-0.000 282 43 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.000 282 43(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 43(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 43| = 0.000 282 43


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 43.

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 43 × 2 = 0 + 0.000 564 86;
  • 2) 0.000 564 86 × 2 = 0 + 0.001 129 72;
  • 3) 0.001 129 72 × 2 = 0 + 0.002 259 44;
  • 4) 0.002 259 44 × 2 = 0 + 0.004 518 88;
  • 5) 0.004 518 88 × 2 = 0 + 0.009 037 76;
  • 6) 0.009 037 76 × 2 = 0 + 0.018 075 52;
  • 7) 0.018 075 52 × 2 = 0 + 0.036 151 04;
  • 8) 0.036 151 04 × 2 = 0 + 0.072 302 08;
  • 9) 0.072 302 08 × 2 = 0 + 0.144 604 16;
  • 10) 0.144 604 16 × 2 = 0 + 0.289 208 32;
  • 11) 0.289 208 32 × 2 = 0 + 0.578 416 64;
  • 12) 0.578 416 64 × 2 = 1 + 0.156 833 28;
  • 13) 0.156 833 28 × 2 = 0 + 0.313 666 56;
  • 14) 0.313 666 56 × 2 = 0 + 0.627 333 12;
  • 15) 0.627 333 12 × 2 = 1 + 0.254 666 24;
  • 16) 0.254 666 24 × 2 = 0 + 0.509 332 48;
  • 17) 0.509 332 48 × 2 = 1 + 0.018 664 96;
  • 18) 0.018 664 96 × 2 = 0 + 0.037 329 92;
  • 19) 0.037 329 92 × 2 = 0 + 0.074 659 84;
  • 20) 0.074 659 84 × 2 = 0 + 0.149 319 68;
  • 21) 0.149 319 68 × 2 = 0 + 0.298 639 36;
  • 22) 0.298 639 36 × 2 = 0 + 0.597 278 72;
  • 23) 0.597 278 72 × 2 = 1 + 0.194 557 44;
  • 24) 0.194 557 44 × 2 = 0 + 0.389 114 88;
  • 25) 0.389 114 88 × 2 = 0 + 0.778 229 76;
  • 26) 0.778 229 76 × 2 = 1 + 0.556 459 52;
  • 27) 0.556 459 52 × 2 = 1 + 0.112 919 04;
  • 28) 0.112 919 04 × 2 = 0 + 0.225 838 08;
  • 29) 0.225 838 08 × 2 = 0 + 0.451 676 16;
  • 30) 0.451 676 16 × 2 = 0 + 0.903 352 32;
  • 31) 0.903 352 32 × 2 = 1 + 0.806 704 64;
  • 32) 0.806 704 64 × 2 = 1 + 0.613 409 28;
  • 33) 0.613 409 28 × 2 = 1 + 0.226 818 56;
  • 34) 0.226 818 56 × 2 = 0 + 0.453 637 12;
  • 35) 0.453 637 12 × 2 = 0 + 0.907 274 24;
  • 36) 0.907 274 24 × 2 = 1 + 0.814 548 48;
  • 37) 0.814 548 48 × 2 = 1 + 0.629 096 96;
  • 38) 0.629 096 96 × 2 = 1 + 0.258 193 92;
  • 39) 0.258 193 92 × 2 = 0 + 0.516 387 84;
  • 40) 0.516 387 84 × 2 = 1 + 0.032 775 68;
  • 41) 0.032 775 68 × 2 = 0 + 0.065 551 36;
  • 42) 0.065 551 36 × 2 = 0 + 0.131 102 72;
  • 43) 0.131 102 72 × 2 = 0 + 0.262 205 44;
  • 44) 0.262 205 44 × 2 = 0 + 0.524 410 88;
  • 45) 0.524 410 88 × 2 = 1 + 0.048 821 76;
  • 46) 0.048 821 76 × 2 = 0 + 0.097 643 52;
  • 47) 0.097 643 52 × 2 = 0 + 0.195 287 04;
  • 48) 0.195 287 04 × 2 = 0 + 0.390 574 08;
  • 49) 0.390 574 08 × 2 = 0 + 0.781 148 16;
  • 50) 0.781 148 16 × 2 = 1 + 0.562 296 32;
  • 51) 0.562 296 32 × 2 = 1 + 0.124 592 64;
  • 52) 0.124 592 64 × 2 = 0 + 0.249 185 28;
  • 53) 0.249 185 28 × 2 = 0 + 0.498 370 56;
  • 54) 0.498 370 56 × 2 = 0 + 0.996 741 12;
  • 55) 0.996 741 12 × 2 = 1 + 0.993 482 24;
  • 56) 0.993 482 24 × 2 = 1 + 0.986 964 48;
  • 57) 0.986 964 48 × 2 = 1 + 0.973 928 96;
  • 58) 0.973 928 96 × 2 = 1 + 0.947 857 92;
  • 59) 0.947 857 92 × 2 = 1 + 0.895 715 84;
  • 60) 0.895 715 84 × 2 = 1 + 0.791 431 68;
  • 61) 0.791 431 68 × 2 = 1 + 0.582 863 36;
  • 62) 0.582 863 36 × 2 = 1 + 0.165 726 72;
  • 63) 0.165 726 72 × 2 = 0 + 0.331 453 44;
  • 64) 0.331 453 44 × 2 = 0 + 0.662 906 88;

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 43(10) =

0.0000 0000 0001 0010 1000 0010 0110 0011 1001 1101 0000 1000 0110 0011 1111 1100(2)

6. Positive number before normalization:

0.000 282 43(10) =

0.0000 0000 0001 0010 1000 0010 0110 0011 1001 1101 0000 1000 0110 0011 1111 1100(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 43(10) =

0.0000 0000 0001 0010 1000 0010 0110 0011 1001 1101 0000 1000 0110 0011 1111 1100(2) =

0.0000 0000 0001 0010 1000 0010 0110 0011 1001 1101 0000 1000 0110 0011 1111 1100(2) × 20 =

1.0010 1000 0010 0110 0011 1001 1101 0000 1000 0110 0011 1111 1100(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 0010 0110 0011 1001 1101 0000 1000 0110 0011 1111 1100


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 0010 0110 0011 1001 1101 0000 1000 0110 0011 1111 1100 =

0010 1000 0010 0110 0011 1001 1101 0000 1000 0110 0011 1111 1100


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 0010 0110 0011 1001 1101 0000 1000 0110 0011 1111 1100


Decimal number -0.000 282 43 converted to 64 bit double precision IEEE 754 binary floating point representation:

1 - 011 1111 0011 - 0010 1000 0010 0110 0011 1001 1101 0000 1000 0110 0011 1111 1100

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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