-0.000 282 258 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.000 282 258(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
-0.000 282 258(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:

|-0.000 282 258| = 0.000 282 258


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: 0.000 282 258.

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) 0.000 282 258 × 2 = 0 + 0.000 564 516;
  • 2) 0.000 564 516 × 2 = 0 + 0.001 129 032;
  • 3) 0.001 129 032 × 2 = 0 + 0.002 258 064;
  • 4) 0.002 258 064 × 2 = 0 + 0.004 516 128;
  • 5) 0.004 516 128 × 2 = 0 + 0.009 032 256;
  • 6) 0.009 032 256 × 2 = 0 + 0.018 064 512;
  • 7) 0.018 064 512 × 2 = 0 + 0.036 129 024;
  • 8) 0.036 129 024 × 2 = 0 + 0.072 258 048;
  • 9) 0.072 258 048 × 2 = 0 + 0.144 516 096;
  • 10) 0.144 516 096 × 2 = 0 + 0.289 032 192;
  • 11) 0.289 032 192 × 2 = 0 + 0.578 064 384;
  • 12) 0.578 064 384 × 2 = 1 + 0.156 128 768;
  • 13) 0.156 128 768 × 2 = 0 + 0.312 257 536;
  • 14) 0.312 257 536 × 2 = 0 + 0.624 515 072;
  • 15) 0.624 515 072 × 2 = 1 + 0.249 030 144;
  • 16) 0.249 030 144 × 2 = 0 + 0.498 060 288;
  • 17) 0.498 060 288 × 2 = 0 + 0.996 120 576;
  • 18) 0.996 120 576 × 2 = 1 + 0.992 241 152;
  • 19) 0.992 241 152 × 2 = 1 + 0.984 482 304;
  • 20) 0.984 482 304 × 2 = 1 + 0.968 964 608;
  • 21) 0.968 964 608 × 2 = 1 + 0.937 929 216;
  • 22) 0.937 929 216 × 2 = 1 + 0.875 858 432;
  • 23) 0.875 858 432 × 2 = 1 + 0.751 716 864;
  • 24) 0.751 716 864 × 2 = 1 + 0.503 433 728;
  • 25) 0.503 433 728 × 2 = 1 + 0.006 867 456;
  • 26) 0.006 867 456 × 2 = 0 + 0.013 734 912;
  • 27) 0.013 734 912 × 2 = 0 + 0.027 469 824;
  • 28) 0.027 469 824 × 2 = 0 + 0.054 939 648;
  • 29) 0.054 939 648 × 2 = 0 + 0.109 879 296;
  • 30) 0.109 879 296 × 2 = 0 + 0.219 758 592;
  • 31) 0.219 758 592 × 2 = 0 + 0.439 517 184;
  • 32) 0.439 517 184 × 2 = 0 + 0.879 034 368;
  • 33) 0.879 034 368 × 2 = 1 + 0.758 068 736;
  • 34) 0.758 068 736 × 2 = 1 + 0.516 137 472;
  • 35) 0.516 137 472 × 2 = 1 + 0.032 274 944;
  • 36) 0.032 274 944 × 2 = 0 + 0.064 549 888;
  • 37) 0.064 549 888 × 2 = 0 + 0.129 099 776;
  • 38) 0.129 099 776 × 2 = 0 + 0.258 199 552;
  • 39) 0.258 199 552 × 2 = 0 + 0.516 399 104;
  • 40) 0.516 399 104 × 2 = 1 + 0.032 798 208;
  • 41) 0.032 798 208 × 2 = 0 + 0.065 596 416;
  • 42) 0.065 596 416 × 2 = 0 + 0.131 192 832;
  • 43) 0.131 192 832 × 2 = 0 + 0.262 385 664;
  • 44) 0.262 385 664 × 2 = 0 + 0.524 771 328;
  • 45) 0.524 771 328 × 2 = 1 + 0.049 542 656;
  • 46) 0.049 542 656 × 2 = 0 + 0.099 085 312;
  • 47) 0.099 085 312 × 2 = 0 + 0.198 170 624;
  • 48) 0.198 170 624 × 2 = 0 + 0.396 341 248;
  • 49) 0.396 341 248 × 2 = 0 + 0.792 682 496;
  • 50) 0.792 682 496 × 2 = 1 + 0.585 364 992;
  • 51) 0.585 364 992 × 2 = 1 + 0.170 729 984;
  • 52) 0.170 729 984 × 2 = 0 + 0.341 459 968;
  • 53) 0.341 459 968 × 2 = 0 + 0.682 919 936;
  • 54) 0.682 919 936 × 2 = 1 + 0.365 839 872;
  • 55) 0.365 839 872 × 2 = 0 + 0.731 679 744;
  • 56) 0.731 679 744 × 2 = 1 + 0.463 359 488;
  • 57) 0.463 359 488 × 2 = 0 + 0.926 718 976;
  • 58) 0.926 718 976 × 2 = 1 + 0.853 437 952;
  • 59) 0.853 437 952 × 2 = 1 + 0.706 875 904;
  • 60) 0.706 875 904 × 2 = 1 + 0.413 751 808;
  • 61) 0.413 751 808 × 2 = 0 + 0.827 503 616;
  • 62) 0.827 503 616 × 2 = 1 + 0.655 007 232;
  • 63) 0.655 007 232 × 2 = 1 + 0.310 014 464;
  • 64) 0.310 014 464 × 2 = 0 + 0.620 028 928;

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:


0.000 282 258(10) =

0.0000 0000 0001 0010 0111 1111 1000 0000 1110 0001 0000 1000 0110 0101 0111 0110(2)

6. Positive number before normalization:

0.000 282 258(10) =

0.0000 0000 0001 0010 0111 1111 1000 0000 1110 0001 0000 1000 0110 0101 0111 0110(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:


0.000 282 258(10) =

0.0000 0000 0001 0010 0111 1111 1000 0000 1110 0001 0000 1000 0110 0101 0111 0110(2) =

0.0000 0000 0001 0010 0111 1111 1000 0000 1110 0001 0000 1000 0110 0101 0111 0110(2) × 20 =

1.0010 0111 1111 1000 0000 1110 0001 0000 1000 0110 0101 0111 0110(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 1111 1000 0000 1110 0001 0000 1000 0110 0101 0111 0110


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 1111 1000 0000 1110 0001 0000 1000 0110 0101 0111 0110 =

0010 0111 1111 1000 0000 1110 0001 0000 1000 0110 0101 0111 0110


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 1111 1000 0000 1110 0001 0000 1000 0110 0101 0111 0110


Decimal number -0.000 282 258 converted to 64 bit double precision IEEE 754 binary floating point representation:

1 - 011 1111 0011 - 0010 0111 1111 1000 0000 1110 0001 0000 1000 0110 0101 0111 0110

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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