 955 485 731 444 975 551 576 037 737 775 668 612 746 345 080 470 741 747 298 599 219 514 275 205 695 270 050 332 672 Jun 03 09:37 UTC (GMT)
0 - 111 1111 1111 - 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = +∞ (Infinity, positive) Jun 03 09:32 UTC (GMT)
0 - 100 0000 1010 - 1010 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 3 328 Jun 03 08:53 UTC (GMT)
1 - 100 0000 1110 - 1000 1001 0001 0100 0000 0100 0000 0010 0000 0000 1000 0000 0000 = -50 314.007 827 773 690 223 693 847 656 25 Jun 03 08:52 UTC (GMT)
1 - 111 1111 1111 - 0001 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = SNaN, Signalling Not a Number Jun 03 08:49 UTC (GMT)
0 - 100 0000 0011 - 1100 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 28 Jun 03 08:37 UTC (GMT)
0 - 100 0001 1000 - 0010 0000 1111 1111 1111 0100 1010 1010 1101 0000 1000 0000 1010 = 37 879 785.334 487 989 544 868 469 238 281 25 Jun 03 08:32 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)