-0.000 282 368 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.000 282 368(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
-0.000 282 368(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:

|-0.000 282 368| = 0.000 282 368


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: 0.000 282 368.

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) 0.000 282 368 × 2 = 0 + 0.000 564 736;
  • 2) 0.000 564 736 × 2 = 0 + 0.001 129 472;
  • 3) 0.001 129 472 × 2 = 0 + 0.002 258 944;
  • 4) 0.002 258 944 × 2 = 0 + 0.004 517 888;
  • 5) 0.004 517 888 × 2 = 0 + 0.009 035 776;
  • 6) 0.009 035 776 × 2 = 0 + 0.018 071 552;
  • 7) 0.018 071 552 × 2 = 0 + 0.036 143 104;
  • 8) 0.036 143 104 × 2 = 0 + 0.072 286 208;
  • 9) 0.072 286 208 × 2 = 0 + 0.144 572 416;
  • 10) 0.144 572 416 × 2 = 0 + 0.289 144 832;
  • 11) 0.289 144 832 × 2 = 0 + 0.578 289 664;
  • 12) 0.578 289 664 × 2 = 1 + 0.156 579 328;
  • 13) 0.156 579 328 × 2 = 0 + 0.313 158 656;
  • 14) 0.313 158 656 × 2 = 0 + 0.626 317 312;
  • 15) 0.626 317 312 × 2 = 1 + 0.252 634 624;
  • 16) 0.252 634 624 × 2 = 0 + 0.505 269 248;
  • 17) 0.505 269 248 × 2 = 1 + 0.010 538 496;
  • 18) 0.010 538 496 × 2 = 0 + 0.021 076 992;
  • 19) 0.021 076 992 × 2 = 0 + 0.042 153 984;
  • 20) 0.042 153 984 × 2 = 0 + 0.084 307 968;
  • 21) 0.084 307 968 × 2 = 0 + 0.168 615 936;
  • 22) 0.168 615 936 × 2 = 0 + 0.337 231 872;
  • 23) 0.337 231 872 × 2 = 0 + 0.674 463 744;
  • 24) 0.674 463 744 × 2 = 1 + 0.348 927 488;
  • 25) 0.348 927 488 × 2 = 0 + 0.697 854 976;
  • 26) 0.697 854 976 × 2 = 1 + 0.395 709 952;
  • 27) 0.395 709 952 × 2 = 0 + 0.791 419 904;
  • 28) 0.791 419 904 × 2 = 1 + 0.582 839 808;
  • 29) 0.582 839 808 × 2 = 1 + 0.165 679 616;
  • 30) 0.165 679 616 × 2 = 0 + 0.331 359 232;
  • 31) 0.331 359 232 × 2 = 0 + 0.662 718 464;
  • 32) 0.662 718 464 × 2 = 1 + 0.325 436 928;
  • 33) 0.325 436 928 × 2 = 0 + 0.650 873 856;
  • 34) 0.650 873 856 × 2 = 1 + 0.301 747 712;
  • 35) 0.301 747 712 × 2 = 0 + 0.603 495 424;
  • 36) 0.603 495 424 × 2 = 1 + 0.206 990 848;
  • 37) 0.206 990 848 × 2 = 0 + 0.413 981 696;
  • 38) 0.413 981 696 × 2 = 0 + 0.827 963 392;
  • 39) 0.827 963 392 × 2 = 1 + 0.655 926 784;
  • 40) 0.655 926 784 × 2 = 1 + 0.311 853 568;
  • 41) 0.311 853 568 × 2 = 0 + 0.623 707 136;
  • 42) 0.623 707 136 × 2 = 1 + 0.247 414 272;
  • 43) 0.247 414 272 × 2 = 0 + 0.494 828 544;
  • 44) 0.494 828 544 × 2 = 0 + 0.989 657 088;
  • 45) 0.989 657 088 × 2 = 1 + 0.979 314 176;
  • 46) 0.979 314 176 × 2 = 1 + 0.958 628 352;
  • 47) 0.958 628 352 × 2 = 1 + 0.917 256 704;
  • 48) 0.917 256 704 × 2 = 1 + 0.834 513 408;
  • 49) 0.834 513 408 × 2 = 1 + 0.669 026 816;
  • 50) 0.669 026 816 × 2 = 1 + 0.338 053 632;
  • 51) 0.338 053 632 × 2 = 0 + 0.676 107 264;
  • 52) 0.676 107 264 × 2 = 1 + 0.352 214 528;
  • 53) 0.352 214 528 × 2 = 0 + 0.704 429 056;
  • 54) 0.704 429 056 × 2 = 1 + 0.408 858 112;
  • 55) 0.408 858 112 × 2 = 0 + 0.817 716 224;
  • 56) 0.817 716 224 × 2 = 1 + 0.635 432 448;
  • 57) 0.635 432 448 × 2 = 1 + 0.270 864 896;
  • 58) 0.270 864 896 × 2 = 0 + 0.541 729 792;
  • 59) 0.541 729 792 × 2 = 1 + 0.083 459 584;
  • 60) 0.083 459 584 × 2 = 0 + 0.166 919 168;
  • 61) 0.166 919 168 × 2 = 0 + 0.333 838 336;
  • 62) 0.333 838 336 × 2 = 0 + 0.667 676 672;
  • 63) 0.667 676 672 × 2 = 1 + 0.335 353 344;
  • 64) 0.335 353 344 × 2 = 0 + 0.670 706 688;

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:


0.000 282 368(10) =

0.0000 0000 0001 0010 1000 0001 0101 1001 0101 0011 0100 1111 1101 0101 1010 0010(2)

6. Positive number before normalization:

0.000 282 368(10) =

0.0000 0000 0001 0010 1000 0001 0101 1001 0101 0011 0100 1111 1101 0101 1010 0010(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:


0.000 282 368(10) =

0.0000 0000 0001 0010 1000 0001 0101 1001 0101 0011 0100 1111 1101 0101 1010 0010(2) =

0.0000 0000 0001 0010 1000 0001 0101 1001 0101 0011 0100 1111 1101 0101 1010 0010(2) × 20 =

1.0010 1000 0001 0101 1001 0101 0011 0100 1111 1101 0101 1010 0010(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 0101 1001 0101 0011 0100 1111 1101 0101 1010 0010


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 0101 1001 0101 0011 0100 1111 1101 0101 1010 0010 =

0010 1000 0001 0101 1001 0101 0011 0100 1111 1101 0101 1010 0010


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 0101 1001 0101 0011 0100 1111 1101 0101 1010 0010


Decimal number -0.000 282 368 converted to 64 bit double precision IEEE 754 binary floating point representation:

1 - 011 1111 0011 - 0010 1000 0001 0101 1001 0101 0011 0100 1111 1101 0101 1010 0010

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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