-0.009 826 508 3 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.009 826 508 3(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
-0.009 826 508 3(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:

|-0.009 826 508 3| = 0.009 826 508 3


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: 0.009 826 508 3.

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) 0.009 826 508 3 × 2 = 0 + 0.019 653 016 6;
  • 2) 0.019 653 016 6 × 2 = 0 + 0.039 306 033 2;
  • 3) 0.039 306 033 2 × 2 = 0 + 0.078 612 066 4;
  • 4) 0.078 612 066 4 × 2 = 0 + 0.157 224 132 8;
  • 5) 0.157 224 132 8 × 2 = 0 + 0.314 448 265 6;
  • 6) 0.314 448 265 6 × 2 = 0 + 0.628 896 531 2;
  • 7) 0.628 896 531 2 × 2 = 1 + 0.257 793 062 4;
  • 8) 0.257 793 062 4 × 2 = 0 + 0.515 586 124 8;
  • 9) 0.515 586 124 8 × 2 = 1 + 0.031 172 249 6;
  • 10) 0.031 172 249 6 × 2 = 0 + 0.062 344 499 2;
  • 11) 0.062 344 499 2 × 2 = 0 + 0.124 688 998 4;
  • 12) 0.124 688 998 4 × 2 = 0 + 0.249 377 996 8;
  • 13) 0.249 377 996 8 × 2 = 0 + 0.498 755 993 6;
  • 14) 0.498 755 993 6 × 2 = 0 + 0.997 511 987 2;
  • 15) 0.997 511 987 2 × 2 = 1 + 0.995 023 974 4;
  • 16) 0.995 023 974 4 × 2 = 1 + 0.990 047 948 8;
  • 17) 0.990 047 948 8 × 2 = 1 + 0.980 095 897 6;
  • 18) 0.980 095 897 6 × 2 = 1 + 0.960 191 795 2;
  • 19) 0.960 191 795 2 × 2 = 1 + 0.920 383 590 4;
  • 20) 0.920 383 590 4 × 2 = 1 + 0.840 767 180 8;
  • 21) 0.840 767 180 8 × 2 = 1 + 0.681 534 361 6;
  • 22) 0.681 534 361 6 × 2 = 1 + 0.363 068 723 2;
  • 23) 0.363 068 723 2 × 2 = 0 + 0.726 137 446 4;
  • 24) 0.726 137 446 4 × 2 = 1 + 0.452 274 892 8;
  • 25) 0.452 274 892 8 × 2 = 0 + 0.904 549 785 6;
  • 26) 0.904 549 785 6 × 2 = 1 + 0.809 099 571 2;
  • 27) 0.809 099 571 2 × 2 = 1 + 0.618 199 142 4;
  • 28) 0.618 199 142 4 × 2 = 1 + 0.236 398 284 8;
  • 29) 0.236 398 284 8 × 2 = 0 + 0.472 796 569 6;
  • 30) 0.472 796 569 6 × 2 = 0 + 0.945 593 139 2;
  • 31) 0.945 593 139 2 × 2 = 1 + 0.891 186 278 4;
  • 32) 0.891 186 278 4 × 2 = 1 + 0.782 372 556 8;
  • 33) 0.782 372 556 8 × 2 = 1 + 0.564 745 113 6;
  • 34) 0.564 745 113 6 × 2 = 1 + 0.129 490 227 2;
  • 35) 0.129 490 227 2 × 2 = 0 + 0.258 980 454 4;
  • 36) 0.258 980 454 4 × 2 = 0 + 0.517 960 908 8;
  • 37) 0.517 960 908 8 × 2 = 1 + 0.035 921 817 6;
  • 38) 0.035 921 817 6 × 2 = 0 + 0.071 843 635 2;
  • 39) 0.071 843 635 2 × 2 = 0 + 0.143 687 270 4;
  • 40) 0.143 687 270 4 × 2 = 0 + 0.287 374 540 8;
  • 41) 0.287 374 540 8 × 2 = 0 + 0.574 749 081 6;
  • 42) 0.574 749 081 6 × 2 = 1 + 0.149 498 163 2;
  • 43) 0.149 498 163 2 × 2 = 0 + 0.298 996 326 4;
  • 44) 0.298 996 326 4 × 2 = 0 + 0.597 992 652 8;
  • 45) 0.597 992 652 8 × 2 = 1 + 0.195 985 305 6;
  • 46) 0.195 985 305 6 × 2 = 0 + 0.391 970 611 2;
  • 47) 0.391 970 611 2 × 2 = 0 + 0.783 941 222 4;
  • 48) 0.783 941 222 4 × 2 = 1 + 0.567 882 444 8;
  • 49) 0.567 882 444 8 × 2 = 1 + 0.135 764 889 6;
  • 50) 0.135 764 889 6 × 2 = 0 + 0.271 529 779 2;
  • 51) 0.271 529 779 2 × 2 = 0 + 0.543 059 558 4;
  • 52) 0.543 059 558 4 × 2 = 1 + 0.086 119 116 8;
  • 53) 0.086 119 116 8 × 2 = 0 + 0.172 238 233 6;
  • 54) 0.172 238 233 6 × 2 = 0 + 0.344 476 467 2;
  • 55) 0.344 476 467 2 × 2 = 0 + 0.688 952 934 4;
  • 56) 0.688 952 934 4 × 2 = 1 + 0.377 905 868 8;
  • 57) 0.377 905 868 8 × 2 = 0 + 0.755 811 737 6;
  • 58) 0.755 811 737 6 × 2 = 1 + 0.511 623 475 2;
  • 59) 0.511 623 475 2 × 2 = 1 + 0.023 246 950 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:


0.009 826 508 3(10) =

0.0000 0010 1000 0011 1111 1101 0111 0011 1100 1000 0100 1001 1001 0001 011(2)

6. Positive number before normalization:

0.009 826 508 3(10) =

0.0000 0010 1000 0011 1111 1101 0111 0011 1100 1000 0100 1001 1001 0001 011(2)

7. Normalize the binary representation of the number.

Shift the decimal mark 7 positions to the right, so that only one non zero digit remains to the left of it:


0.009 826 508 3(10) =

0.0000 0010 1000 0011 1111 1101 0111 0011 1100 1000 0100 1001 1001 0001 011(2) =

0.0000 0010 1000 0011 1111 1101 0111 0011 1100 1000 0100 1001 1001 0001 011(2) × 20 =

1.0100 0001 1111 1110 1011 1001 1110 0100 0010 0100 1100 1000 1011(2) × 2-7


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): -7

Mantissa (not normalized):
1.0100 0001 1111 1110 1011 1001 1110 0100 0010 0100 1100 1000 1011


9. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

-7 + 2(11-1) - 1 =

(-7 + 1 023)(10) =

1 016(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;
  • 1 016 ÷ 2 = 508 + 0;
  • 508 ÷ 2 = 254 + 0;
  • 254 ÷ 2 = 127 + 0;
  • 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) =

1016(10) =

011 1111 1000(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 0001 1111 1110 1011 1001 1110 0100 0010 0100 1100 1000 1011 =

0100 0001 1111 1110 1011 1001 1110 0100 0010 0100 1100 1000 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 1000

Mantissa (52 bits) =
0100 0001 1111 1110 1011 1001 1110 0100 0010 0100 1100 1000 1011


Decimal number -0.009 826 508 3 converted to 64 bit double precision IEEE 754 binary floating point representation:

1 - 011 1111 1000 - 0100 0001 1111 1110 1011 1001 1110 0100 0010 0100 1100 1000 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