-0.003 901 8 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.003 901 8(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
-0.003 901 8(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:

|-0.003 901 8| = 0.003 901 8


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: 0.003 901 8.

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) 0.003 901 8 × 2 = 0 + 0.007 803 6;
  • 2) 0.007 803 6 × 2 = 0 + 0.015 607 2;
  • 3) 0.015 607 2 × 2 = 0 + 0.031 214 4;
  • 4) 0.031 214 4 × 2 = 0 + 0.062 428 8;
  • 5) 0.062 428 8 × 2 = 0 + 0.124 857 6;
  • 6) 0.124 857 6 × 2 = 0 + 0.249 715 2;
  • 7) 0.249 715 2 × 2 = 0 + 0.499 430 4;
  • 8) 0.499 430 4 × 2 = 0 + 0.998 860 8;
  • 9) 0.998 860 8 × 2 = 1 + 0.997 721 6;
  • 10) 0.997 721 6 × 2 = 1 + 0.995 443 2;
  • 11) 0.995 443 2 × 2 = 1 + 0.990 886 4;
  • 12) 0.990 886 4 × 2 = 1 + 0.981 772 8;
  • 13) 0.981 772 8 × 2 = 1 + 0.963 545 6;
  • 14) 0.963 545 6 × 2 = 1 + 0.927 091 2;
  • 15) 0.927 091 2 × 2 = 1 + 0.854 182 4;
  • 16) 0.854 182 4 × 2 = 1 + 0.708 364 8;
  • 17) 0.708 364 8 × 2 = 1 + 0.416 729 6;
  • 18) 0.416 729 6 × 2 = 0 + 0.833 459 2;
  • 19) 0.833 459 2 × 2 = 1 + 0.666 918 4;
  • 20) 0.666 918 4 × 2 = 1 + 0.333 836 8;
  • 21) 0.333 836 8 × 2 = 0 + 0.667 673 6;
  • 22) 0.667 673 6 × 2 = 1 + 0.335 347 2;
  • 23) 0.335 347 2 × 2 = 0 + 0.670 694 4;
  • 24) 0.670 694 4 × 2 = 1 + 0.341 388 8;
  • 25) 0.341 388 8 × 2 = 0 + 0.682 777 6;
  • 26) 0.682 777 6 × 2 = 1 + 0.365 555 2;
  • 27) 0.365 555 2 × 2 = 0 + 0.731 110 4;
  • 28) 0.731 110 4 × 2 = 1 + 0.462 220 8;
  • 29) 0.462 220 8 × 2 = 0 + 0.924 441 6;
  • 30) 0.924 441 6 × 2 = 1 + 0.848 883 2;
  • 31) 0.848 883 2 × 2 = 1 + 0.697 766 4;
  • 32) 0.697 766 4 × 2 = 1 + 0.395 532 8;
  • 33) 0.395 532 8 × 2 = 0 + 0.791 065 6;
  • 34) 0.791 065 6 × 2 = 1 + 0.582 131 2;
  • 35) 0.582 131 2 × 2 = 1 + 0.164 262 4;
  • 36) 0.164 262 4 × 2 = 0 + 0.328 524 8;
  • 37) 0.328 524 8 × 2 = 0 + 0.657 049 6;
  • 38) 0.657 049 6 × 2 = 1 + 0.314 099 2;
  • 39) 0.314 099 2 × 2 = 0 + 0.628 198 4;
  • 40) 0.628 198 4 × 2 = 1 + 0.256 396 8;
  • 41) 0.256 396 8 × 2 = 0 + 0.512 793 6;
  • 42) 0.512 793 6 × 2 = 1 + 0.025 587 2;
  • 43) 0.025 587 2 × 2 = 0 + 0.051 174 4;
  • 44) 0.051 174 4 × 2 = 0 + 0.102 348 8;
  • 45) 0.102 348 8 × 2 = 0 + 0.204 697 6;
  • 46) 0.204 697 6 × 2 = 0 + 0.409 395 2;
  • 47) 0.409 395 2 × 2 = 0 + 0.818 790 4;
  • 48) 0.818 790 4 × 2 = 1 + 0.637 580 8;
  • 49) 0.637 580 8 × 2 = 1 + 0.275 161 6;
  • 50) 0.275 161 6 × 2 = 0 + 0.550 323 2;
  • 51) 0.550 323 2 × 2 = 1 + 0.100 646 4;
  • 52) 0.100 646 4 × 2 = 0 + 0.201 292 8;
  • 53) 0.201 292 8 × 2 = 0 + 0.402 585 6;
  • 54) 0.402 585 6 × 2 = 0 + 0.805 171 2;
  • 55) 0.805 171 2 × 2 = 1 + 0.610 342 4;
  • 56) 0.610 342 4 × 2 = 1 + 0.220 684 8;
  • 57) 0.220 684 8 × 2 = 0 + 0.441 369 6;
  • 58) 0.441 369 6 × 2 = 0 + 0.882 739 2;
  • 59) 0.882 739 2 × 2 = 1 + 0.765 478 4;
  • 60) 0.765 478 4 × 2 = 1 + 0.530 956 8;
  • 61) 0.530 956 8 × 2 = 1 + 0.061 913 6;

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:


0.003 901 8(10) =

0.0000 0000 1111 1111 1011 0101 0101 0111 0110 0101 0100 0001 1010 0011 0011 1(2)

6. Positive number before normalization:

0.003 901 8(10) =

0.0000 0000 1111 1111 1011 0101 0101 0111 0110 0101 0100 0001 1010 0011 0011 1(2)

7. Normalize the binary representation of the number.

Shift the decimal mark 9 positions to the right, so that only one non zero digit remains to the left of it:


0.003 901 8(10) =

0.0000 0000 1111 1111 1011 0101 0101 0111 0110 0101 0100 0001 1010 0011 0011 1(2) =

0.0000 0000 1111 1111 1011 0101 0101 0111 0110 0101 0100 0001 1010 0011 0011 1(2) × 20 =

1.1111 1111 0110 1010 1010 1110 1100 1010 1000 0011 0100 0110 0111(2) × 2-9


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): -9

Mantissa (not normalized):
1.1111 1111 0110 1010 1010 1110 1100 1010 1000 0011 0100 0110 0111


9. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

-9 + 2(11-1) - 1 =

(-9 + 1 023)(10) =

1 014(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;
  • 1 014 ÷ 2 = 507 + 0;
  • 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) =

1014(10) =

011 1111 0110(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. 1111 1111 0110 1010 1010 1110 1100 1010 1000 0011 0100 0110 0111 =

1111 1111 0110 1010 1010 1110 1100 1010 1000 0011 0100 0110 0111


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 0110

Mantissa (52 bits) =
1111 1111 0110 1010 1010 1110 1100 1010 1000 0011 0100 0110 0111


Decimal number -0.003 901 8 converted to 64 bit double precision IEEE 754 binary floating point representation:

1 - 011 1111 0110 - 1111 1111 0110 1010 1010 1110 1100 1010 1000 0011 0100 0110 0111

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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