-0.000 282 384 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.000 282 384(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
-0.000 282 384(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:

|-0.000 282 384| = 0.000 282 384


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: 0.000 282 384.

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) 0.000 282 384 × 2 = 0 + 0.000 564 768;
  • 2) 0.000 564 768 × 2 = 0 + 0.001 129 536;
  • 3) 0.001 129 536 × 2 = 0 + 0.002 259 072;
  • 4) 0.002 259 072 × 2 = 0 + 0.004 518 144;
  • 5) 0.004 518 144 × 2 = 0 + 0.009 036 288;
  • 6) 0.009 036 288 × 2 = 0 + 0.018 072 576;
  • 7) 0.018 072 576 × 2 = 0 + 0.036 145 152;
  • 8) 0.036 145 152 × 2 = 0 + 0.072 290 304;
  • 9) 0.072 290 304 × 2 = 0 + 0.144 580 608;
  • 10) 0.144 580 608 × 2 = 0 + 0.289 161 216;
  • 11) 0.289 161 216 × 2 = 0 + 0.578 322 432;
  • 12) 0.578 322 432 × 2 = 1 + 0.156 644 864;
  • 13) 0.156 644 864 × 2 = 0 + 0.313 289 728;
  • 14) 0.313 289 728 × 2 = 0 + 0.626 579 456;
  • 15) 0.626 579 456 × 2 = 1 + 0.253 158 912;
  • 16) 0.253 158 912 × 2 = 0 + 0.506 317 824;
  • 17) 0.506 317 824 × 2 = 1 + 0.012 635 648;
  • 18) 0.012 635 648 × 2 = 0 + 0.025 271 296;
  • 19) 0.025 271 296 × 2 = 0 + 0.050 542 592;
  • 20) 0.050 542 592 × 2 = 0 + 0.101 085 184;
  • 21) 0.101 085 184 × 2 = 0 + 0.202 170 368;
  • 22) 0.202 170 368 × 2 = 0 + 0.404 340 736;
  • 23) 0.404 340 736 × 2 = 0 + 0.808 681 472;
  • 24) 0.808 681 472 × 2 = 1 + 0.617 362 944;
  • 25) 0.617 362 944 × 2 = 1 + 0.234 725 888;
  • 26) 0.234 725 888 × 2 = 0 + 0.469 451 776;
  • 27) 0.469 451 776 × 2 = 0 + 0.938 903 552;
  • 28) 0.938 903 552 × 2 = 1 + 0.877 807 104;
  • 29) 0.877 807 104 × 2 = 1 + 0.755 614 208;
  • 30) 0.755 614 208 × 2 = 1 + 0.511 228 416;
  • 31) 0.511 228 416 × 2 = 1 + 0.022 456 832;
  • 32) 0.022 456 832 × 2 = 0 + 0.044 913 664;
  • 33) 0.044 913 664 × 2 = 0 + 0.089 827 328;
  • 34) 0.089 827 328 × 2 = 0 + 0.179 654 656;
  • 35) 0.179 654 656 × 2 = 0 + 0.359 309 312;
  • 36) 0.359 309 312 × 2 = 0 + 0.718 618 624;
  • 37) 0.718 618 624 × 2 = 1 + 0.437 237 248;
  • 38) 0.437 237 248 × 2 = 0 + 0.874 474 496;
  • 39) 0.874 474 496 × 2 = 1 + 0.748 948 992;
  • 40) 0.748 948 992 × 2 = 1 + 0.497 897 984;
  • 41) 0.497 897 984 × 2 = 0 + 0.995 795 968;
  • 42) 0.995 795 968 × 2 = 1 + 0.991 591 936;
  • 43) 0.991 591 936 × 2 = 1 + 0.983 183 872;
  • 44) 0.983 183 872 × 2 = 1 + 0.966 367 744;
  • 45) 0.966 367 744 × 2 = 1 + 0.932 735 488;
  • 46) 0.932 735 488 × 2 = 1 + 0.865 470 976;
  • 47) 0.865 470 976 × 2 = 1 + 0.730 941 952;
  • 48) 0.730 941 952 × 2 = 1 + 0.461 883 904;
  • 49) 0.461 883 904 × 2 = 0 + 0.923 767 808;
  • 50) 0.923 767 808 × 2 = 1 + 0.847 535 616;
  • 51) 0.847 535 616 × 2 = 1 + 0.695 071 232;
  • 52) 0.695 071 232 × 2 = 1 + 0.390 142 464;
  • 53) 0.390 142 464 × 2 = 0 + 0.780 284 928;
  • 54) 0.780 284 928 × 2 = 1 + 0.560 569 856;
  • 55) 0.560 569 856 × 2 = 1 + 0.121 139 712;
  • 56) 0.121 139 712 × 2 = 0 + 0.242 279 424;
  • 57) 0.242 279 424 × 2 = 0 + 0.484 558 848;
  • 58) 0.484 558 848 × 2 = 0 + 0.969 117 696;
  • 59) 0.969 117 696 × 2 = 1 + 0.938 235 392;
  • 60) 0.938 235 392 × 2 = 1 + 0.876 470 784;
  • 61) 0.876 470 784 × 2 = 1 + 0.752 941 568;
  • 62) 0.752 941 568 × 2 = 1 + 0.505 883 136;
  • 63) 0.505 883 136 × 2 = 1 + 0.011 766 272;
  • 64) 0.011 766 272 × 2 = 0 + 0.023 532 544;

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:


0.000 282 384(10) =

0.0000 0000 0001 0010 1000 0001 1001 1110 0000 1011 0111 1111 0111 0110 0011 1110(2)

6. Positive number before normalization:

0.000 282 384(10) =

0.0000 0000 0001 0010 1000 0001 1001 1110 0000 1011 0111 1111 0111 0110 0011 1110(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:


0.000 282 384(10) =

0.0000 0000 0001 0010 1000 0001 1001 1110 0000 1011 0111 1111 0111 0110 0011 1110(2) =

0.0000 0000 0001 0010 1000 0001 1001 1110 0000 1011 0111 1111 0111 0110 0011 1110(2) × 20 =

1.0010 1000 0001 1001 1110 0000 1011 0111 1111 0111 0110 0011 1110(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 1001 1110 0000 1011 0111 1111 0111 0110 0011 1110


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 1001 1110 0000 1011 0111 1111 0111 0110 0011 1110 =

0010 1000 0001 1001 1110 0000 1011 0111 1111 0111 0110 0011 1110


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 1001 1110 0000 1011 0111 1111 0111 0110 0011 1110


Decimal number -0.000 282 384 converted to 64 bit double precision IEEE 754 binary floating point representation:

1 - 011 1111 0011 - 0010 1000 0001 1001 1110 0000 1011 0111 1111 0111 0110 0011 1110

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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