-0.140 032 4 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.140 032 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
-0.140 032 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:

|-0.140 032 4| = 0.140 032 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: 0.140 032 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) 0.140 032 4 × 2 = 0 + 0.280 064 8;
  • 2) 0.280 064 8 × 2 = 0 + 0.560 129 6;
  • 3) 0.560 129 6 × 2 = 1 + 0.120 259 2;
  • 4) 0.120 259 2 × 2 = 0 + 0.240 518 4;
  • 5) 0.240 518 4 × 2 = 0 + 0.481 036 8;
  • 6) 0.481 036 8 × 2 = 0 + 0.962 073 6;
  • 7) 0.962 073 6 × 2 = 1 + 0.924 147 2;
  • 8) 0.924 147 2 × 2 = 1 + 0.848 294 4;
  • 9) 0.848 294 4 × 2 = 1 + 0.696 588 8;
  • 10) 0.696 588 8 × 2 = 1 + 0.393 177 6;
  • 11) 0.393 177 6 × 2 = 0 + 0.786 355 2;
  • 12) 0.786 355 2 × 2 = 1 + 0.572 710 4;
  • 13) 0.572 710 4 × 2 = 1 + 0.145 420 8;
  • 14) 0.145 420 8 × 2 = 0 + 0.290 841 6;
  • 15) 0.290 841 6 × 2 = 0 + 0.581 683 2;
  • 16) 0.581 683 2 × 2 = 1 + 0.163 366 4;
  • 17) 0.163 366 4 × 2 = 0 + 0.326 732 8;
  • 18) 0.326 732 8 × 2 = 0 + 0.653 465 6;
  • 19) 0.653 465 6 × 2 = 1 + 0.306 931 2;
  • 20) 0.306 931 2 × 2 = 0 + 0.613 862 4;
  • 21) 0.613 862 4 × 2 = 1 + 0.227 724 8;
  • 22) 0.227 724 8 × 2 = 0 + 0.455 449 6;
  • 23) 0.455 449 6 × 2 = 0 + 0.910 899 2;
  • 24) 0.910 899 2 × 2 = 1 + 0.821 798 4;
  • 25) 0.821 798 4 × 2 = 1 + 0.643 596 8;
  • 26) 0.643 596 8 × 2 = 1 + 0.287 193 6;
  • 27) 0.287 193 6 × 2 = 0 + 0.574 387 2;
  • 28) 0.574 387 2 × 2 = 1 + 0.148 774 4;
  • 29) 0.148 774 4 × 2 = 0 + 0.297 548 8;
  • 30) 0.297 548 8 × 2 = 0 + 0.595 097 6;
  • 31) 0.595 097 6 × 2 = 1 + 0.190 195 2;
  • 32) 0.190 195 2 × 2 = 0 + 0.380 390 4;
  • 33) 0.380 390 4 × 2 = 0 + 0.760 780 8;
  • 34) 0.760 780 8 × 2 = 1 + 0.521 561 6;
  • 35) 0.521 561 6 × 2 = 1 + 0.043 123 2;
  • 36) 0.043 123 2 × 2 = 0 + 0.086 246 4;
  • 37) 0.086 246 4 × 2 = 0 + 0.172 492 8;
  • 38) 0.172 492 8 × 2 = 0 + 0.344 985 6;
  • 39) 0.344 985 6 × 2 = 0 + 0.689 971 2;
  • 40) 0.689 971 2 × 2 = 1 + 0.379 942 4;
  • 41) 0.379 942 4 × 2 = 0 + 0.759 884 8;
  • 42) 0.759 884 8 × 2 = 1 + 0.519 769 6;
  • 43) 0.519 769 6 × 2 = 1 + 0.039 539 2;
  • 44) 0.039 539 2 × 2 = 0 + 0.079 078 4;
  • 45) 0.079 078 4 × 2 = 0 + 0.158 156 8;
  • 46) 0.158 156 8 × 2 = 0 + 0.316 313 6;
  • 47) 0.316 313 6 × 2 = 0 + 0.632 627 2;
  • 48) 0.632 627 2 × 2 = 1 + 0.265 254 4;
  • 49) 0.265 254 4 × 2 = 0 + 0.530 508 8;
  • 50) 0.530 508 8 × 2 = 1 + 0.061 017 6;
  • 51) 0.061 017 6 × 2 = 0 + 0.122 035 2;
  • 52) 0.122 035 2 × 2 = 0 + 0.244 070 4;
  • 53) 0.244 070 4 × 2 = 0 + 0.488 140 8;
  • 54) 0.488 140 8 × 2 = 0 + 0.976 281 6;
  • 55) 0.976 281 6 × 2 = 1 + 0.952 563 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:


0.140 032 4(10) =

0.0010 0011 1101 1001 0010 1001 1101 0010 0110 0001 0110 0001 0100 001(2)

6. Positive number before normalization:

0.140 032 4(10) =

0.0010 0011 1101 1001 0010 1001 1101 0010 0110 0001 0110 0001 0100 001(2)

7. Normalize the binary representation of the number.

Shift the decimal mark 3 positions to the right, so that only one non zero digit remains to the left of it:


0.140 032 4(10) =

0.0010 0011 1101 1001 0010 1001 1101 0010 0110 0001 0110 0001 0100 001(2) =

0.0010 0011 1101 1001 0010 1001 1101 0010 0110 0001 0110 0001 0100 001(2) × 20 =

1.0001 1110 1100 1001 0100 1110 1001 0011 0000 1011 0000 1010 0001(2) × 2-3


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): -3

Mantissa (not normalized):
1.0001 1110 1100 1001 0100 1110 1001 0011 0000 1011 0000 1010 0001


9. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

-3 + 2(11-1) - 1 =

(-3 + 1 023)(10) =

1 020(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;
  • 1 020 ÷ 2 = 510 + 0;
  • 510 ÷ 2 = 255 + 0;
  • 255 ÷ 2 = 127 + 1;
  • 127 ÷ 2 = 63 + 1;
  • 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) =

1020(10) =

011 1111 1100(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. 0001 1110 1100 1001 0100 1110 1001 0011 0000 1011 0000 1010 0001 =

0001 1110 1100 1001 0100 1110 1001 0011 0000 1011 0000 1010 0001


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 1100

Mantissa (52 bits) =
0001 1110 1100 1001 0100 1110 1001 0011 0000 1011 0000 1010 0001


Decimal number -0.140 032 4 converted to 64 bit double precision IEEE 754 binary floating point representation:

1 - 011 1111 1100 - 0001 1110 1100 1001 0100 1110 1001 0011 0000 1011 0000 1010 0001

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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