-0.000 282 132 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.000 282 132(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
-0.000 282 132(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:

|-0.000 282 132| = 0.000 282 132


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: 0.000 282 132.

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) 0.000 282 132 × 2 = 0 + 0.000 564 264;
  • 2) 0.000 564 264 × 2 = 0 + 0.001 128 528;
  • 3) 0.001 128 528 × 2 = 0 + 0.002 257 056;
  • 4) 0.002 257 056 × 2 = 0 + 0.004 514 112;
  • 5) 0.004 514 112 × 2 = 0 + 0.009 028 224;
  • 6) 0.009 028 224 × 2 = 0 + 0.018 056 448;
  • 7) 0.018 056 448 × 2 = 0 + 0.036 112 896;
  • 8) 0.036 112 896 × 2 = 0 + 0.072 225 792;
  • 9) 0.072 225 792 × 2 = 0 + 0.144 451 584;
  • 10) 0.144 451 584 × 2 = 0 + 0.288 903 168;
  • 11) 0.288 903 168 × 2 = 0 + 0.577 806 336;
  • 12) 0.577 806 336 × 2 = 1 + 0.155 612 672;
  • 13) 0.155 612 672 × 2 = 0 + 0.311 225 344;
  • 14) 0.311 225 344 × 2 = 0 + 0.622 450 688;
  • 15) 0.622 450 688 × 2 = 1 + 0.244 901 376;
  • 16) 0.244 901 376 × 2 = 0 + 0.489 802 752;
  • 17) 0.489 802 752 × 2 = 0 + 0.979 605 504;
  • 18) 0.979 605 504 × 2 = 1 + 0.959 211 008;
  • 19) 0.959 211 008 × 2 = 1 + 0.918 422 016;
  • 20) 0.918 422 016 × 2 = 1 + 0.836 844 032;
  • 21) 0.836 844 032 × 2 = 1 + 0.673 688 064;
  • 22) 0.673 688 064 × 2 = 1 + 0.347 376 128;
  • 23) 0.347 376 128 × 2 = 0 + 0.694 752 256;
  • 24) 0.694 752 256 × 2 = 1 + 0.389 504 512;
  • 25) 0.389 504 512 × 2 = 0 + 0.779 009 024;
  • 26) 0.779 009 024 × 2 = 1 + 0.558 018 048;
  • 27) 0.558 018 048 × 2 = 1 + 0.116 036 096;
  • 28) 0.116 036 096 × 2 = 0 + 0.232 072 192;
  • 29) 0.232 072 192 × 2 = 0 + 0.464 144 384;
  • 30) 0.464 144 384 × 2 = 0 + 0.928 288 768;
  • 31) 0.928 288 768 × 2 = 1 + 0.856 577 536;
  • 32) 0.856 577 536 × 2 = 1 + 0.713 155 072;
  • 33) 0.713 155 072 × 2 = 1 + 0.426 310 144;
  • 34) 0.426 310 144 × 2 = 0 + 0.852 620 288;
  • 35) 0.852 620 288 × 2 = 1 + 0.705 240 576;
  • 36) 0.705 240 576 × 2 = 1 + 0.410 481 152;
  • 37) 0.410 481 152 × 2 = 0 + 0.820 962 304;
  • 38) 0.820 962 304 × 2 = 1 + 0.641 924 608;
  • 39) 0.641 924 608 × 2 = 1 + 0.283 849 216;
  • 40) 0.283 849 216 × 2 = 0 + 0.567 698 432;
  • 41) 0.567 698 432 × 2 = 1 + 0.135 396 864;
  • 42) 0.135 396 864 × 2 = 0 + 0.270 793 728;
  • 43) 0.270 793 728 × 2 = 0 + 0.541 587 456;
  • 44) 0.541 587 456 × 2 = 1 + 0.083 174 912;
  • 45) 0.083 174 912 × 2 = 0 + 0.166 349 824;
  • 46) 0.166 349 824 × 2 = 0 + 0.332 699 648;
  • 47) 0.332 699 648 × 2 = 0 + 0.665 399 296;
  • 48) 0.665 399 296 × 2 = 1 + 0.330 798 592;
  • 49) 0.330 798 592 × 2 = 0 + 0.661 597 184;
  • 50) 0.661 597 184 × 2 = 1 + 0.323 194 368;
  • 51) 0.323 194 368 × 2 = 0 + 0.646 388 736;
  • 52) 0.646 388 736 × 2 = 1 + 0.292 777 472;
  • 53) 0.292 777 472 × 2 = 0 + 0.585 554 944;
  • 54) 0.585 554 944 × 2 = 1 + 0.171 109 888;
  • 55) 0.171 109 888 × 2 = 0 + 0.342 219 776;
  • 56) 0.342 219 776 × 2 = 0 + 0.684 439 552;
  • 57) 0.684 439 552 × 2 = 1 + 0.368 879 104;
  • 58) 0.368 879 104 × 2 = 0 + 0.737 758 208;
  • 59) 0.737 758 208 × 2 = 1 + 0.475 516 416;
  • 60) 0.475 516 416 × 2 = 0 + 0.951 032 832;
  • 61) 0.951 032 832 × 2 = 1 + 0.902 065 664;
  • 62) 0.902 065 664 × 2 = 1 + 0.804 131 328;
  • 63) 0.804 131 328 × 2 = 1 + 0.608 262 656;
  • 64) 0.608 262 656 × 2 = 1 + 0.216 525 312;

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:


0.000 282 132(10) =

0.0000 0000 0001 0010 0111 1101 0110 0011 1011 0110 1001 0001 0101 0100 1010 1111(2)

6. Positive number before normalization:

0.000 282 132(10) =

0.0000 0000 0001 0010 0111 1101 0110 0011 1011 0110 1001 0001 0101 0100 1010 1111(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:


0.000 282 132(10) =

0.0000 0000 0001 0010 0111 1101 0110 0011 1011 0110 1001 0001 0101 0100 1010 1111(2) =

0.0000 0000 0001 0010 0111 1101 0110 0011 1011 0110 1001 0001 0101 0100 1010 1111(2) × 20 =

1.0010 0111 1101 0110 0011 1011 0110 1001 0001 0101 0100 1010 1111(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 1101 0110 0011 1011 0110 1001 0001 0101 0100 1010 1111


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 1101 0110 0011 1011 0110 1001 0001 0101 0100 1010 1111 =

0010 0111 1101 0110 0011 1011 0110 1001 0001 0101 0100 1010 1111


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 1101 0110 0011 1011 0110 1001 0001 0101 0100 1010 1111


Decimal number -0.000 282 132 converted to 64 bit double precision IEEE 754 binary floating point representation:

1 - 011 1111 0011 - 0010 0111 1101 0110 0011 1011 0110 1001 0001 0101 0100 1010 1111

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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