-0.000 281 96 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.000 281 96(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
-0.000 281 96(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:

|-0.000 281 96| = 0.000 281 96


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: 0.000 281 96.

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) 0.000 281 96 × 2 = 0 + 0.000 563 92;
  • 2) 0.000 563 92 × 2 = 0 + 0.001 127 84;
  • 3) 0.001 127 84 × 2 = 0 + 0.002 255 68;
  • 4) 0.002 255 68 × 2 = 0 + 0.004 511 36;
  • 5) 0.004 511 36 × 2 = 0 + 0.009 022 72;
  • 6) 0.009 022 72 × 2 = 0 + 0.018 045 44;
  • 7) 0.018 045 44 × 2 = 0 + 0.036 090 88;
  • 8) 0.036 090 88 × 2 = 0 + 0.072 181 76;
  • 9) 0.072 181 76 × 2 = 0 + 0.144 363 52;
  • 10) 0.144 363 52 × 2 = 0 + 0.288 727 04;
  • 11) 0.288 727 04 × 2 = 0 + 0.577 454 08;
  • 12) 0.577 454 08 × 2 = 1 + 0.154 908 16;
  • 13) 0.154 908 16 × 2 = 0 + 0.309 816 32;
  • 14) 0.309 816 32 × 2 = 0 + 0.619 632 64;
  • 15) 0.619 632 64 × 2 = 1 + 0.239 265 28;
  • 16) 0.239 265 28 × 2 = 0 + 0.478 530 56;
  • 17) 0.478 530 56 × 2 = 0 + 0.957 061 12;
  • 18) 0.957 061 12 × 2 = 1 + 0.914 122 24;
  • 19) 0.914 122 24 × 2 = 1 + 0.828 244 48;
  • 20) 0.828 244 48 × 2 = 1 + 0.656 488 96;
  • 21) 0.656 488 96 × 2 = 1 + 0.312 977 92;
  • 22) 0.312 977 92 × 2 = 0 + 0.625 955 84;
  • 23) 0.625 955 84 × 2 = 1 + 0.251 911 68;
  • 24) 0.251 911 68 × 2 = 0 + 0.503 823 36;
  • 25) 0.503 823 36 × 2 = 1 + 0.007 646 72;
  • 26) 0.007 646 72 × 2 = 0 + 0.015 293 44;
  • 27) 0.015 293 44 × 2 = 0 + 0.030 586 88;
  • 28) 0.030 586 88 × 2 = 0 + 0.061 173 76;
  • 29) 0.061 173 76 × 2 = 0 + 0.122 347 52;
  • 30) 0.122 347 52 × 2 = 0 + 0.244 695 04;
  • 31) 0.244 695 04 × 2 = 0 + 0.489 390 08;
  • 32) 0.489 390 08 × 2 = 0 + 0.978 780 16;
  • 33) 0.978 780 16 × 2 = 1 + 0.957 560 32;
  • 34) 0.957 560 32 × 2 = 1 + 0.915 120 64;
  • 35) 0.915 120 64 × 2 = 1 + 0.830 241 28;
  • 36) 0.830 241 28 × 2 = 1 + 0.660 482 56;
  • 37) 0.660 482 56 × 2 = 1 + 0.320 965 12;
  • 38) 0.320 965 12 × 2 = 0 + 0.641 930 24;
  • 39) 0.641 930 24 × 2 = 1 + 0.283 860 48;
  • 40) 0.283 860 48 × 2 = 0 + 0.567 720 96;
  • 41) 0.567 720 96 × 2 = 1 + 0.135 441 92;
  • 42) 0.135 441 92 × 2 = 0 + 0.270 883 84;
  • 43) 0.270 883 84 × 2 = 0 + 0.541 767 68;
  • 44) 0.541 767 68 × 2 = 1 + 0.083 535 36;
  • 45) 0.083 535 36 × 2 = 0 + 0.167 070 72;
  • 46) 0.167 070 72 × 2 = 0 + 0.334 141 44;
  • 47) 0.334 141 44 × 2 = 0 + 0.668 282 88;
  • 48) 0.668 282 88 × 2 = 1 + 0.336 565 76;
  • 49) 0.336 565 76 × 2 = 0 + 0.673 131 52;
  • 50) 0.673 131 52 × 2 = 1 + 0.346 263 04;
  • 51) 0.346 263 04 × 2 = 0 + 0.692 526 08;
  • 52) 0.692 526 08 × 2 = 1 + 0.385 052 16;
  • 53) 0.385 052 16 × 2 = 0 + 0.770 104 32;
  • 54) 0.770 104 32 × 2 = 1 + 0.540 208 64;
  • 55) 0.540 208 64 × 2 = 1 + 0.080 417 28;
  • 56) 0.080 417 28 × 2 = 0 + 0.160 834 56;
  • 57) 0.160 834 56 × 2 = 0 + 0.321 669 12;
  • 58) 0.321 669 12 × 2 = 0 + 0.643 338 24;
  • 59) 0.643 338 24 × 2 = 1 + 0.286 676 48;
  • 60) 0.286 676 48 × 2 = 0 + 0.573 352 96;
  • 61) 0.573 352 96 × 2 = 1 + 0.146 705 92;
  • 62) 0.146 705 92 × 2 = 0 + 0.293 411 84;
  • 63) 0.293 411 84 × 2 = 0 + 0.586 823 68;
  • 64) 0.586 823 68 × 2 = 1 + 0.173 647 36;

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:


0.000 281 96(10) =

0.0000 0000 0001 0010 0111 1010 1000 0000 1111 1010 1001 0001 0101 0110 0010 1001(2)

6. Positive number before normalization:

0.000 281 96(10) =

0.0000 0000 0001 0010 0111 1010 1000 0000 1111 1010 1001 0001 0101 0110 0010 1001(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:


0.000 281 96(10) =

0.0000 0000 0001 0010 0111 1010 1000 0000 1111 1010 1001 0001 0101 0110 0010 1001(2) =

0.0000 0000 0001 0010 0111 1010 1000 0000 1111 1010 1001 0001 0101 0110 0010 1001(2) × 20 =

1.0010 0111 1010 1000 0000 1111 1010 1001 0001 0101 0110 0010 1001(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 1010 1000 0000 1111 1010 1001 0001 0101 0110 0010 1001


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 1010 1000 0000 1111 1010 1001 0001 0101 0110 0010 1001 =

0010 0111 1010 1000 0000 1111 1010 1001 0001 0101 0110 0010 1001


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 1010 1000 0000 1111 1010 1001 0001 0101 0110 0010 1001


Decimal number -0.000 281 96 converted to 64 bit double precision IEEE 754 binary floating point representation:

1 - 011 1111 0011 - 0010 0111 1010 1000 0000 1111 1010 1001 0001 0101 0110 0010 1001

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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