24.777 777 777 777 777 777 780 4 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 24.777 777 777 777 777 777 780 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
24.777 777 777 777 777 777 780 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. First, convert to binary (in base 2) the integer part: 24.
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;
  • 24 ÷ 2 = 12 + 0;
  • 12 ÷ 2 = 6 + 0;
  • 6 ÷ 2 = 3 + 0;
  • 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.

24(10) =

1 1000(2)


3. Convert to binary (base 2) the fractional part: 0.777 777 777 777 777 777 780 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) 0.777 777 777 777 777 777 780 4 × 2 = 1 + 0.555 555 555 555 555 555 560 8;
  • 2) 0.555 555 555 555 555 555 560 8 × 2 = 1 + 0.111 111 111 111 111 111 121 6;
  • 3) 0.111 111 111 111 111 111 121 6 × 2 = 0 + 0.222 222 222 222 222 222 243 2;
  • 4) 0.222 222 222 222 222 222 243 2 × 2 = 0 + 0.444 444 444 444 444 444 486 4;
  • 5) 0.444 444 444 444 444 444 486 4 × 2 = 0 + 0.888 888 888 888 888 888 972 8;
  • 6) 0.888 888 888 888 888 888 972 8 × 2 = 1 + 0.777 777 777 777 777 777 945 6;
  • 7) 0.777 777 777 777 777 777 945 6 × 2 = 1 + 0.555 555 555 555 555 555 891 2;
  • 8) 0.555 555 555 555 555 555 891 2 × 2 = 1 + 0.111 111 111 111 111 111 782 4;
  • 9) 0.111 111 111 111 111 111 782 4 × 2 = 0 + 0.222 222 222 222 222 223 564 8;
  • 10) 0.222 222 222 222 222 223 564 8 × 2 = 0 + 0.444 444 444 444 444 447 129 6;
  • 11) 0.444 444 444 444 444 447 129 6 × 2 = 0 + 0.888 888 888 888 888 894 259 2;
  • 12) 0.888 888 888 888 888 894 259 2 × 2 = 1 + 0.777 777 777 777 777 788 518 4;
  • 13) 0.777 777 777 777 777 788 518 4 × 2 = 1 + 0.555 555 555 555 555 577 036 8;
  • 14) 0.555 555 555 555 555 577 036 8 × 2 = 1 + 0.111 111 111 111 111 154 073 6;
  • 15) 0.111 111 111 111 111 154 073 6 × 2 = 0 + 0.222 222 222 222 222 308 147 2;
  • 16) 0.222 222 222 222 222 308 147 2 × 2 = 0 + 0.444 444 444 444 444 616 294 4;
  • 17) 0.444 444 444 444 444 616 294 4 × 2 = 0 + 0.888 888 888 888 889 232 588 8;
  • 18) 0.888 888 888 888 889 232 588 8 × 2 = 1 + 0.777 777 777 777 778 465 177 6;
  • 19) 0.777 777 777 777 778 465 177 6 × 2 = 1 + 0.555 555 555 555 556 930 355 2;
  • 20) 0.555 555 555 555 556 930 355 2 × 2 = 1 + 0.111 111 111 111 113 860 710 4;
  • 21) 0.111 111 111 111 113 860 710 4 × 2 = 0 + 0.222 222 222 222 227 721 420 8;
  • 22) 0.222 222 222 222 227 721 420 8 × 2 = 0 + 0.444 444 444 444 455 442 841 6;
  • 23) 0.444 444 444 444 455 442 841 6 × 2 = 0 + 0.888 888 888 888 910 885 683 2;
  • 24) 0.888 888 888 888 910 885 683 2 × 2 = 1 + 0.777 777 777 777 821 771 366 4;
  • 25) 0.777 777 777 777 821 771 366 4 × 2 = 1 + 0.555 555 555 555 643 542 732 8;
  • 26) 0.555 555 555 555 643 542 732 8 × 2 = 1 + 0.111 111 111 111 287 085 465 6;
  • 27) 0.111 111 111 111 287 085 465 6 × 2 = 0 + 0.222 222 222 222 574 170 931 2;
  • 28) 0.222 222 222 222 574 170 931 2 × 2 = 0 + 0.444 444 444 445 148 341 862 4;
  • 29) 0.444 444 444 445 148 341 862 4 × 2 = 0 + 0.888 888 888 890 296 683 724 8;
  • 30) 0.888 888 888 890 296 683 724 8 × 2 = 1 + 0.777 777 777 780 593 367 449 6;
  • 31) 0.777 777 777 780 593 367 449 6 × 2 = 1 + 0.555 555 555 561 186 734 899 2;
  • 32) 0.555 555 555 561 186 734 899 2 × 2 = 1 + 0.111 111 111 122 373 469 798 4;
  • 33) 0.111 111 111 122 373 469 798 4 × 2 = 0 + 0.222 222 222 244 746 939 596 8;
  • 34) 0.222 222 222 244 746 939 596 8 × 2 = 0 + 0.444 444 444 489 493 879 193 6;
  • 35) 0.444 444 444 489 493 879 193 6 × 2 = 0 + 0.888 888 888 978 987 758 387 2;
  • 36) 0.888 888 888 978 987 758 387 2 × 2 = 1 + 0.777 777 777 957 975 516 774 4;
  • 37) 0.777 777 777 957 975 516 774 4 × 2 = 1 + 0.555 555 555 915 951 033 548 8;
  • 38) 0.555 555 555 915 951 033 548 8 × 2 = 1 + 0.111 111 111 831 902 067 097 6;
  • 39) 0.111 111 111 831 902 067 097 6 × 2 = 0 + 0.222 222 223 663 804 134 195 2;
  • 40) 0.222 222 223 663 804 134 195 2 × 2 = 0 + 0.444 444 447 327 608 268 390 4;
  • 41) 0.444 444 447 327 608 268 390 4 × 2 = 0 + 0.888 888 894 655 216 536 780 8;
  • 42) 0.888 888 894 655 216 536 780 8 × 2 = 1 + 0.777 777 789 310 433 073 561 6;
  • 43) 0.777 777 789 310 433 073 561 6 × 2 = 1 + 0.555 555 578 620 866 147 123 2;
  • 44) 0.555 555 578 620 866 147 123 2 × 2 = 1 + 0.111 111 157 241 732 294 246 4;
  • 45) 0.111 111 157 241 732 294 246 4 × 2 = 0 + 0.222 222 314 483 464 588 492 8;
  • 46) 0.222 222 314 483 464 588 492 8 × 2 = 0 + 0.444 444 628 966 929 176 985 6;
  • 47) 0.444 444 628 966 929 176 985 6 × 2 = 0 + 0.888 889 257 933 858 353 971 2;
  • 48) 0.888 889 257 933 858 353 971 2 × 2 = 1 + 0.777 778 515 867 716 707 942 4;
  • 49) 0.777 778 515 867 716 707 942 4 × 2 = 1 + 0.555 557 031 735 433 415 884 8;
  • 50) 0.555 557 031 735 433 415 884 8 × 2 = 1 + 0.111 114 063 470 866 831 769 6;
  • 51) 0.111 114 063 470 866 831 769 6 × 2 = 0 + 0.222 228 126 941 733 663 539 2;
  • 52) 0.222 228 126 941 733 663 539 2 × 2 = 0 + 0.444 456 253 883 467 327 078 4;
  • 53) 0.444 456 253 883 467 327 078 4 × 2 = 0 + 0.888 912 507 766 934 654 156 8;

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:


0.777 777 777 777 777 777 780 4(10) =

0.1100 0111 0001 1100 0111 0001 1100 0111 0001 1100 0111 0001 1100 0(2)

5. Positive number before normalization:

24.777 777 777 777 777 777 780 4(10) =

1 1000.1100 0111 0001 1100 0111 0001 1100 0111 0001 1100 0111 0001 1100 0(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 4 positions to the left, so that only one non zero digit remains to the left of it:


24.777 777 777 777 777 777 780 4(10) =

1 1000.1100 0111 0001 1100 0111 0001 1100 0111 0001 1100 0111 0001 1100 0(2) =

1 1000.1100 0111 0001 1100 0111 0001 1100 0111 0001 1100 0111 0001 1100 0(2) × 20 =

1.1000 1100 0111 0001 1100 0111 0001 1100 0111 0001 1100 0111 0001 1100 0(2) × 24


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): 4

Mantissa (not normalized):
1.1000 1100 0111 0001 1100 0111 0001 1100 0111 0001 1100 0111 0001 1100 0


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

4 + 2(11-1) - 1 =

(4 + 1 023)(10) =

1 027(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;
  • 1 027 ÷ 2 = 513 + 1;
  • 513 ÷ 2 = 256 + 1;
  • 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) =

1027(10) =

100 0000 0011(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. 1000 1100 0111 0001 1100 0111 0001 1100 0111 0001 1100 0111 0001 1 1000 =

1000 1100 0111 0001 1100 0111 0001 1100 0111 0001 1100 0111 0001


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 0011

Mantissa (52 bits) =
1000 1100 0111 0001 1100 0111 0001 1100 0111 0001 1100 0111 0001


Decimal number 24.777 777 777 777 777 777 780 4 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 100 0000 0011 - 1000 1100 0111 0001 1100 0111 0001 1100 0111 0001 1100 0111 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