24.777 777 777 777 777 761 7 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 24.777 777 777 777 777 761 7(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
24.777 777 777 777 777 761 7(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: 0.777 777 777 777 777 761 7.

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) 0.777 777 777 777 777 761 7 × 2 = 1 + 0.555 555 555 555 555 523 4;
  • 2) 0.555 555 555 555 555 523 4 × 2 = 1 + 0.111 111 111 111 111 046 8;
  • 3) 0.111 111 111 111 111 046 8 × 2 = 0 + 0.222 222 222 222 222 093 6;
  • 4) 0.222 222 222 222 222 093 6 × 2 = 0 + 0.444 444 444 444 444 187 2;
  • 5) 0.444 444 444 444 444 187 2 × 2 = 0 + 0.888 888 888 888 888 374 4;
  • 6) 0.888 888 888 888 888 374 4 × 2 = 1 + 0.777 777 777 777 776 748 8;
  • 7) 0.777 777 777 777 776 748 8 × 2 = 1 + 0.555 555 555 555 553 497 6;
  • 8) 0.555 555 555 555 553 497 6 × 2 = 1 + 0.111 111 111 111 106 995 2;
  • 9) 0.111 111 111 111 106 995 2 × 2 = 0 + 0.222 222 222 222 213 990 4;
  • 10) 0.222 222 222 222 213 990 4 × 2 = 0 + 0.444 444 444 444 427 980 8;
  • 11) 0.444 444 444 444 427 980 8 × 2 = 0 + 0.888 888 888 888 855 961 6;
  • 12) 0.888 888 888 888 855 961 6 × 2 = 1 + 0.777 777 777 777 711 923 2;
  • 13) 0.777 777 777 777 711 923 2 × 2 = 1 + 0.555 555 555 555 423 846 4;
  • 14) 0.555 555 555 555 423 846 4 × 2 = 1 + 0.111 111 111 110 847 692 8;
  • 15) 0.111 111 111 110 847 692 8 × 2 = 0 + 0.222 222 222 221 695 385 6;
  • 16) 0.222 222 222 221 695 385 6 × 2 = 0 + 0.444 444 444 443 390 771 2;
  • 17) 0.444 444 444 443 390 771 2 × 2 = 0 + 0.888 888 888 886 781 542 4;
  • 18) 0.888 888 888 886 781 542 4 × 2 = 1 + 0.777 777 777 773 563 084 8;
  • 19) 0.777 777 777 773 563 084 8 × 2 = 1 + 0.555 555 555 547 126 169 6;
  • 20) 0.555 555 555 547 126 169 6 × 2 = 1 + 0.111 111 111 094 252 339 2;
  • 21) 0.111 111 111 094 252 339 2 × 2 = 0 + 0.222 222 222 188 504 678 4;
  • 22) 0.222 222 222 188 504 678 4 × 2 = 0 + 0.444 444 444 377 009 356 8;
  • 23) 0.444 444 444 377 009 356 8 × 2 = 0 + 0.888 888 888 754 018 713 6;
  • 24) 0.888 888 888 754 018 713 6 × 2 = 1 + 0.777 777 777 508 037 427 2;
  • 25) 0.777 777 777 508 037 427 2 × 2 = 1 + 0.555 555 555 016 074 854 4;
  • 26) 0.555 555 555 016 074 854 4 × 2 = 1 + 0.111 111 110 032 149 708 8;
  • 27) 0.111 111 110 032 149 708 8 × 2 = 0 + 0.222 222 220 064 299 417 6;
  • 28) 0.222 222 220 064 299 417 6 × 2 = 0 + 0.444 444 440 128 598 835 2;
  • 29) 0.444 444 440 128 598 835 2 × 2 = 0 + 0.888 888 880 257 197 670 4;
  • 30) 0.888 888 880 257 197 670 4 × 2 = 1 + 0.777 777 760 514 395 340 8;
  • 31) 0.777 777 760 514 395 340 8 × 2 = 1 + 0.555 555 521 028 790 681 6;
  • 32) 0.555 555 521 028 790 681 6 × 2 = 1 + 0.111 111 042 057 581 363 2;
  • 33) 0.111 111 042 057 581 363 2 × 2 = 0 + 0.222 222 084 115 162 726 4;
  • 34) 0.222 222 084 115 162 726 4 × 2 = 0 + 0.444 444 168 230 325 452 8;
  • 35) 0.444 444 168 230 325 452 8 × 2 = 0 + 0.888 888 336 460 650 905 6;
  • 36) 0.888 888 336 460 650 905 6 × 2 = 1 + 0.777 776 672 921 301 811 2;
  • 37) 0.777 776 672 921 301 811 2 × 2 = 1 + 0.555 553 345 842 603 622 4;
  • 38) 0.555 553 345 842 603 622 4 × 2 = 1 + 0.111 106 691 685 207 244 8;
  • 39) 0.111 106 691 685 207 244 8 × 2 = 0 + 0.222 213 383 370 414 489 6;
  • 40) 0.222 213 383 370 414 489 6 × 2 = 0 + 0.444 426 766 740 828 979 2;
  • 41) 0.444 426 766 740 828 979 2 × 2 = 0 + 0.888 853 533 481 657 958 4;
  • 42) 0.888 853 533 481 657 958 4 × 2 = 1 + 0.777 707 066 963 315 916 8;
  • 43) 0.777 707 066 963 315 916 8 × 2 = 1 + 0.555 414 133 926 631 833 6;
  • 44) 0.555 414 133 926 631 833 6 × 2 = 1 + 0.110 828 267 853 263 667 2;
  • 45) 0.110 828 267 853 263 667 2 × 2 = 0 + 0.221 656 535 706 527 334 4;
  • 46) 0.221 656 535 706 527 334 4 × 2 = 0 + 0.443 313 071 413 054 668 8;
  • 47) 0.443 313 071 413 054 668 8 × 2 = 0 + 0.886 626 142 826 109 337 6;
  • 48) 0.886 626 142 826 109 337 6 × 2 = 1 + 0.773 252 285 652 218 675 2;
  • 49) 0.773 252 285 652 218 675 2 × 2 = 1 + 0.546 504 571 304 437 350 4;
  • 50) 0.546 504 571 304 437 350 4 × 2 = 1 + 0.093 009 142 608 874 700 8;
  • 51) 0.093 009 142 608 874 700 8 × 2 = 0 + 0.186 018 285 217 749 401 6;
  • 52) 0.186 018 285 217 749 401 6 × 2 = 0 + 0.372 036 570 435 498 803 2;
  • 53) 0.372 036 570 435 498 803 2 × 2 = 0 + 0.744 073 140 870 997 606 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:


0.777 777 777 777 777 761 7(10) =

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

5. Positive number before normalization:

24.777 777 777 777 777 761 7(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:


24.777 777 777 777 777 761 7(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 24.777 777 777 777 777 761 7 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