24.777 777 777 777 777 777 777 937 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 24.777 777 777 777 777 777 777 937(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
24.777 777 777 777 777 777 777 937(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: 0.777 777 777 777 777 777 777 937.

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) 0.777 777 777 777 777 777 777 937 × 2 = 1 + 0.555 555 555 555 555 555 555 874;
  • 2) 0.555 555 555 555 555 555 555 874 × 2 = 1 + 0.111 111 111 111 111 111 111 748;
  • 3) 0.111 111 111 111 111 111 111 748 × 2 = 0 + 0.222 222 222 222 222 222 223 496;
  • 4) 0.222 222 222 222 222 222 223 496 × 2 = 0 + 0.444 444 444 444 444 444 446 992;
  • 5) 0.444 444 444 444 444 444 446 992 × 2 = 0 + 0.888 888 888 888 888 888 893 984;
  • 6) 0.888 888 888 888 888 888 893 984 × 2 = 1 + 0.777 777 777 777 777 777 787 968;
  • 7) 0.777 777 777 777 777 777 787 968 × 2 = 1 + 0.555 555 555 555 555 555 575 936;
  • 8) 0.555 555 555 555 555 555 575 936 × 2 = 1 + 0.111 111 111 111 111 111 151 872;
  • 9) 0.111 111 111 111 111 111 151 872 × 2 = 0 + 0.222 222 222 222 222 222 303 744;
  • 10) 0.222 222 222 222 222 222 303 744 × 2 = 0 + 0.444 444 444 444 444 444 607 488;
  • 11) 0.444 444 444 444 444 444 607 488 × 2 = 0 + 0.888 888 888 888 888 889 214 976;
  • 12) 0.888 888 888 888 888 889 214 976 × 2 = 1 + 0.777 777 777 777 777 778 429 952;
  • 13) 0.777 777 777 777 777 778 429 952 × 2 = 1 + 0.555 555 555 555 555 556 859 904;
  • 14) 0.555 555 555 555 555 556 859 904 × 2 = 1 + 0.111 111 111 111 111 113 719 808;
  • 15) 0.111 111 111 111 111 113 719 808 × 2 = 0 + 0.222 222 222 222 222 227 439 616;
  • 16) 0.222 222 222 222 222 227 439 616 × 2 = 0 + 0.444 444 444 444 444 454 879 232;
  • 17) 0.444 444 444 444 444 454 879 232 × 2 = 0 + 0.888 888 888 888 888 909 758 464;
  • 18) 0.888 888 888 888 888 909 758 464 × 2 = 1 + 0.777 777 777 777 777 819 516 928;
  • 19) 0.777 777 777 777 777 819 516 928 × 2 = 1 + 0.555 555 555 555 555 639 033 856;
  • 20) 0.555 555 555 555 555 639 033 856 × 2 = 1 + 0.111 111 111 111 111 278 067 712;
  • 21) 0.111 111 111 111 111 278 067 712 × 2 = 0 + 0.222 222 222 222 222 556 135 424;
  • 22) 0.222 222 222 222 222 556 135 424 × 2 = 0 + 0.444 444 444 444 445 112 270 848;
  • 23) 0.444 444 444 444 445 112 270 848 × 2 = 0 + 0.888 888 888 888 890 224 541 696;
  • 24) 0.888 888 888 888 890 224 541 696 × 2 = 1 + 0.777 777 777 777 780 449 083 392;
  • 25) 0.777 777 777 777 780 449 083 392 × 2 = 1 + 0.555 555 555 555 560 898 166 784;
  • 26) 0.555 555 555 555 560 898 166 784 × 2 = 1 + 0.111 111 111 111 121 796 333 568;
  • 27) 0.111 111 111 111 121 796 333 568 × 2 = 0 + 0.222 222 222 222 243 592 667 136;
  • 28) 0.222 222 222 222 243 592 667 136 × 2 = 0 + 0.444 444 444 444 487 185 334 272;
  • 29) 0.444 444 444 444 487 185 334 272 × 2 = 0 + 0.888 888 888 888 974 370 668 544;
  • 30) 0.888 888 888 888 974 370 668 544 × 2 = 1 + 0.777 777 777 777 948 741 337 088;
  • 31) 0.777 777 777 777 948 741 337 088 × 2 = 1 + 0.555 555 555 555 897 482 674 176;
  • 32) 0.555 555 555 555 897 482 674 176 × 2 = 1 + 0.111 111 111 111 794 965 348 352;
  • 33) 0.111 111 111 111 794 965 348 352 × 2 = 0 + 0.222 222 222 223 589 930 696 704;
  • 34) 0.222 222 222 223 589 930 696 704 × 2 = 0 + 0.444 444 444 447 179 861 393 408;
  • 35) 0.444 444 444 447 179 861 393 408 × 2 = 0 + 0.888 888 888 894 359 722 786 816;
  • 36) 0.888 888 888 894 359 722 786 816 × 2 = 1 + 0.777 777 777 788 719 445 573 632;
  • 37) 0.777 777 777 788 719 445 573 632 × 2 = 1 + 0.555 555 555 577 438 891 147 264;
  • 38) 0.555 555 555 577 438 891 147 264 × 2 = 1 + 0.111 111 111 154 877 782 294 528;
  • 39) 0.111 111 111 154 877 782 294 528 × 2 = 0 + 0.222 222 222 309 755 564 589 056;
  • 40) 0.222 222 222 309 755 564 589 056 × 2 = 0 + 0.444 444 444 619 511 129 178 112;
  • 41) 0.444 444 444 619 511 129 178 112 × 2 = 0 + 0.888 888 889 239 022 258 356 224;
  • 42) 0.888 888 889 239 022 258 356 224 × 2 = 1 + 0.777 777 778 478 044 516 712 448;
  • 43) 0.777 777 778 478 044 516 712 448 × 2 = 1 + 0.555 555 556 956 089 033 424 896;
  • 44) 0.555 555 556 956 089 033 424 896 × 2 = 1 + 0.111 111 113 912 178 066 849 792;
  • 45) 0.111 111 113 912 178 066 849 792 × 2 = 0 + 0.222 222 227 824 356 133 699 584;
  • 46) 0.222 222 227 824 356 133 699 584 × 2 = 0 + 0.444 444 455 648 712 267 399 168;
  • 47) 0.444 444 455 648 712 267 399 168 × 2 = 0 + 0.888 888 911 297 424 534 798 336;
  • 48) 0.888 888 911 297 424 534 798 336 × 2 = 1 + 0.777 777 822 594 849 069 596 672;
  • 49) 0.777 777 822 594 849 069 596 672 × 2 = 1 + 0.555 555 645 189 698 139 193 344;
  • 50) 0.555 555 645 189 698 139 193 344 × 2 = 1 + 0.111 111 290 379 396 278 386 688;
  • 51) 0.111 111 290 379 396 278 386 688 × 2 = 0 + 0.222 222 580 758 792 556 773 376;
  • 52) 0.222 222 580 758 792 556 773 376 × 2 = 0 + 0.444 445 161 517 585 113 546 752;
  • 53) 0.444 445 161 517 585 113 546 752 × 2 = 0 + 0.888 890 323 035 170 227 093 504;

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:


0.777 777 777 777 777 777 777 937(10) =

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

5. Positive number before normalization:

24.777 777 777 777 777 777 777 937(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:


24.777 777 777 777 777 777 777 937(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 24.777 777 777 777 777 777 777 937 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