3.141 592 571 3 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 3.141 592 571 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
3.141 592 571 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. First, convert to binary (in base 2) the integer part: 3.
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;
  • 3 ÷ 2 = 1 + 1;
  • 1 ÷ 2 = 0 + 1;

2. 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.

3(10) =

11(2)


3. Convert to binary (base 2) the fractional part: 0.141 592 571 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) 0.141 592 571 3 × 2 = 0 + 0.283 185 142 6;
  • 2) 0.283 185 142 6 × 2 = 0 + 0.566 370 285 2;
  • 3) 0.566 370 285 2 × 2 = 1 + 0.132 740 570 4;
  • 4) 0.132 740 570 4 × 2 = 0 + 0.265 481 140 8;
  • 5) 0.265 481 140 8 × 2 = 0 + 0.530 962 281 6;
  • 6) 0.530 962 281 6 × 2 = 1 + 0.061 924 563 2;
  • 7) 0.061 924 563 2 × 2 = 0 + 0.123 849 126 4;
  • 8) 0.123 849 126 4 × 2 = 0 + 0.247 698 252 8;
  • 9) 0.247 698 252 8 × 2 = 0 + 0.495 396 505 6;
  • 10) 0.495 396 505 6 × 2 = 0 + 0.990 793 011 2;
  • 11) 0.990 793 011 2 × 2 = 1 + 0.981 586 022 4;
  • 12) 0.981 586 022 4 × 2 = 1 + 0.963 172 044 8;
  • 13) 0.963 172 044 8 × 2 = 1 + 0.926 344 089 6;
  • 14) 0.926 344 089 6 × 2 = 1 + 0.852 688 179 2;
  • 15) 0.852 688 179 2 × 2 = 1 + 0.705 376 358 4;
  • 16) 0.705 376 358 4 × 2 = 1 + 0.410 752 716 8;
  • 17) 0.410 752 716 8 × 2 = 0 + 0.821 505 433 6;
  • 18) 0.821 505 433 6 × 2 = 1 + 0.643 010 867 2;
  • 19) 0.643 010 867 2 × 2 = 1 + 0.286 021 734 4;
  • 20) 0.286 021 734 4 × 2 = 0 + 0.572 043 468 8;
  • 21) 0.572 043 468 8 × 2 = 1 + 0.144 086 937 6;
  • 22) 0.144 086 937 6 × 2 = 0 + 0.288 173 875 2;
  • 23) 0.288 173 875 2 × 2 = 0 + 0.576 347 750 4;
  • 24) 0.576 347 750 4 × 2 = 1 + 0.152 695 500 8;
  • 25) 0.152 695 500 8 × 2 = 0 + 0.305 391 001 6;
  • 26) 0.305 391 001 6 × 2 = 0 + 0.610 782 003 2;
  • 27) 0.610 782 003 2 × 2 = 1 + 0.221 564 006 4;
  • 28) 0.221 564 006 4 × 2 = 0 + 0.443 128 012 8;
  • 29) 0.443 128 012 8 × 2 = 0 + 0.886 256 025 6;
  • 30) 0.886 256 025 6 × 2 = 1 + 0.772 512 051 2;
  • 31) 0.772 512 051 2 × 2 = 1 + 0.545 024 102 4;
  • 32) 0.545 024 102 4 × 2 = 1 + 0.090 048 204 8;
  • 33) 0.090 048 204 8 × 2 = 0 + 0.180 096 409 6;
  • 34) 0.180 096 409 6 × 2 = 0 + 0.360 192 819 2;
  • 35) 0.360 192 819 2 × 2 = 0 + 0.720 385 638 4;
  • 36) 0.720 385 638 4 × 2 = 1 + 0.440 771 276 8;
  • 37) 0.440 771 276 8 × 2 = 0 + 0.881 542 553 6;
  • 38) 0.881 542 553 6 × 2 = 1 + 0.763 085 107 2;
  • 39) 0.763 085 107 2 × 2 = 1 + 0.526 170 214 4;
  • 40) 0.526 170 214 4 × 2 = 1 + 0.052 340 428 8;
  • 41) 0.052 340 428 8 × 2 = 0 + 0.104 680 857 6;
  • 42) 0.104 680 857 6 × 2 = 0 + 0.209 361 715 2;
  • 43) 0.209 361 715 2 × 2 = 0 + 0.418 723 430 4;
  • 44) 0.418 723 430 4 × 2 = 0 + 0.837 446 860 8;
  • 45) 0.837 446 860 8 × 2 = 1 + 0.674 893 721 6;
  • 46) 0.674 893 721 6 × 2 = 1 + 0.349 787 443 2;
  • 47) 0.349 787 443 2 × 2 = 0 + 0.699 574 886 4;
  • 48) 0.699 574 886 4 × 2 = 1 + 0.399 149 772 8;
  • 49) 0.399 149 772 8 × 2 = 0 + 0.798 299 545 6;
  • 50) 0.798 299 545 6 × 2 = 1 + 0.596 599 091 2;
  • 51) 0.596 599 091 2 × 2 = 1 + 0.193 198 182 4;
  • 52) 0.193 198 182 4 × 2 = 0 + 0.386 396 364 8;
  • 53) 0.386 396 364 8 × 2 = 0 + 0.772 792 729 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).


4. 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:


0.141 592 571 3(10) =

0.0010 0100 0011 1111 0110 1001 0010 0111 0001 0111 0000 1101 0110 0(2)

5. Positive number before normalization:

3.141 592 571 3(10) =

11.0010 0100 0011 1111 0110 1001 0010 0111 0001 0111 0000 1101 0110 0(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 1 positions to the left, so that only one non zero digit remains to the left of it:


3.141 592 571 3(10) =

11.0010 0100 0011 1111 0110 1001 0010 0111 0001 0111 0000 1101 0110 0(2) =

11.0010 0100 0011 1111 0110 1001 0010 0111 0001 0111 0000 1101 0110 0(2) × 20 =

1.1001 0010 0001 1111 1011 0100 1001 0011 1000 1011 1000 0110 1011 00(2) × 21


7. 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 0 (a positive number)

Exponent (unadjusted): 1

Mantissa (not normalized):
1.1001 0010 0001 1111 1011 0100 1001 0011 1000 1011 1000 0110 1011 00


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

1 + 2(11-1) - 1 =

(1 + 1 023)(10) =

1 024(10)


9. 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;
  • 1 024 ÷ 2 = 512 + 0;
  • 512 ÷ 2 = 256 + 0;
  • 256 ÷ 2 = 128 + 0;
  • 128 ÷ 2 = 64 + 0;
  • 64 ÷ 2 = 32 + 0;
  • 32 ÷ 2 = 16 + 0;
  • 16 ÷ 2 = 8 + 0;
  • 8 ÷ 2 = 4 + 0;
  • 4 ÷ 2 = 2 + 0;
  • 2 ÷ 2 = 1 + 0;
  • 1 ÷ 2 = 0 + 1;

10. Construct the base 2 representation of the adjusted exponent.

Take all the remainders starting from the bottom of the list constructed above.


Exponent (adjusted) =

1024(10) =

100 0000 0000(2)


11. 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, by removing the excess bits, from the right (if any of the excess bits is set on 1, we are losing precision...).


Mantissa (normalized) =

1. 1001 0010 0001 1111 1011 0100 1001 0011 1000 1011 1000 0110 1011 00 =

1001 0010 0001 1111 1011 0100 1001 0011 1000 1011 1000 0110 1011


12. The three elements that make up the number's 64 bit double precision IEEE 754 binary floating point representation:

Sign (1 bit) =
0 (a positive number)

Exponent (11 bits) =
100 0000 0000

Mantissa (52 bits) =
1001 0010 0001 1111 1011 0100 1001 0011 1000 1011 1000 0110 1011


Decimal number 3.141 592 571 3 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 100 0000 0000 - 1001 0010 0001 1111 1011 0100 1001 0011 1000 1011 1000 0110 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