24.777 777 777 777 904 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 24.777 777 777 777 904(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
24.777 777 777 777 904(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: 0.777 777 777 777 904.

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) 0.777 777 777 777 904 × 2 = 1 + 0.555 555 555 555 808;
  • 2) 0.555 555 555 555 808 × 2 = 1 + 0.111 111 111 111 616;
  • 3) 0.111 111 111 111 616 × 2 = 0 + 0.222 222 222 223 232;
  • 4) 0.222 222 222 223 232 × 2 = 0 + 0.444 444 444 446 464;
  • 5) 0.444 444 444 446 464 × 2 = 0 + 0.888 888 888 892 928;
  • 6) 0.888 888 888 892 928 × 2 = 1 + 0.777 777 777 785 856;
  • 7) 0.777 777 777 785 856 × 2 = 1 + 0.555 555 555 571 712;
  • 8) 0.555 555 555 571 712 × 2 = 1 + 0.111 111 111 143 424;
  • 9) 0.111 111 111 143 424 × 2 = 0 + 0.222 222 222 286 848;
  • 10) 0.222 222 222 286 848 × 2 = 0 + 0.444 444 444 573 696;
  • 11) 0.444 444 444 573 696 × 2 = 0 + 0.888 888 889 147 392;
  • 12) 0.888 888 889 147 392 × 2 = 1 + 0.777 777 778 294 784;
  • 13) 0.777 777 778 294 784 × 2 = 1 + 0.555 555 556 589 568;
  • 14) 0.555 555 556 589 568 × 2 = 1 + 0.111 111 113 179 136;
  • 15) 0.111 111 113 179 136 × 2 = 0 + 0.222 222 226 358 272;
  • 16) 0.222 222 226 358 272 × 2 = 0 + 0.444 444 452 716 544;
  • 17) 0.444 444 452 716 544 × 2 = 0 + 0.888 888 905 433 088;
  • 18) 0.888 888 905 433 088 × 2 = 1 + 0.777 777 810 866 176;
  • 19) 0.777 777 810 866 176 × 2 = 1 + 0.555 555 621 732 352;
  • 20) 0.555 555 621 732 352 × 2 = 1 + 0.111 111 243 464 704;
  • 21) 0.111 111 243 464 704 × 2 = 0 + 0.222 222 486 929 408;
  • 22) 0.222 222 486 929 408 × 2 = 0 + 0.444 444 973 858 816;
  • 23) 0.444 444 973 858 816 × 2 = 0 + 0.888 889 947 717 632;
  • 24) 0.888 889 947 717 632 × 2 = 1 + 0.777 779 895 435 264;
  • 25) 0.777 779 895 435 264 × 2 = 1 + 0.555 559 790 870 528;
  • 26) 0.555 559 790 870 528 × 2 = 1 + 0.111 119 581 741 056;
  • 27) 0.111 119 581 741 056 × 2 = 0 + 0.222 239 163 482 112;
  • 28) 0.222 239 163 482 112 × 2 = 0 + 0.444 478 326 964 224;
  • 29) 0.444 478 326 964 224 × 2 = 0 + 0.888 956 653 928 448;
  • 30) 0.888 956 653 928 448 × 2 = 1 + 0.777 913 307 856 896;
  • 31) 0.777 913 307 856 896 × 2 = 1 + 0.555 826 615 713 792;
  • 32) 0.555 826 615 713 792 × 2 = 1 + 0.111 653 231 427 584;
  • 33) 0.111 653 231 427 584 × 2 = 0 + 0.223 306 462 855 168;
  • 34) 0.223 306 462 855 168 × 2 = 0 + 0.446 612 925 710 336;
  • 35) 0.446 612 925 710 336 × 2 = 0 + 0.893 225 851 420 672;
  • 36) 0.893 225 851 420 672 × 2 = 1 + 0.786 451 702 841 344;
  • 37) 0.786 451 702 841 344 × 2 = 1 + 0.572 903 405 682 688;
  • 38) 0.572 903 405 682 688 × 2 = 1 + 0.145 806 811 365 376;
  • 39) 0.145 806 811 365 376 × 2 = 0 + 0.291 613 622 730 752;
  • 40) 0.291 613 622 730 752 × 2 = 0 + 0.583 227 245 461 504;
  • 41) 0.583 227 245 461 504 × 2 = 1 + 0.166 454 490 923 008;
  • 42) 0.166 454 490 923 008 × 2 = 0 + 0.332 908 981 846 016;
  • 43) 0.332 908 981 846 016 × 2 = 0 + 0.665 817 963 692 032;
  • 44) 0.665 817 963 692 032 × 2 = 1 + 0.331 635 927 384 064;
  • 45) 0.331 635 927 384 064 × 2 = 0 + 0.663 271 854 768 128;
  • 46) 0.663 271 854 768 128 × 2 = 1 + 0.326 543 709 536 256;
  • 47) 0.326 543 709 536 256 × 2 = 0 + 0.653 087 419 072 512;
  • 48) 0.653 087 419 072 512 × 2 = 1 + 0.306 174 838 145 024;
  • 49) 0.306 174 838 145 024 × 2 = 0 + 0.612 349 676 290 048;
  • 50) 0.612 349 676 290 048 × 2 = 1 + 0.224 699 352 580 096;
  • 51) 0.224 699 352 580 096 × 2 = 0 + 0.449 398 705 160 192;
  • 52) 0.449 398 705 160 192 × 2 = 0 + 0.898 797 410 320 384;
  • 53) 0.898 797 410 320 384 × 2 = 1 + 0.797 594 820 640 768;

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:


0.777 777 777 777 904(10) =

0.1100 0111 0001 1100 0111 0001 1100 0111 0001 1100 1001 0101 0100 1(2)

5. Positive number before normalization:

24.777 777 777 777 904(10) =

1 1000.1100 0111 0001 1100 0111 0001 1100 0111 0001 1100 1001 0101 0100 1(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:


24.777 777 777 777 904(10) =

1 1000.1100 0111 0001 1100 0111 0001 1100 0111 0001 1100 1001 0101 0100 1(2) =

1 1000.1100 0111 0001 1100 0111 0001 1100 0111 0001 1100 1001 0101 0100 1(2) × 20 =

1.1000 1100 0111 0001 1100 0111 0001 1100 0111 0001 1100 1001 0101 0100 1(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 1001 0101 0100 1


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 1001 0101 0 1001 =

1000 1100 0111 0001 1100 0111 0001 1100 0111 0001 1100 1001 0101


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 1001 0101


Decimal number 24.777 777 777 777 904 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 1001 0101

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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