-0.004 999 985 4 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.004 999 985 4(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
-0.004 999 985 4(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:

|-0.004 999 985 4| = 0.004 999 985 4


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: 0.004 999 985 4.

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) 0.004 999 985 4 × 2 = 0 + 0.009 999 970 8;
  • 2) 0.009 999 970 8 × 2 = 0 + 0.019 999 941 6;
  • 3) 0.019 999 941 6 × 2 = 0 + 0.039 999 883 2;
  • 4) 0.039 999 883 2 × 2 = 0 + 0.079 999 766 4;
  • 5) 0.079 999 766 4 × 2 = 0 + 0.159 999 532 8;
  • 6) 0.159 999 532 8 × 2 = 0 + 0.319 999 065 6;
  • 7) 0.319 999 065 6 × 2 = 0 + 0.639 998 131 2;
  • 8) 0.639 998 131 2 × 2 = 1 + 0.279 996 262 4;
  • 9) 0.279 996 262 4 × 2 = 0 + 0.559 992 524 8;
  • 10) 0.559 992 524 8 × 2 = 1 + 0.119 985 049 6;
  • 11) 0.119 985 049 6 × 2 = 0 + 0.239 970 099 2;
  • 12) 0.239 970 099 2 × 2 = 0 + 0.479 940 198 4;
  • 13) 0.479 940 198 4 × 2 = 0 + 0.959 880 396 8;
  • 14) 0.959 880 396 8 × 2 = 1 + 0.919 760 793 6;
  • 15) 0.919 760 793 6 × 2 = 1 + 0.839 521 587 2;
  • 16) 0.839 521 587 2 × 2 = 1 + 0.679 043 174 4;
  • 17) 0.679 043 174 4 × 2 = 1 + 0.358 086 348 8;
  • 18) 0.358 086 348 8 × 2 = 0 + 0.716 172 697 6;
  • 19) 0.716 172 697 6 × 2 = 1 + 0.432 345 395 2;
  • 20) 0.432 345 395 2 × 2 = 0 + 0.864 690 790 4;
  • 21) 0.864 690 790 4 × 2 = 1 + 0.729 381 580 8;
  • 22) 0.729 381 580 8 × 2 = 1 + 0.458 763 161 6;
  • 23) 0.458 763 161 6 × 2 = 0 + 0.917 526 323 2;
  • 24) 0.917 526 323 2 × 2 = 1 + 0.835 052 646 4;
  • 25) 0.835 052 646 4 × 2 = 1 + 0.670 105 292 8;
  • 26) 0.670 105 292 8 × 2 = 1 + 0.340 210 585 6;
  • 27) 0.340 210 585 6 × 2 = 0 + 0.680 421 171 2;
  • 28) 0.680 421 171 2 × 2 = 1 + 0.360 842 342 4;
  • 29) 0.360 842 342 4 × 2 = 0 + 0.721 684 684 8;
  • 30) 0.721 684 684 8 × 2 = 1 + 0.443 369 369 6;
  • 31) 0.443 369 369 6 × 2 = 0 + 0.886 738 739 2;
  • 32) 0.886 738 739 2 × 2 = 1 + 0.773 477 478 4;
  • 33) 0.773 477 478 4 × 2 = 1 + 0.546 954 956 8;
  • 34) 0.546 954 956 8 × 2 = 1 + 0.093 909 913 6;
  • 35) 0.093 909 913 6 × 2 = 0 + 0.187 819 827 2;
  • 36) 0.187 819 827 2 × 2 = 0 + 0.375 639 654 4;
  • 37) 0.375 639 654 4 × 2 = 0 + 0.751 279 308 8;
  • 38) 0.751 279 308 8 × 2 = 1 + 0.502 558 617 6;
  • 39) 0.502 558 617 6 × 2 = 1 + 0.005 117 235 2;
  • 40) 0.005 117 235 2 × 2 = 0 + 0.010 234 470 4;
  • 41) 0.010 234 470 4 × 2 = 0 + 0.020 468 940 8;
  • 42) 0.020 468 940 8 × 2 = 0 + 0.040 937 881 6;
  • 43) 0.040 937 881 6 × 2 = 0 + 0.081 875 763 2;
  • 44) 0.081 875 763 2 × 2 = 0 + 0.163 751 526 4;
  • 45) 0.163 751 526 4 × 2 = 0 + 0.327 503 052 8;
  • 46) 0.327 503 052 8 × 2 = 0 + 0.655 006 105 6;
  • 47) 0.655 006 105 6 × 2 = 1 + 0.310 012 211 2;
  • 48) 0.310 012 211 2 × 2 = 0 + 0.620 024 422 4;
  • 49) 0.620 024 422 4 × 2 = 1 + 0.240 048 844 8;
  • 50) 0.240 048 844 8 × 2 = 0 + 0.480 097 689 6;
  • 51) 0.480 097 689 6 × 2 = 0 + 0.960 195 379 2;
  • 52) 0.960 195 379 2 × 2 = 1 + 0.920 390 758 4;
  • 53) 0.920 390 758 4 × 2 = 1 + 0.840 781 516 8;
  • 54) 0.840 781 516 8 × 2 = 1 + 0.681 563 033 6;
  • 55) 0.681 563 033 6 × 2 = 1 + 0.363 126 067 2;
  • 56) 0.363 126 067 2 × 2 = 0 + 0.726 252 134 4;
  • 57) 0.726 252 134 4 × 2 = 1 + 0.452 504 268 8;
  • 58) 0.452 504 268 8 × 2 = 0 + 0.905 008 537 6;
  • 59) 0.905 008 537 6 × 2 = 1 + 0.810 017 075 2;
  • 60) 0.810 017 075 2 × 2 = 1 + 0.620 034 150 4;

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:


0.004 999 985 4(10) =

0.0000 0001 0100 0111 1010 1101 1101 0101 1100 0110 0000 0010 1001 1110 1011(2)

6. Positive number before normalization:

0.004 999 985 4(10) =

0.0000 0001 0100 0111 1010 1101 1101 0101 1100 0110 0000 0010 1001 1110 1011(2)

7. Normalize the binary representation of the number.

Shift the decimal mark 8 positions to the right, so that only one non zero digit remains to the left of it:


0.004 999 985 4(10) =

0.0000 0001 0100 0111 1010 1101 1101 0101 1100 0110 0000 0010 1001 1110 1011(2) =

0.0000 0001 0100 0111 1010 1101 1101 0101 1100 0110 0000 0010 1001 1110 1011(2) × 20 =

1.0100 0111 1010 1101 1101 0101 1100 0110 0000 0010 1001 1110 1011(2) × 2-8


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): -8

Mantissa (not normalized):
1.0100 0111 1010 1101 1101 0101 1100 0110 0000 0010 1001 1110 1011


9. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

Exponent (unadjusted) + 2(11-1) - 1 =

-8 + 2(11-1) - 1 =

(-8 + 1 023)(10) =

1 015(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;
  • 1 015 ÷ 2 = 507 + 1;
  • 507 ÷ 2 = 253 + 1;
  • 253 ÷ 2 = 126 + 1;
  • 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) =

1015(10) =

011 1111 0111(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. 0100 0111 1010 1101 1101 0101 1100 0110 0000 0010 1001 1110 1011 =

0100 0111 1010 1101 1101 0101 1100 0110 0000 0010 1001 1110 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 0111

Mantissa (52 bits) =
0100 0111 1010 1101 1101 0101 1100 0110 0000 0010 1001 1110 1011


Decimal number -0.004 999 985 4 converted to 64 bit double precision IEEE 754 binary floating point representation:

1 - 011 1111 0111 - 0100 0111 1010 1101 1101 0101 1100 0110 0000 0010 1001 1110 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