24.777 777 777 777 777 777 780 9 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 24.777 777 777 777 777 777 780 9(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 9(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 9.

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 9 × 2 = 1 + 0.555 555 555 555 555 555 561 8;
  • 2) 0.555 555 555 555 555 555 561 8 × 2 = 1 + 0.111 111 111 111 111 111 123 6;
  • 3) 0.111 111 111 111 111 111 123 6 × 2 = 0 + 0.222 222 222 222 222 222 247 2;
  • 4) 0.222 222 222 222 222 222 247 2 × 2 = 0 + 0.444 444 444 444 444 444 494 4;
  • 5) 0.444 444 444 444 444 444 494 4 × 2 = 0 + 0.888 888 888 888 888 888 988 8;
  • 6) 0.888 888 888 888 888 888 988 8 × 2 = 1 + 0.777 777 777 777 777 777 977 6;
  • 7) 0.777 777 777 777 777 777 977 6 × 2 = 1 + 0.555 555 555 555 555 555 955 2;
  • 8) 0.555 555 555 555 555 555 955 2 × 2 = 1 + 0.111 111 111 111 111 111 910 4;
  • 9) 0.111 111 111 111 111 111 910 4 × 2 = 0 + 0.222 222 222 222 222 223 820 8;
  • 10) 0.222 222 222 222 222 223 820 8 × 2 = 0 + 0.444 444 444 444 444 447 641 6;
  • 11) 0.444 444 444 444 444 447 641 6 × 2 = 0 + 0.888 888 888 888 888 895 283 2;
  • 12) 0.888 888 888 888 888 895 283 2 × 2 = 1 + 0.777 777 777 777 777 790 566 4;
  • 13) 0.777 777 777 777 777 790 566 4 × 2 = 1 + 0.555 555 555 555 555 581 132 8;
  • 14) 0.555 555 555 555 555 581 132 8 × 2 = 1 + 0.111 111 111 111 111 162 265 6;
  • 15) 0.111 111 111 111 111 162 265 6 × 2 = 0 + 0.222 222 222 222 222 324 531 2;
  • 16) 0.222 222 222 222 222 324 531 2 × 2 = 0 + 0.444 444 444 444 444 649 062 4;
  • 17) 0.444 444 444 444 444 649 062 4 × 2 = 0 + 0.888 888 888 888 889 298 124 8;
  • 18) 0.888 888 888 888 889 298 124 8 × 2 = 1 + 0.777 777 777 777 778 596 249 6;
  • 19) 0.777 777 777 777 778 596 249 6 × 2 = 1 + 0.555 555 555 555 557 192 499 2;
  • 20) 0.555 555 555 555 557 192 499 2 × 2 = 1 + 0.111 111 111 111 114 384 998 4;
  • 21) 0.111 111 111 111 114 384 998 4 × 2 = 0 + 0.222 222 222 222 228 769 996 8;
  • 22) 0.222 222 222 222 228 769 996 8 × 2 = 0 + 0.444 444 444 444 457 539 993 6;
  • 23) 0.444 444 444 444 457 539 993 6 × 2 = 0 + 0.888 888 888 888 915 079 987 2;
  • 24) 0.888 888 888 888 915 079 987 2 × 2 = 1 + 0.777 777 777 777 830 159 974 4;
  • 25) 0.777 777 777 777 830 159 974 4 × 2 = 1 + 0.555 555 555 555 660 319 948 8;
  • 26) 0.555 555 555 555 660 319 948 8 × 2 = 1 + 0.111 111 111 111 320 639 897 6;
  • 27) 0.111 111 111 111 320 639 897 6 × 2 = 0 + 0.222 222 222 222 641 279 795 2;
  • 28) 0.222 222 222 222 641 279 795 2 × 2 = 0 + 0.444 444 444 445 282 559 590 4;
  • 29) 0.444 444 444 445 282 559 590 4 × 2 = 0 + 0.888 888 888 890 565 119 180 8;
  • 30) 0.888 888 888 890 565 119 180 8 × 2 = 1 + 0.777 777 777 781 130 238 361 6;
  • 31) 0.777 777 777 781 130 238 361 6 × 2 = 1 + 0.555 555 555 562 260 476 723 2;
  • 32) 0.555 555 555 562 260 476 723 2 × 2 = 1 + 0.111 111 111 124 520 953 446 4;
  • 33) 0.111 111 111 124 520 953 446 4 × 2 = 0 + 0.222 222 222 249 041 906 892 8;
  • 34) 0.222 222 222 249 041 906 892 8 × 2 = 0 + 0.444 444 444 498 083 813 785 6;
  • 35) 0.444 444 444 498 083 813 785 6 × 2 = 0 + 0.888 888 888 996 167 627 571 2;
  • 36) 0.888 888 888 996 167 627 571 2 × 2 = 1 + 0.777 777 777 992 335 255 142 4;
  • 37) 0.777 777 777 992 335 255 142 4 × 2 = 1 + 0.555 555 555 984 670 510 284 8;
  • 38) 0.555 555 555 984 670 510 284 8 × 2 = 1 + 0.111 111 111 969 341 020 569 6;
  • 39) 0.111 111 111 969 341 020 569 6 × 2 = 0 + 0.222 222 223 938 682 041 139 2;
  • 40) 0.222 222 223 938 682 041 139 2 × 2 = 0 + 0.444 444 447 877 364 082 278 4;
  • 41) 0.444 444 447 877 364 082 278 4 × 2 = 0 + 0.888 888 895 754 728 164 556 8;
  • 42) 0.888 888 895 754 728 164 556 8 × 2 = 1 + 0.777 777 791 509 456 329 113 6;
  • 43) 0.777 777 791 509 456 329 113 6 × 2 = 1 + 0.555 555 583 018 912 658 227 2;
  • 44) 0.555 555 583 018 912 658 227 2 × 2 = 1 + 0.111 111 166 037 825 316 454 4;
  • 45) 0.111 111 166 037 825 316 454 4 × 2 = 0 + 0.222 222 332 075 650 632 908 8;
  • 46) 0.222 222 332 075 650 632 908 8 × 2 = 0 + 0.444 444 664 151 301 265 817 6;
  • 47) 0.444 444 664 151 301 265 817 6 × 2 = 0 + 0.888 889 328 302 602 531 635 2;
  • 48) 0.888 889 328 302 602 531 635 2 × 2 = 1 + 0.777 778 656 605 205 063 270 4;
  • 49) 0.777 778 656 605 205 063 270 4 × 2 = 1 + 0.555 557 313 210 410 126 540 8;
  • 50) 0.555 557 313 210 410 126 540 8 × 2 = 1 + 0.111 114 626 420 820 253 081 6;
  • 51) 0.111 114 626 420 820 253 081 6 × 2 = 0 + 0.222 229 252 841 640 506 163 2;
  • 52) 0.222 229 252 841 640 506 163 2 × 2 = 0 + 0.444 458 505 683 281 012 326 4;
  • 53) 0.444 458 505 683 281 012 326 4 × 2 = 0 + 0.888 917 011 366 562 024 652 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 9(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 9(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 9(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 9 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