Binary ↘ Double: The 64 Bit Double Precision IEEE 754 Binary Floating Point Standard Representation Number 1 - 111 1001 0101 - 1011 1111 1100 0110 0001 0000 1111 0110 0000 0000 0000 0000 0000 Converted and Written as a Base Ten Decimal System Number (as a Double)

1 - 111 1001 0101 - 1011 1111 1100 0110 0001 0000 1111 0110 0000 0000 0000 0000 0000: 64 bit double precision IEEE 754 binary floating point standard representation number converted to decimal system (base ten)

1. Identify the elements that make up the binary representation of the number:

The first bit (the leftmost) indicates the sign,
1 = negative, 0 = positive.
1


The next 11 bits contain the exponent:
111 1001 0101


The last 52 bits contain the mantissa:
1011 1111 1100 0110 0001 0000 1111 0110 0000 0000 0000 0000 0000


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

The exponent is allways a positive integer.

111 1001 0101(2) =


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


1,024 + 512 + 256 + 128 + 0 + 0 + 16 + 0 + 4 + 0 + 1 =


1,024 + 512 + 256 + 128 + 16 + 4 + 1 =


1,941(10)

3. Adjust the exponent.

Subtract the excess bits: 2(11 - 1) - 1 = 1023,

that is due to the 11 bit excess/bias notation.


The exponent, adjusted = 1,941 - 1023 = 918


4. Convert the mantissa from binary (from base 2) to decimal (in base 10).

The mantissa represents the fractional part of the number (what comes after the whole part of the number, separated from it by a comma).


1011 1111 1100 0110 0001 0000 1111 0110 0000 0000 0000 0000 0000(2) =

1 × 2-1 + 0 × 2-2 + 1 × 2-3 + 1 × 2-4 + 1 × 2-5 + 1 × 2-6 + 1 × 2-7 + 1 × 2-8 + 1 × 2-9 + 1 × 2-10 + 0 × 2-11 + 0 × 2-12 + 0 × 2-13 + 1 × 2-14 + 1 × 2-15 + 0 × 2-16 + 0 × 2-17 + 0 × 2-18 + 0 × 2-19 + 1 × 2-20 + 0 × 2-21 + 0 × 2-22 + 0 × 2-23 + 0 × 2-24 + 1 × 2-25 + 1 × 2-26 + 1 × 2-27 + 1 × 2-28 + 0 × 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 + 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 + 0.125 + 0.062 5 + 0.031 25 + 0.015 625 + 0.007 812 5 + 0.003 906 25 + 0.001 953 125 + 0.000 976 562 5 + 0 + 0 + 0 + 0.000 061 035 156 25 + 0.000 030 517 578 125 + 0 + 0 + 0 + 0 + 0.000 000 953 674 316 406 25 + 0 + 0 + 0 + 0 + 0.000 000 029 802 322 387 695 312 5 + 0.000 000 014 901 161 193 847 656 25 + 0.000 000 007 450 580 596 923 828 125 + 0.000 000 003 725 290 298 461 914 062 5 + 0 + 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 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 =


0.5 + 0.125 + 0.062 5 + 0.031 25 + 0.015 625 + 0.007 812 5 + 0.003 906 25 + 0.001 953 125 + 0.000 976 562 5 + 0.000 061 035 156 25 + 0.000 030 517 578 125 + 0.000 000 953 674 316 406 25 + 0.000 000 029 802 322 387 695 312 5 + 0.000 000 014 901 161 193 847 656 25 + 0.000 000 007 450 580 596 923 828 125 + 0.000 000 003 725 290 298 461 914 062 5 + 0.000 000 000 931 322 574 615 478 515 625 + 0.000 000 000 465 661 287 307 739 257 812 5 =


0.749 116 001 185 029 745 101 928 710 937 5(10)

5. Put all the numbers into expression to calculate the double precision floating point decimal value:

(-1)Sign × (1 + Mantissa) × 2(Adjusted exponent) =


(-1)1 × (1 + 0.749 116 001 185 029 745 101 928 710 937 5) × 2918 =


-1.749 116 001 185 029 745 101 928 710 937 5 × 2918 =


-3 875 739 974 753 834 706 075 163 486 913 256 793 856 570 234 510 836 280 412 770 513 148 715 349 080 425 257 484 493 363 182 218 688 710 603 984 309 181 221 006 228 382 042 997 444 461 286 993 983 555 596 632 805 592 952 453 024 263 932 572 099 783 602 225 529 645 252 313 717 991 116 420 093 794 541 292 813 610 327 893 705 685 088 582 916 171 632 511 843 259 383 808

1 - 111 1001 0101 - 1011 1111 1100 0110 0001 0000 1111 0110 0000 0000 0000 0000 0000 converted from a 64 bit double precision IEEE 754 binary floating point standard representation number to a decimal system number, written in base ten (double) = -3 875 739 974 753 834 706 075 163 486 913 256 793 856 570 234 510 836 280 412 770 513 148 715 349 080 425 257 484 493 363 182 218 688 710 603 984 309 181 221 006 228 382 042 997 444 461 286 993 983 555 596 632 805 592 952 453 024 263 932 572 099 783 602 225 529 645 252 313 717 991 116 420 093 794 541 292 813 610 327 893 705 685 088 582 916 171 632 511 843 259 383 808(10)

Spaces were used to group digits: for binary, by 4, for decimal, by 3.

The latest 64 bit double precision IEEE 754 floating point binary standard numbers converted and written as decimal system numbers (in base ten, double)

The number 1 - 111 1001 0101 - 1011 1111 1100 0110 0001 0000 1111 0110 0000 0000 0000 0000 0000 converted from 64 bit double precision IEEE 754 binary floating point system and written as a decimal number (double) written in base ten = ? Mar 29 05:07 UTC (GMT)
The number 0 - 011 1111 1110 - 1111 1110 0000 1101 0011 1011 0100 0000 1011 0111 1001 0111 1010 converted from 64 bit double precision IEEE 754 binary floating point system and written as a decimal number (double) written in base ten = ? Mar 29 05:07 UTC (GMT)
The number 1 - 100 0001 0000 - 1000 0010 1010 1010 0101 1111 1010 1010 1010 1010 0010 0011 0010 converted from 64 bit double precision IEEE 754 binary floating point system and written as a decimal number (double) written in base ten = ? Mar 29 05:06 UTC (GMT)
The number 0 - 100 0000 0010 - 0011 0001 1000 1001 0011 0111 0100 1011 1100 0110 1010 0111 0011 converted from 64 bit double precision IEEE 754 binary floating point system and written as a decimal number (double) written in base ten = ? Mar 29 05:06 UTC (GMT)
The number 1 - 000 0000 0000 - 1111 1101 1100 1101 1101 0111 1101 0110 1111 0000 0111 1011 0010 converted from 64 bit double precision IEEE 754 binary floating point system and written as a decimal number (double) written in base ten = ? Mar 29 05:04 UTC (GMT)
The number 0 - 111 1100 0000 - 1110 1001 0011 1000 0000 0000 0000 0000 0000 0000 0000 0000 0000 converted from 64 bit double precision IEEE 754 binary floating point system and written as a decimal number (double) written in base ten = ? Mar 29 05:04 UTC (GMT)
The number 0 - 100 0001 1000 - 1010 1011 0011 1110 1111 1111 1111 1111 1111 1111 1111 1011 0101 converted from 64 bit double precision IEEE 754 binary floating point system and written as a decimal number (double) written in base ten = ? Mar 29 05:04 UTC (GMT)
All 64 bit double precision IEEE 754 binary floating point representation numbers converted to base ten decimal numbers (double)

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)