-0.000 282 027 7 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.000 282 027 7(10) to 64 bit double precision IEEE 754 binary floating point representation standard (1 bit for sign, 11 bits for exponent, 52 bits for mantissa)

What are the steps to convert decimal number
-0.000 282 027 7(10) to 64 bit double precision IEEE 754 binary floating point representation (1 bit for sign, 11 bits for exponent, 52 bits for mantissa)

1. Start with the positive version of the number:

|-0.000 282 027 7| = 0.000 282 027 7


2. First, convert to binary (in base 2) the integer part: 0.
Divide the number repeatedly by 2.

Keep track of each remainder.

We stop when we get a quotient that is equal to zero.


  • division = quotient + remainder;
  • 0 ÷ 2 = 0 + 0;

3. Construct the base 2 representation of the integer part of the number.

Take all the remainders starting from the bottom of the list constructed above.

0(10) =

0(2)


4. Convert to binary (base 2) the fractional part: 0.000 282 027 7.

Multiply it repeatedly by 2.

Keep track of each integer part of the results.

Stop when we get a fractional part that is equal to zero.


  • #) multiplying = integer + fractional part;
  • 1) 0.000 282 027 7 × 2 = 0 + 0.000 564 055 4;
  • 2) 0.000 564 055 4 × 2 = 0 + 0.001 128 110 8;
  • 3) 0.001 128 110 8 × 2 = 0 + 0.002 256 221 6;
  • 4) 0.002 256 221 6 × 2 = 0 + 0.004 512 443 2;
  • 5) 0.004 512 443 2 × 2 = 0 + 0.009 024 886 4;
  • 6) 0.009 024 886 4 × 2 = 0 + 0.018 049 772 8;
  • 7) 0.018 049 772 8 × 2 = 0 + 0.036 099 545 6;
  • 8) 0.036 099 545 6 × 2 = 0 + 0.072 199 091 2;
  • 9) 0.072 199 091 2 × 2 = 0 + 0.144 398 182 4;
  • 10) 0.144 398 182 4 × 2 = 0 + 0.288 796 364 8;
  • 11) 0.288 796 364 8 × 2 = 0 + 0.577 592 729 6;
  • 12) 0.577 592 729 6 × 2 = 1 + 0.155 185 459 2;
  • 13) 0.155 185 459 2 × 2 = 0 + 0.310 370 918 4;
  • 14) 0.310 370 918 4 × 2 = 0 + 0.620 741 836 8;
  • 15) 0.620 741 836 8 × 2 = 1 + 0.241 483 673 6;
  • 16) 0.241 483 673 6 × 2 = 0 + 0.482 967 347 2;
  • 17) 0.482 967 347 2 × 2 = 0 + 0.965 934 694 4;
  • 18) 0.965 934 694 4 × 2 = 1 + 0.931 869 388 8;
  • 19) 0.931 869 388 8 × 2 = 1 + 0.863 738 777 6;
  • 20) 0.863 738 777 6 × 2 = 1 + 0.727 477 555 2;
  • 21) 0.727 477 555 2 × 2 = 1 + 0.454 955 110 4;
  • 22) 0.454 955 110 4 × 2 = 0 + 0.909 910 220 8;
  • 23) 0.909 910 220 8 × 2 = 1 + 0.819 820 441 6;
  • 24) 0.819 820 441 6 × 2 = 1 + 0.639 640 883 2;
  • 25) 0.639 640 883 2 × 2 = 1 + 0.279 281 766 4;
  • 26) 0.279 281 766 4 × 2 = 0 + 0.558 563 532 8;
  • 27) 0.558 563 532 8 × 2 = 1 + 0.117 127 065 6;
  • 28) 0.117 127 065 6 × 2 = 0 + 0.234 254 131 2;
  • 29) 0.234 254 131 2 × 2 = 0 + 0.468 508 262 4;
  • 30) 0.468 508 262 4 × 2 = 0 + 0.937 016 524 8;
  • 31) 0.937 016 524 8 × 2 = 1 + 0.874 033 049 6;
  • 32) 0.874 033 049 6 × 2 = 1 + 0.748 066 099 2;
  • 33) 0.748 066 099 2 × 2 = 1 + 0.496 132 198 4;
  • 34) 0.496 132 198 4 × 2 = 0 + 0.992 264 396 8;
  • 35) 0.992 264 396 8 × 2 = 1 + 0.984 528 793 6;
  • 36) 0.984 528 793 6 × 2 = 1 + 0.969 057 587 2;
  • 37) 0.969 057 587 2 × 2 = 1 + 0.938 115 174 4;
  • 38) 0.938 115 174 4 × 2 = 1 + 0.876 230 348 8;
  • 39) 0.876 230 348 8 × 2 = 1 + 0.752 460 697 6;
  • 40) 0.752 460 697 6 × 2 = 1 + 0.504 921 395 2;
  • 41) 0.504 921 395 2 × 2 = 1 + 0.009 842 790 4;
  • 42) 0.009 842 790 4 × 2 = 0 + 0.019 685 580 8;
  • 43) 0.019 685 580 8 × 2 = 0 + 0.039 371 161 6;
  • 44) 0.039 371 161 6 × 2 = 0 + 0.078 742 323 2;
  • 45) 0.078 742 323 2 × 2 = 0 + 0.157 484 646 4;
  • 46) 0.157 484 646 4 × 2 = 0 + 0.314 969 292 8;
  • 47) 0.314 969 292 8 × 2 = 0 + 0.629 938 585 6;
  • 48) 0.629 938 585 6 × 2 = 1 + 0.259 877 171 2;
  • 49) 0.259 877 171 2 × 2 = 0 + 0.519 754 342 4;
  • 50) 0.519 754 342 4 × 2 = 1 + 0.039 508 684 8;
  • 51) 0.039 508 684 8 × 2 = 0 + 0.079 017 369 6;
  • 52) 0.079 017 369 6 × 2 = 0 + 0.158 034 739 2;
  • 53) 0.158 034 739 2 × 2 = 0 + 0.316 069 478 4;
  • 54) 0.316 069 478 4 × 2 = 0 + 0.632 138 956 8;
  • 55) 0.632 138 956 8 × 2 = 1 + 0.264 277 913 6;
  • 56) 0.264 277 913 6 × 2 = 0 + 0.528 555 827 2;
  • 57) 0.528 555 827 2 × 2 = 1 + 0.057 111 654 4;
  • 58) 0.057 111 654 4 × 2 = 0 + 0.114 223 308 8;
  • 59) 0.114 223 308 8 × 2 = 0 + 0.228 446 617 6;
  • 60) 0.228 446 617 6 × 2 = 0 + 0.456 893 235 2;
  • 61) 0.456 893 235 2 × 2 = 0 + 0.913 786 470 4;
  • 62) 0.913 786 470 4 × 2 = 1 + 0.827 572 940 8;
  • 63) 0.827 572 940 8 × 2 = 1 + 0.655 145 881 6;
  • 64) 0.655 145 881 6 × 2 = 1 + 0.310 291 763 2;

We didn't get any fractional part that was equal to zero. But we had enough iterations (over Mantissa limit) and at least one integer that was different from zero => FULL STOP (Losing precision - the converted number we get in the end will be just a very good approximation of the initial one).


5. Construct the base 2 representation of the fractional part of the number.

Take all the integer parts of the multiplying operations, starting from the top of the constructed list above:


0.000 282 027 7(10) =

0.0000 0000 0001 0010 0111 1011 1010 0011 1011 1111 1000 0001 0100 0010 1000 0111(2)

6. Positive number before normalization:

0.000 282 027 7(10) =

0.0000 0000 0001 0010 0111 1011 1010 0011 1011 1111 1000 0001 0100 0010 1000 0111(2)

7. Normalize the binary representation of the number.

Shift the decimal mark 12 positions to the right, so that only one non zero digit remains to the left of it:


0.000 282 027 7(10) =

0.0000 0000 0001 0010 0111 1011 1010 0011 1011 1111 1000 0001 0100 0010 1000 0111(2) =

0.0000 0000 0001 0010 0111 1011 1010 0011 1011 1111 1000 0001 0100 0010 1000 0111(2) × 20 =

1.0010 0111 1011 1010 0011 1011 1111 1000 0001 0100 0010 1000 0111(2) × 2-12


8. Up to this moment, there are the following elements that would feed into the 64 bit double precision IEEE 754 binary floating point representation:

Sign 1 (a negative number)

Exponent (unadjusted): -12

Mantissa (not normalized):
1.0010 0111 1011 1010 0011 1011 1111 1000 0001 0100 0010 1000 0111


9. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

Exponent (unadjusted) + 2(11-1) - 1 =

-12 + 2(11-1) - 1 =

(-12 + 1 023)(10) =

1 011(10)


10. Convert the adjusted exponent from the decimal (base 10) to 11 bit binary.

Use the same technique of repeatedly dividing by 2:


  • division = quotient + remainder;
  • 1 011 ÷ 2 = 505 + 1;
  • 505 ÷ 2 = 252 + 1;
  • 252 ÷ 2 = 126 + 0;
  • 126 ÷ 2 = 63 + 0;
  • 63 ÷ 2 = 31 + 1;
  • 31 ÷ 2 = 15 + 1;
  • 15 ÷ 2 = 7 + 1;
  • 7 ÷ 2 = 3 + 1;
  • 3 ÷ 2 = 1 + 1;
  • 1 ÷ 2 = 0 + 1;

11. Construct the base 2 representation of the adjusted exponent.

Take all the remainders starting from the bottom of the list constructed above.


Exponent (adjusted) =

1011(10) =

011 1111 0011(2)


12. Normalize the mantissa.

a) Remove the leading (the leftmost) bit, since it's allways 1, and the decimal point, if the case.

b) Adjust its length to 52 bits, only if necessary (not the case here).


Mantissa (normalized) =

1. 0010 0111 1011 1010 0011 1011 1111 1000 0001 0100 0010 1000 0111 =

0010 0111 1011 1010 0011 1011 1111 1000 0001 0100 0010 1000 0111


13. The three elements that make up the number's 64 bit double precision IEEE 754 binary floating point representation:

Sign (1 bit) =
1 (a negative number)

Exponent (11 bits) =
011 1111 0011

Mantissa (52 bits) =
0010 0111 1011 1010 0011 1011 1111 1000 0001 0100 0010 1000 0111


Decimal number -0.000 282 027 7 converted to 64 bit double precision IEEE 754 binary floating point representation:

1 - 011 1111 0011 - 0010 0111 1011 1010 0011 1011 1111 1000 0001 0100 0010 1000 0111

Share

» Convert decimal -0.000 282 032 8 to 64 bit double precision IEEE 754 binary floating point representation standard

» Calculations Performed by Our Visitors: Decimal Numbers Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard. Data organized on a Monthly Basis

» Month 05, 2026 [May]: Decimal Numbers Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard. Calculations performed during the month of: May - by our visitors

» Improve your social skills: The Attention Economy - The Battlefield For Your Focus. NotebookLM YouTube Video of 7.50 minutes

How to convert numbers from the decimal system (base ten) to 64 bit double precision IEEE 754 binary floating point standard

Follow the steps below to convert a base 10 decimal number to 64 bit double precision IEEE 754 binary floating point:

  • 1. If the number to be converted is negative, start with its the positive version.
  • 2. First convert the integer part. Divide repeatedly by 2 the positive representation of the integer number that is to be converted to binary, until we get a quotient that is equal to zero, keeping track of each remainder.
  • 3. Construct the base 2 representation of the positive integer part of the number, by taking all the remainders from the previous operations, starting from the bottom of the list constructed above. Thus, the last remainder of the divisions becomes the first symbol (the leftmost) of the base two number, while the first remainder becomes the last symbol (the rightmost).
  • 4. Then convert the fractional part. Multiply the number repeatedly by 2, until we get a fractional part that is equal to zero, keeping track of each integer part of the results.
  • 5. Construct the base 2 representation of the fractional part of the number, by taking all the integer parts of the multiplying operations, starting from the top of the list constructed above (they should appear in the binary representation, from left to right, in the order they have been calculated).
  • 6. Normalize the binary representation of the number, shifting the decimal mark (the decimal point) "n" positions either to the left, or to the right, so that only one non zero digit remains to the left of the decimal mark.
  • 7. Adjust the exponent in 11 bit excess/bias notation and then convert it from decimal (base 10) to 11 bit binary, by using the same technique of repeatedly dividing by 2, as shown above:
    Exponent (adjusted) = Exponent (unadjusted) + 2(11-1) - 1
  • 8. Normalize mantissa, remove the leading (leftmost) bit, since it's allways '1' (and the decimal mark, if the case) and adjust its length to 52 bits, either by removing the excess bits from the right (losing precision...) or by adding extra bits set on '0' to the right.
  • 9. Sign (it takes 1 bit) is either 1 for a negative or 0 for a positive number.

Example: convert the negative number -31.640 215 from the decimal system (base ten) to 64 bit double precision IEEE 754 binary floating point:

  • 1. Start with the positive version of the number:

    |-31.640 215| = 31.640 215

  • 2. First convert the integer part, 31. Divide it repeatedly by 2, keeping track of each remainder, until we get a quotient that is equal to zero:
    • division = quotient + remainder;
    • 31 ÷ 2 = 15 + 1;
    • 15 ÷ 2 = 7 + 1;
    • 7 ÷ 2 = 3 + 1;
    • 3 ÷ 2 = 1 + 1;
    • 1 ÷ 2 = 0 + 1;
    • We have encountered a quotient that is ZERO => FULL STOP
  • 3. Construct the base 2 representation of the integer part of the number by taking all the remainders of the previous dividing operations, starting from the bottom of the list constructed above:

    31(10) = 1 1111(2)

  • 4. Then, convert the fractional part, 0.640 215. Multiply repeatedly by 2, keeping track of each integer part of the results, until we get a fractional part that is equal to zero:
    • #) multiplying = integer + fractional part;
    • 1) 0.640 215 × 2 = 1 + 0.280 43;
    • 2) 0.280 43 × 2 = 0 + 0.560 86;
    • 3) 0.560 86 × 2 = 1 + 0.121 72;
    • 4) 0.121 72 × 2 = 0 + 0.243 44;
    • 5) 0.243 44 × 2 = 0 + 0.486 88;
    • 6) 0.486 88 × 2 = 0 + 0.973 76;
    • 7) 0.973 76 × 2 = 1 + 0.947 52;
    • 8) 0.947 52 × 2 = 1 + 0.895 04;
    • 9) 0.895 04 × 2 = 1 + 0.790 08;
    • 10) 0.790 08 × 2 = 1 + 0.580 16;
    • 11) 0.580 16 × 2 = 1 + 0.160 32;
    • 12) 0.160 32 × 2 = 0 + 0.320 64;
    • 13) 0.320 64 × 2 = 0 + 0.641 28;
    • 14) 0.641 28 × 2 = 1 + 0.282 56;
    • 15) 0.282 56 × 2 = 0 + 0.565 12;
    • 16) 0.565 12 × 2 = 1 + 0.130 24;
    • 17) 0.130 24 × 2 = 0 + 0.260 48;
    • 18) 0.260 48 × 2 = 0 + 0.520 96;
    • 19) 0.520 96 × 2 = 1 + 0.041 92;
    • 20) 0.041 92 × 2 = 0 + 0.083 84;
    • 21) 0.083 84 × 2 = 0 + 0.167 68;
    • 22) 0.167 68 × 2 = 0 + 0.335 36;
    • 23) 0.335 36 × 2 = 0 + 0.670 72;
    • 24) 0.670 72 × 2 = 1 + 0.341 44;
    • 25) 0.341 44 × 2 = 0 + 0.682 88;
    • 26) 0.682 88 × 2 = 1 + 0.365 76;
    • 27) 0.365 76 × 2 = 0 + 0.731 52;
    • 28) 0.731 52 × 2 = 1 + 0.463 04;
    • 29) 0.463 04 × 2 = 0 + 0.926 08;
    • 30) 0.926 08 × 2 = 1 + 0.852 16;
    • 31) 0.852 16 × 2 = 1 + 0.704 32;
    • 32) 0.704 32 × 2 = 1 + 0.408 64;
    • 33) 0.408 64 × 2 = 0 + 0.817 28;
    • 34) 0.817 28 × 2 = 1 + 0.634 56;
    • 35) 0.634 56 × 2 = 1 + 0.269 12;
    • 36) 0.269 12 × 2 = 0 + 0.538 24;
    • 37) 0.538 24 × 2 = 1 + 0.076 48;
    • 38) 0.076 48 × 2 = 0 + 0.152 96;
    • 39) 0.152 96 × 2 = 0 + 0.305 92;
    • 40) 0.305 92 × 2 = 0 + 0.611 84;
    • 41) 0.611 84 × 2 = 1 + 0.223 68;
    • 42) 0.223 68 × 2 = 0 + 0.447 36;
    • 43) 0.447 36 × 2 = 0 + 0.894 72;
    • 44) 0.894 72 × 2 = 1 + 0.789 44;
    • 45) 0.789 44 × 2 = 1 + 0.578 88;
    • 46) 0.578 88 × 2 = 1 + 0.157 76;
    • 47) 0.157 76 × 2 = 0 + 0.315 52;
    • 48) 0.315 52 × 2 = 0 + 0.631 04;
    • 49) 0.631 04 × 2 = 1 + 0.262 08;
    • 50) 0.262 08 × 2 = 0 + 0.524 16;
    • 51) 0.524 16 × 2 = 1 + 0.048 32;
    • 52) 0.048 32 × 2 = 0 + 0.096 64;
    • 53) 0.096 64 × 2 = 0 + 0.193 28;
    • We didn't get any fractional part that was equal to zero. But we had enough iterations (over Mantissa limit = 52) and at least one integer part that was different from zero => FULL STOP (losing precision...).
  • 5. Construct the base 2 representation of the fractional part of the number, by taking all the integer parts of the previous multiplying operations, starting from the top of the constructed list above:

    0.640 215(10) = 0.1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2)

  • 6. Summarizing - the positive number before normalization:

    31.640 215(10) = 1 1111.1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2)

  • 7. Normalize the binary representation of the number, shifting the decimal mark 4 positions to the left so that only one non-zero digit stays to the left of the decimal mark:

    31.640 215(10) =
    1 1111.1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2) =
    1 1111.1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2) × 20 =
    1.1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2) × 24

  • 8. Up to this moment, there are the following elements that would feed into the 64 bit double precision IEEE 754 binary floating point representation:

    Sign: 1 (a negative number)

    Exponent (unadjusted): 4

    Mantissa (not-normalized): 1.1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0

  • 9. Adjust the exponent in 11 bit excess/bias notation and then convert it from decimal (base 10) to 11 bit binary (base 2), by using the same technique of repeatedly dividing it by 2, as shown above:

    Exponent (adjusted) = Exponent (unadjusted) + 2(11-1) - 1 = (4 + 1023)(10) = 1027(10) =
    100 0000 0011(2)

  • 10. Normalize mantissa, remove the leading (leftmost) bit, since it's allways '1' (and the decimal sign) and adjust its length to 52 bits, by removing the excess bits, from the right (losing precision...):

    Mantissa (not-normalized): 1.1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0

    Mantissa (normalized): 1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100

  • Conclusion:

    Sign (1 bit) = 1 (a negative number)

    Exponent (8 bits) = 100 0000 0011

    Mantissa (52 bits) = 1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100

  • Number -31.640 215, converted from decimal system (base 10) to 64 bit double precision IEEE 754 binary floating point =
    1 - 100 0000 0011 - 1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100