-0.000 282 034 7 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.000 282 034 7(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
-0.000 282 034 7(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:

|-0.000 282 034 7| = 0.000 282 034 7


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: 0.000 282 034 7.

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) 0.000 282 034 7 × 2 = 0 + 0.000 564 069 4;
  • 2) 0.000 564 069 4 × 2 = 0 + 0.001 128 138 8;
  • 3) 0.001 128 138 8 × 2 = 0 + 0.002 256 277 6;
  • 4) 0.002 256 277 6 × 2 = 0 + 0.004 512 555 2;
  • 5) 0.004 512 555 2 × 2 = 0 + 0.009 025 110 4;
  • 6) 0.009 025 110 4 × 2 = 0 + 0.018 050 220 8;
  • 7) 0.018 050 220 8 × 2 = 0 + 0.036 100 441 6;
  • 8) 0.036 100 441 6 × 2 = 0 + 0.072 200 883 2;
  • 9) 0.072 200 883 2 × 2 = 0 + 0.144 401 766 4;
  • 10) 0.144 401 766 4 × 2 = 0 + 0.288 803 532 8;
  • 11) 0.288 803 532 8 × 2 = 0 + 0.577 607 065 6;
  • 12) 0.577 607 065 6 × 2 = 1 + 0.155 214 131 2;
  • 13) 0.155 214 131 2 × 2 = 0 + 0.310 428 262 4;
  • 14) 0.310 428 262 4 × 2 = 0 + 0.620 856 524 8;
  • 15) 0.620 856 524 8 × 2 = 1 + 0.241 713 049 6;
  • 16) 0.241 713 049 6 × 2 = 0 + 0.483 426 099 2;
  • 17) 0.483 426 099 2 × 2 = 0 + 0.966 852 198 4;
  • 18) 0.966 852 198 4 × 2 = 1 + 0.933 704 396 8;
  • 19) 0.933 704 396 8 × 2 = 1 + 0.867 408 793 6;
  • 20) 0.867 408 793 6 × 2 = 1 + 0.734 817 587 2;
  • 21) 0.734 817 587 2 × 2 = 1 + 0.469 635 174 4;
  • 22) 0.469 635 174 4 × 2 = 0 + 0.939 270 348 8;
  • 23) 0.939 270 348 8 × 2 = 1 + 0.878 540 697 6;
  • 24) 0.878 540 697 6 × 2 = 1 + 0.757 081 395 2;
  • 25) 0.757 081 395 2 × 2 = 1 + 0.514 162 790 4;
  • 26) 0.514 162 790 4 × 2 = 1 + 0.028 325 580 8;
  • 27) 0.028 325 580 8 × 2 = 0 + 0.056 651 161 6;
  • 28) 0.056 651 161 6 × 2 = 0 + 0.113 302 323 2;
  • 29) 0.113 302 323 2 × 2 = 0 + 0.226 604 646 4;
  • 30) 0.226 604 646 4 × 2 = 0 + 0.453 209 292 8;
  • 31) 0.453 209 292 8 × 2 = 0 + 0.906 418 585 6;
  • 32) 0.906 418 585 6 × 2 = 1 + 0.812 837 171 2;
  • 33) 0.812 837 171 2 × 2 = 1 + 0.625 674 342 4;
  • 34) 0.625 674 342 4 × 2 = 1 + 0.251 348 684 8;
  • 35) 0.251 348 684 8 × 2 = 0 + 0.502 697 369 6;
  • 36) 0.502 697 369 6 × 2 = 1 + 0.005 394 739 2;
  • 37) 0.005 394 739 2 × 2 = 0 + 0.010 789 478 4;
  • 38) 0.010 789 478 4 × 2 = 0 + 0.021 578 956 8;
  • 39) 0.021 578 956 8 × 2 = 0 + 0.043 157 913 6;
  • 40) 0.043 157 913 6 × 2 = 0 + 0.086 315 827 2;
  • 41) 0.086 315 827 2 × 2 = 0 + 0.172 631 654 4;
  • 42) 0.172 631 654 4 × 2 = 0 + 0.345 263 308 8;
  • 43) 0.345 263 308 8 × 2 = 0 + 0.690 526 617 6;
  • 44) 0.690 526 617 6 × 2 = 1 + 0.381 053 235 2;
  • 45) 0.381 053 235 2 × 2 = 0 + 0.762 106 470 4;
  • 46) 0.762 106 470 4 × 2 = 1 + 0.524 212 940 8;
  • 47) 0.524 212 940 8 × 2 = 1 + 0.048 425 881 6;
  • 48) 0.048 425 881 6 × 2 = 0 + 0.096 851 763 2;
  • 49) 0.096 851 763 2 × 2 = 0 + 0.193 703 526 4;
  • 50) 0.193 703 526 4 × 2 = 0 + 0.387 407 052 8;
  • 51) 0.387 407 052 8 × 2 = 0 + 0.774 814 105 6;
  • 52) 0.774 814 105 6 × 2 = 1 + 0.549 628 211 2;
  • 53) 0.549 628 211 2 × 2 = 1 + 0.099 256 422 4;
  • 54) 0.099 256 422 4 × 2 = 0 + 0.198 512 844 8;
  • 55) 0.198 512 844 8 × 2 = 0 + 0.397 025 689 6;
  • 56) 0.397 025 689 6 × 2 = 0 + 0.794 051 379 2;
  • 57) 0.794 051 379 2 × 2 = 1 + 0.588 102 758 4;
  • 58) 0.588 102 758 4 × 2 = 1 + 0.176 205 516 8;
  • 59) 0.176 205 516 8 × 2 = 0 + 0.352 411 033 6;
  • 60) 0.352 411 033 6 × 2 = 0 + 0.704 822 067 2;
  • 61) 0.704 822 067 2 × 2 = 1 + 0.409 644 134 4;
  • 62) 0.409 644 134 4 × 2 = 0 + 0.819 288 268 8;
  • 63) 0.819 288 268 8 × 2 = 1 + 0.638 576 537 6;
  • 64) 0.638 576 537 6 × 2 = 1 + 0.277 153 075 2;

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:


0.000 282 034 7(10) =

0.0000 0000 0001 0010 0111 1011 1100 0001 1101 0000 0001 0110 0001 1000 1100 1011(2)

6. Positive number before normalization:

0.000 282 034 7(10) =

0.0000 0000 0001 0010 0111 1011 1100 0001 1101 0000 0001 0110 0001 1000 1100 1011(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:


0.000 282 034 7(10) =

0.0000 0000 0001 0010 0111 1011 1100 0001 1101 0000 0001 0110 0001 1000 1100 1011(2) =

0.0000 0000 0001 0010 0111 1011 1100 0001 1101 0000 0001 0110 0001 1000 1100 1011(2) × 20 =

1.0010 0111 1011 1100 0001 1101 0000 0001 0110 0001 1000 1100 1011(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 1011 1100 0001 1101 0000 0001 0110 0001 1000 1100 1011


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 1011 1100 0001 1101 0000 0001 0110 0001 1000 1100 1011 =

0010 0111 1011 1100 0001 1101 0000 0001 0110 0001 1000 1100 1011


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 1011 1100 0001 1101 0000 0001 0110 0001 1000 1100 1011


Decimal number -0.000 282 034 7 converted to 64 bit double precision IEEE 754 binary floating point representation:

1 - 011 1111 0011 - 0010 0111 1011 1100 0001 1101 0000 0001 0110 0001 1000 1100 1011

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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