64 bit double precision IEEE 754 binary floating point number 1 - 100 0000 0101 - 0101 0110 0100 0000 0000 0000 0000 0000 0000 0000 0000 0000 0001 converted to decimal base ten (double)

64 bit double precision IEEE 754 binary floating point 1 - 100 0000 0101 - 0101 0110 0100 0000 0000 0000 0000 0000 0000 0000 0000 0000 0001 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:
100 0000 0101


The last 52 bits contain the mantissa:
0101 0110 0100 0000 0000 0000 0000 0000 0000 0000 0000 0000 0001

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

The exponent is allways a positive integer.

100 0000 0101(2) =


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


1,024 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 4 + 0 + 1 =


1,024 + 4 + 1 =


1,029(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 = 1,029 - 1023 = 6


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)

0101 0110 0100 0000 0000 0000 0000 0000 0000 0000 0000 0000 0001(2) =

0 × 2-1 + 1 × 2-2 + 0 × 2-3 + 1 × 2-4 + 0 × 2-5 + 1 × 2-6 + 1 × 2-7 + 0 × 2-8 + 0 × 2-9 + 1 × 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 + 1 × 2-52 =


0 + 0.25 + 0 + 0.062 5 + 0 + 0.015 625 + 0.007 812 5 + 0 + 0 + 0.000 976 562 5 + 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 + 0.000 000 000 000 000 222 044 604 925 031 308 084 726 333 618 164 062 5 =


0.25 + 0.062 5 + 0.015 625 + 0.007 812 5 + 0.000 976 562 5 + 0.000 000 000 000 000 222 044 604 925 031 308 084 726 333 618 164 062 5 =


0.336 914 062 500 000 222 044 604 925 031 308 084 726 333 618 164 062 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 + 0.336 914 062 500 000 222 044 604 925 031 308 084 726 333 618 164 062 5) × 26 =


-1.336 914 062 500 000 222 044 604 925 031 308 084 726 333 618 164 062 5 × 26 =


-85.562 500 000 000 014 210 854 715 202 003 717 422 485 351 562 5

1 - 100 0000 0101 - 0101 0110 0100 0000 0000 0000 0000 0000 0000 0000 0000 0000 0001 converted from 64 bit double precision IEEE 754 binary floating point to base ten decimal system (double) =
-85.562 500 000 000 014 210 854 715 202 003 717 422 485 351 562 5(10)

More operations of this kind:

1 - 100 0000 0101 - 0101 0110 0100 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = ?

1 - 100 0000 0101 - 0101 0110 0100 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 = ?


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 - 100 0000 0101 - 0101 0110 0100 0000 0000 0000 0000 0000 0000 0000 0000 0000 0001 = ? Apr 14 10:18 UTC (GMT)
0 - 100 0000 0101 - 0101 0101 1111 1111 1111 1111 1001 0000 0000 0000 0000 0000 0000 = ? Apr 14 10:17 UTC (GMT)
0 - 010 0000 0100 - 0001 1000 1010 1110 1010 1100 1010 1111 1111 0110 1101 0011 0011 = ? Apr 14 10:17 UTC (GMT)
0 - 000 0000 0000 - 0000 0000 0000 0000 0000 0100 0000 1011 0100 0110 1101 0000 0110 = ? Apr 14 10:16 UTC (GMT)
0 - 100 0001 0111 - 0001 0010 0111 0010 0000 0011 1110 1100 0111 0101 1100 1101 1110 = ? Apr 14 10:14 UTC (GMT)
0 - 100 0000 0011 - 0011 0110 0110 0110 0110 0110 0110 0110 0110 0110 0110 0110 0101 = ? Apr 14 10:14 UTC (GMT)
1 - 011 0001 0001 - 0010 0010 0110 1110 0000 1111 1100 0010 1010 0000 1111 1000 0000 = ? Apr 14 10:14 UTC (GMT)
1 - 100 0000 0000 - 1000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = ? Apr 14 10:13 UTC (GMT)
1 - 011 1110 1011 - 0100 1111 1000 1011 0101 1000 1000 1110 0011 0110 1000 1111 0000 = ? Apr 14 10:12 UTC (GMT)
0 - 011 1111 1011 - 1011 1000 0101 0001 1110 1011 1000 0101 0001 1110 1011 1000 0000 = ? Apr 14 10:10 UTC (GMT)
0 - 100 0000 0101 - 0011 1001 1101 0000 0000 0000 0000 0000 0000 0000 0000 0000 0001 = ? Apr 14 10:10 UTC (GMT)
0 - 100 0000 0110 - 0101 1001 0000 1111 1111 1111 1111 1111 1111 1111 1111 1111 1001 = ? Apr 14 10:10 UTC (GMT)
1 - 111 1111 1111 - 1111 1111 1111 1111 1111 1111 1111 1111 1111 1001 0100 0111 0011 = ? Apr 14 10:10 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)