-0.000 282 351 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.000 282 351(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
-0.000 282 351(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:

|-0.000 282 351| = 0.000 282 351


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: 0.000 282 351.

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) 0.000 282 351 × 2 = 0 + 0.000 564 702;
  • 2) 0.000 564 702 × 2 = 0 + 0.001 129 404;
  • 3) 0.001 129 404 × 2 = 0 + 0.002 258 808;
  • 4) 0.002 258 808 × 2 = 0 + 0.004 517 616;
  • 5) 0.004 517 616 × 2 = 0 + 0.009 035 232;
  • 6) 0.009 035 232 × 2 = 0 + 0.018 070 464;
  • 7) 0.018 070 464 × 2 = 0 + 0.036 140 928;
  • 8) 0.036 140 928 × 2 = 0 + 0.072 281 856;
  • 9) 0.072 281 856 × 2 = 0 + 0.144 563 712;
  • 10) 0.144 563 712 × 2 = 0 + 0.289 127 424;
  • 11) 0.289 127 424 × 2 = 0 + 0.578 254 848;
  • 12) 0.578 254 848 × 2 = 1 + 0.156 509 696;
  • 13) 0.156 509 696 × 2 = 0 + 0.313 019 392;
  • 14) 0.313 019 392 × 2 = 0 + 0.626 038 784;
  • 15) 0.626 038 784 × 2 = 1 + 0.252 077 568;
  • 16) 0.252 077 568 × 2 = 0 + 0.504 155 136;
  • 17) 0.504 155 136 × 2 = 1 + 0.008 310 272;
  • 18) 0.008 310 272 × 2 = 0 + 0.016 620 544;
  • 19) 0.016 620 544 × 2 = 0 + 0.033 241 088;
  • 20) 0.033 241 088 × 2 = 0 + 0.066 482 176;
  • 21) 0.066 482 176 × 2 = 0 + 0.132 964 352;
  • 22) 0.132 964 352 × 2 = 0 + 0.265 928 704;
  • 23) 0.265 928 704 × 2 = 0 + 0.531 857 408;
  • 24) 0.531 857 408 × 2 = 1 + 0.063 714 816;
  • 25) 0.063 714 816 × 2 = 0 + 0.127 429 632;
  • 26) 0.127 429 632 × 2 = 0 + 0.254 859 264;
  • 27) 0.254 859 264 × 2 = 0 + 0.509 718 528;
  • 28) 0.509 718 528 × 2 = 1 + 0.019 437 056;
  • 29) 0.019 437 056 × 2 = 0 + 0.038 874 112;
  • 30) 0.038 874 112 × 2 = 0 + 0.077 748 224;
  • 31) 0.077 748 224 × 2 = 0 + 0.155 496 448;
  • 32) 0.155 496 448 × 2 = 0 + 0.310 992 896;
  • 33) 0.310 992 896 × 2 = 0 + 0.621 985 792;
  • 34) 0.621 985 792 × 2 = 1 + 0.243 971 584;
  • 35) 0.243 971 584 × 2 = 0 + 0.487 943 168;
  • 36) 0.487 943 168 × 2 = 0 + 0.975 886 336;
  • 37) 0.975 886 336 × 2 = 1 + 0.951 772 672;
  • 38) 0.951 772 672 × 2 = 1 + 0.903 545 344;
  • 39) 0.903 545 344 × 2 = 1 + 0.807 090 688;
  • 40) 0.807 090 688 × 2 = 1 + 0.614 181 376;
  • 41) 0.614 181 376 × 2 = 1 + 0.228 362 752;
  • 42) 0.228 362 752 × 2 = 0 + 0.456 725 504;
  • 43) 0.456 725 504 × 2 = 0 + 0.913 451 008;
  • 44) 0.913 451 008 × 2 = 1 + 0.826 902 016;
  • 45) 0.826 902 016 × 2 = 1 + 0.653 804 032;
  • 46) 0.653 804 032 × 2 = 1 + 0.307 608 064;
  • 47) 0.307 608 064 × 2 = 0 + 0.615 216 128;
  • 48) 0.615 216 128 × 2 = 1 + 0.230 432 256;
  • 49) 0.230 432 256 × 2 = 0 + 0.460 864 512;
  • 50) 0.460 864 512 × 2 = 0 + 0.921 729 024;
  • 51) 0.921 729 024 × 2 = 1 + 0.843 458 048;
  • 52) 0.843 458 048 × 2 = 1 + 0.686 916 096;
  • 53) 0.686 916 096 × 2 = 1 + 0.373 832 192;
  • 54) 0.373 832 192 × 2 = 0 + 0.747 664 384;
  • 55) 0.747 664 384 × 2 = 1 + 0.495 328 768;
  • 56) 0.495 328 768 × 2 = 0 + 0.990 657 536;
  • 57) 0.990 657 536 × 2 = 1 + 0.981 315 072;
  • 58) 0.981 315 072 × 2 = 1 + 0.962 630 144;
  • 59) 0.962 630 144 × 2 = 1 + 0.925 260 288;
  • 60) 0.925 260 288 × 2 = 1 + 0.850 520 576;
  • 61) 0.850 520 576 × 2 = 1 + 0.701 041 152;
  • 62) 0.701 041 152 × 2 = 1 + 0.402 082 304;
  • 63) 0.402 082 304 × 2 = 0 + 0.804 164 608;
  • 64) 0.804 164 608 × 2 = 1 + 0.608 329 216;

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:


0.000 282 351(10) =

0.0000 0000 0001 0010 1000 0001 0001 0000 0100 1111 1001 1101 0011 1010 1111 1101(2)

6. Positive number before normalization:

0.000 282 351(10) =

0.0000 0000 0001 0010 1000 0001 0001 0000 0100 1111 1001 1101 0011 1010 1111 1101(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:


0.000 282 351(10) =

0.0000 0000 0001 0010 1000 0001 0001 0000 0100 1111 1001 1101 0011 1010 1111 1101(2) =

0.0000 0000 0001 0010 1000 0001 0001 0000 0100 1111 1001 1101 0011 1010 1111 1101(2) × 20 =

1.0010 1000 0001 0001 0000 0100 1111 1001 1101 0011 1010 1111 1101(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 1000 0001 0001 0000 0100 1111 1001 1101 0011 1010 1111 1101


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 1000 0001 0001 0000 0100 1111 1001 1101 0011 1010 1111 1101 =

0010 1000 0001 0001 0000 0100 1111 1001 1101 0011 1010 1111 1101


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 1000 0001 0001 0000 0100 1111 1001 1101 0011 1010 1111 1101


Decimal number -0.000 282 351 converted to 64 bit double precision IEEE 754 binary floating point representation:

1 - 011 1111 0011 - 0010 1000 0001 0001 0000 0100 1111 1001 1101 0011 1010 1111 1101

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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