24.777 777 777 777 777 777 784 2 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 24.777 777 777 777 777 777 784 2(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
24.777 777 777 777 777 777 784 2(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: 0.777 777 777 777 777 777 784 2.

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) 0.777 777 777 777 777 777 784 2 × 2 = 1 + 0.555 555 555 555 555 555 568 4;
  • 2) 0.555 555 555 555 555 555 568 4 × 2 = 1 + 0.111 111 111 111 111 111 136 8;
  • 3) 0.111 111 111 111 111 111 136 8 × 2 = 0 + 0.222 222 222 222 222 222 273 6;
  • 4) 0.222 222 222 222 222 222 273 6 × 2 = 0 + 0.444 444 444 444 444 444 547 2;
  • 5) 0.444 444 444 444 444 444 547 2 × 2 = 0 + 0.888 888 888 888 888 889 094 4;
  • 6) 0.888 888 888 888 888 889 094 4 × 2 = 1 + 0.777 777 777 777 777 778 188 8;
  • 7) 0.777 777 777 777 777 778 188 8 × 2 = 1 + 0.555 555 555 555 555 556 377 6;
  • 8) 0.555 555 555 555 555 556 377 6 × 2 = 1 + 0.111 111 111 111 111 112 755 2;
  • 9) 0.111 111 111 111 111 112 755 2 × 2 = 0 + 0.222 222 222 222 222 225 510 4;
  • 10) 0.222 222 222 222 222 225 510 4 × 2 = 0 + 0.444 444 444 444 444 451 020 8;
  • 11) 0.444 444 444 444 444 451 020 8 × 2 = 0 + 0.888 888 888 888 888 902 041 6;
  • 12) 0.888 888 888 888 888 902 041 6 × 2 = 1 + 0.777 777 777 777 777 804 083 2;
  • 13) 0.777 777 777 777 777 804 083 2 × 2 = 1 + 0.555 555 555 555 555 608 166 4;
  • 14) 0.555 555 555 555 555 608 166 4 × 2 = 1 + 0.111 111 111 111 111 216 332 8;
  • 15) 0.111 111 111 111 111 216 332 8 × 2 = 0 + 0.222 222 222 222 222 432 665 6;
  • 16) 0.222 222 222 222 222 432 665 6 × 2 = 0 + 0.444 444 444 444 444 865 331 2;
  • 17) 0.444 444 444 444 444 865 331 2 × 2 = 0 + 0.888 888 888 888 889 730 662 4;
  • 18) 0.888 888 888 888 889 730 662 4 × 2 = 1 + 0.777 777 777 777 779 461 324 8;
  • 19) 0.777 777 777 777 779 461 324 8 × 2 = 1 + 0.555 555 555 555 558 922 649 6;
  • 20) 0.555 555 555 555 558 922 649 6 × 2 = 1 + 0.111 111 111 111 117 845 299 2;
  • 21) 0.111 111 111 111 117 845 299 2 × 2 = 0 + 0.222 222 222 222 235 690 598 4;
  • 22) 0.222 222 222 222 235 690 598 4 × 2 = 0 + 0.444 444 444 444 471 381 196 8;
  • 23) 0.444 444 444 444 471 381 196 8 × 2 = 0 + 0.888 888 888 888 942 762 393 6;
  • 24) 0.888 888 888 888 942 762 393 6 × 2 = 1 + 0.777 777 777 777 885 524 787 2;
  • 25) 0.777 777 777 777 885 524 787 2 × 2 = 1 + 0.555 555 555 555 771 049 574 4;
  • 26) 0.555 555 555 555 771 049 574 4 × 2 = 1 + 0.111 111 111 111 542 099 148 8;
  • 27) 0.111 111 111 111 542 099 148 8 × 2 = 0 + 0.222 222 222 223 084 198 297 6;
  • 28) 0.222 222 222 223 084 198 297 6 × 2 = 0 + 0.444 444 444 446 168 396 595 2;
  • 29) 0.444 444 444 446 168 396 595 2 × 2 = 0 + 0.888 888 888 892 336 793 190 4;
  • 30) 0.888 888 888 892 336 793 190 4 × 2 = 1 + 0.777 777 777 784 673 586 380 8;
  • 31) 0.777 777 777 784 673 586 380 8 × 2 = 1 + 0.555 555 555 569 347 172 761 6;
  • 32) 0.555 555 555 569 347 172 761 6 × 2 = 1 + 0.111 111 111 138 694 345 523 2;
  • 33) 0.111 111 111 138 694 345 523 2 × 2 = 0 + 0.222 222 222 277 388 691 046 4;
  • 34) 0.222 222 222 277 388 691 046 4 × 2 = 0 + 0.444 444 444 554 777 382 092 8;
  • 35) 0.444 444 444 554 777 382 092 8 × 2 = 0 + 0.888 888 889 109 554 764 185 6;
  • 36) 0.888 888 889 109 554 764 185 6 × 2 = 1 + 0.777 777 778 219 109 528 371 2;
  • 37) 0.777 777 778 219 109 528 371 2 × 2 = 1 + 0.555 555 556 438 219 056 742 4;
  • 38) 0.555 555 556 438 219 056 742 4 × 2 = 1 + 0.111 111 112 876 438 113 484 8;
  • 39) 0.111 111 112 876 438 113 484 8 × 2 = 0 + 0.222 222 225 752 876 226 969 6;
  • 40) 0.222 222 225 752 876 226 969 6 × 2 = 0 + 0.444 444 451 505 752 453 939 2;
  • 41) 0.444 444 451 505 752 453 939 2 × 2 = 0 + 0.888 888 903 011 504 907 878 4;
  • 42) 0.888 888 903 011 504 907 878 4 × 2 = 1 + 0.777 777 806 023 009 815 756 8;
  • 43) 0.777 777 806 023 009 815 756 8 × 2 = 1 + 0.555 555 612 046 019 631 513 6;
  • 44) 0.555 555 612 046 019 631 513 6 × 2 = 1 + 0.111 111 224 092 039 263 027 2;
  • 45) 0.111 111 224 092 039 263 027 2 × 2 = 0 + 0.222 222 448 184 078 526 054 4;
  • 46) 0.222 222 448 184 078 526 054 4 × 2 = 0 + 0.444 444 896 368 157 052 108 8;
  • 47) 0.444 444 896 368 157 052 108 8 × 2 = 0 + 0.888 889 792 736 314 104 217 6;
  • 48) 0.888 889 792 736 314 104 217 6 × 2 = 1 + 0.777 779 585 472 628 208 435 2;
  • 49) 0.777 779 585 472 628 208 435 2 × 2 = 1 + 0.555 559 170 945 256 416 870 4;
  • 50) 0.555 559 170 945 256 416 870 4 × 2 = 1 + 0.111 118 341 890 512 833 740 8;
  • 51) 0.111 118 341 890 512 833 740 8 × 2 = 0 + 0.222 236 683 781 025 667 481 6;
  • 52) 0.222 236 683 781 025 667 481 6 × 2 = 0 + 0.444 473 367 562 051 334 963 2;
  • 53) 0.444 473 367 562 051 334 963 2 × 2 = 0 + 0.888 946 735 124 102 669 926 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).


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:


0.777 777 777 777 777 777 784 2(10) =

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

5. Positive number before normalization:

24.777 777 777 777 777 777 784 2(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:


24.777 777 777 777 777 777 784 2(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 24.777 777 777 777 777 777 784 2 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