-0.000 282 292 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.000 282 292(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
-0.000 282 292(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:

|-0.000 282 292| = 0.000 282 292


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: 0.000 282 292.

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) 0.000 282 292 × 2 = 0 + 0.000 564 584;
  • 2) 0.000 564 584 × 2 = 0 + 0.001 129 168;
  • 3) 0.001 129 168 × 2 = 0 + 0.002 258 336;
  • 4) 0.002 258 336 × 2 = 0 + 0.004 516 672;
  • 5) 0.004 516 672 × 2 = 0 + 0.009 033 344;
  • 6) 0.009 033 344 × 2 = 0 + 0.018 066 688;
  • 7) 0.018 066 688 × 2 = 0 + 0.036 133 376;
  • 8) 0.036 133 376 × 2 = 0 + 0.072 266 752;
  • 9) 0.072 266 752 × 2 = 0 + 0.144 533 504;
  • 10) 0.144 533 504 × 2 = 0 + 0.289 067 008;
  • 11) 0.289 067 008 × 2 = 0 + 0.578 134 016;
  • 12) 0.578 134 016 × 2 = 1 + 0.156 268 032;
  • 13) 0.156 268 032 × 2 = 0 + 0.312 536 064;
  • 14) 0.312 536 064 × 2 = 0 + 0.625 072 128;
  • 15) 0.625 072 128 × 2 = 1 + 0.250 144 256;
  • 16) 0.250 144 256 × 2 = 0 + 0.500 288 512;
  • 17) 0.500 288 512 × 2 = 1 + 0.000 577 024;
  • 18) 0.000 577 024 × 2 = 0 + 0.001 154 048;
  • 19) 0.001 154 048 × 2 = 0 + 0.002 308 096;
  • 20) 0.002 308 096 × 2 = 0 + 0.004 616 192;
  • 21) 0.004 616 192 × 2 = 0 + 0.009 232 384;
  • 22) 0.009 232 384 × 2 = 0 + 0.018 464 768;
  • 23) 0.018 464 768 × 2 = 0 + 0.036 929 536;
  • 24) 0.036 929 536 × 2 = 0 + 0.073 859 072;
  • 25) 0.073 859 072 × 2 = 0 + 0.147 718 144;
  • 26) 0.147 718 144 × 2 = 0 + 0.295 436 288;
  • 27) 0.295 436 288 × 2 = 0 + 0.590 872 576;
  • 28) 0.590 872 576 × 2 = 1 + 0.181 745 152;
  • 29) 0.181 745 152 × 2 = 0 + 0.363 490 304;
  • 30) 0.363 490 304 × 2 = 0 + 0.726 980 608;
  • 31) 0.726 980 608 × 2 = 1 + 0.453 961 216;
  • 32) 0.453 961 216 × 2 = 0 + 0.907 922 432;
  • 33) 0.907 922 432 × 2 = 1 + 0.815 844 864;
  • 34) 0.815 844 864 × 2 = 1 + 0.631 689 728;
  • 35) 0.631 689 728 × 2 = 1 + 0.263 379 456;
  • 36) 0.263 379 456 × 2 = 0 + 0.526 758 912;
  • 37) 0.526 758 912 × 2 = 1 + 0.053 517 824;
  • 38) 0.053 517 824 × 2 = 0 + 0.107 035 648;
  • 39) 0.107 035 648 × 2 = 0 + 0.214 071 296;
  • 40) 0.214 071 296 × 2 = 0 + 0.428 142 592;
  • 41) 0.428 142 592 × 2 = 0 + 0.856 285 184;
  • 42) 0.856 285 184 × 2 = 1 + 0.712 570 368;
  • 43) 0.712 570 368 × 2 = 1 + 0.425 140 736;
  • 44) 0.425 140 736 × 2 = 0 + 0.850 281 472;
  • 45) 0.850 281 472 × 2 = 1 + 0.700 562 944;
  • 46) 0.700 562 944 × 2 = 1 + 0.401 125 888;
  • 47) 0.401 125 888 × 2 = 0 + 0.802 251 776;
  • 48) 0.802 251 776 × 2 = 1 + 0.604 503 552;
  • 49) 0.604 503 552 × 2 = 1 + 0.209 007 104;
  • 50) 0.209 007 104 × 2 = 0 + 0.418 014 208;
  • 51) 0.418 014 208 × 2 = 0 + 0.836 028 416;
  • 52) 0.836 028 416 × 2 = 1 + 0.672 056 832;
  • 53) 0.672 056 832 × 2 = 1 + 0.344 113 664;
  • 54) 0.344 113 664 × 2 = 0 + 0.688 227 328;
  • 55) 0.688 227 328 × 2 = 1 + 0.376 454 656;
  • 56) 0.376 454 656 × 2 = 0 + 0.752 909 312;
  • 57) 0.752 909 312 × 2 = 1 + 0.505 818 624;
  • 58) 0.505 818 624 × 2 = 1 + 0.011 637 248;
  • 59) 0.011 637 248 × 2 = 0 + 0.023 274 496;
  • 60) 0.023 274 496 × 2 = 0 + 0.046 548 992;
  • 61) 0.046 548 992 × 2 = 0 + 0.093 097 984;
  • 62) 0.093 097 984 × 2 = 0 + 0.186 195 968;
  • 63) 0.186 195 968 × 2 = 0 + 0.372 391 936;
  • 64) 0.372 391 936 × 2 = 0 + 0.744 783 872;

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:


0.000 282 292(10) =

0.0000 0000 0001 0010 1000 0000 0001 0010 1110 1000 0110 1101 1001 1010 1100 0000(2)

6. Positive number before normalization:

0.000 282 292(10) =

0.0000 0000 0001 0010 1000 0000 0001 0010 1110 1000 0110 1101 1001 1010 1100 0000(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:


0.000 282 292(10) =

0.0000 0000 0001 0010 1000 0000 0001 0010 1110 1000 0110 1101 1001 1010 1100 0000(2) =

0.0000 0000 0001 0010 1000 0000 0001 0010 1110 1000 0110 1101 1001 1010 1100 0000(2) × 20 =

1.0010 1000 0000 0001 0010 1110 1000 0110 1101 1001 1010 1100 0000(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 0000 0001 0010 1110 1000 0110 1101 1001 1010 1100 0000


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 0000 0001 0010 1110 1000 0110 1101 1001 1010 1100 0000 =

0010 1000 0000 0001 0010 1110 1000 0110 1101 1001 1010 1100 0000


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 0000 0001 0010 1110 1000 0110 1101 1001 1010 1100 0000


Decimal number -0.000 282 292 converted to 64 bit double precision IEEE 754 binary floating point representation:

1 - 011 1111 0011 - 0010 1000 0000 0001 0010 1110 1000 0110 1101 1001 1010 1100 0000

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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