24.777 777 777 777 777 777 780 8 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 24.777 777 777 777 777 777 780 8(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 8(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 8.

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 8 × 2 = 1 + 0.555 555 555 555 555 555 561 6;
  • 2) 0.555 555 555 555 555 555 561 6 × 2 = 1 + 0.111 111 111 111 111 111 123 2;
  • 3) 0.111 111 111 111 111 111 123 2 × 2 = 0 + 0.222 222 222 222 222 222 246 4;
  • 4) 0.222 222 222 222 222 222 246 4 × 2 = 0 + 0.444 444 444 444 444 444 492 8;
  • 5) 0.444 444 444 444 444 444 492 8 × 2 = 0 + 0.888 888 888 888 888 888 985 6;
  • 6) 0.888 888 888 888 888 888 985 6 × 2 = 1 + 0.777 777 777 777 777 777 971 2;
  • 7) 0.777 777 777 777 777 777 971 2 × 2 = 1 + 0.555 555 555 555 555 555 942 4;
  • 8) 0.555 555 555 555 555 555 942 4 × 2 = 1 + 0.111 111 111 111 111 111 884 8;
  • 9) 0.111 111 111 111 111 111 884 8 × 2 = 0 + 0.222 222 222 222 222 223 769 6;
  • 10) 0.222 222 222 222 222 223 769 6 × 2 = 0 + 0.444 444 444 444 444 447 539 2;
  • 11) 0.444 444 444 444 444 447 539 2 × 2 = 0 + 0.888 888 888 888 888 895 078 4;
  • 12) 0.888 888 888 888 888 895 078 4 × 2 = 1 + 0.777 777 777 777 777 790 156 8;
  • 13) 0.777 777 777 777 777 790 156 8 × 2 = 1 + 0.555 555 555 555 555 580 313 6;
  • 14) 0.555 555 555 555 555 580 313 6 × 2 = 1 + 0.111 111 111 111 111 160 627 2;
  • 15) 0.111 111 111 111 111 160 627 2 × 2 = 0 + 0.222 222 222 222 222 321 254 4;
  • 16) 0.222 222 222 222 222 321 254 4 × 2 = 0 + 0.444 444 444 444 444 642 508 8;
  • 17) 0.444 444 444 444 444 642 508 8 × 2 = 0 + 0.888 888 888 888 889 285 017 6;
  • 18) 0.888 888 888 888 889 285 017 6 × 2 = 1 + 0.777 777 777 777 778 570 035 2;
  • 19) 0.777 777 777 777 778 570 035 2 × 2 = 1 + 0.555 555 555 555 557 140 070 4;
  • 20) 0.555 555 555 555 557 140 070 4 × 2 = 1 + 0.111 111 111 111 114 280 140 8;
  • 21) 0.111 111 111 111 114 280 140 8 × 2 = 0 + 0.222 222 222 222 228 560 281 6;
  • 22) 0.222 222 222 222 228 560 281 6 × 2 = 0 + 0.444 444 444 444 457 120 563 2;
  • 23) 0.444 444 444 444 457 120 563 2 × 2 = 0 + 0.888 888 888 888 914 241 126 4;
  • 24) 0.888 888 888 888 914 241 126 4 × 2 = 1 + 0.777 777 777 777 828 482 252 8;
  • 25) 0.777 777 777 777 828 482 252 8 × 2 = 1 + 0.555 555 555 555 656 964 505 6;
  • 26) 0.555 555 555 555 656 964 505 6 × 2 = 1 + 0.111 111 111 111 313 929 011 2;
  • 27) 0.111 111 111 111 313 929 011 2 × 2 = 0 + 0.222 222 222 222 627 858 022 4;
  • 28) 0.222 222 222 222 627 858 022 4 × 2 = 0 + 0.444 444 444 445 255 716 044 8;
  • 29) 0.444 444 444 445 255 716 044 8 × 2 = 0 + 0.888 888 888 890 511 432 089 6;
  • 30) 0.888 888 888 890 511 432 089 6 × 2 = 1 + 0.777 777 777 781 022 864 179 2;
  • 31) 0.777 777 777 781 022 864 179 2 × 2 = 1 + 0.555 555 555 562 045 728 358 4;
  • 32) 0.555 555 555 562 045 728 358 4 × 2 = 1 + 0.111 111 111 124 091 456 716 8;
  • 33) 0.111 111 111 124 091 456 716 8 × 2 = 0 + 0.222 222 222 248 182 913 433 6;
  • 34) 0.222 222 222 248 182 913 433 6 × 2 = 0 + 0.444 444 444 496 365 826 867 2;
  • 35) 0.444 444 444 496 365 826 867 2 × 2 = 0 + 0.888 888 888 992 731 653 734 4;
  • 36) 0.888 888 888 992 731 653 734 4 × 2 = 1 + 0.777 777 777 985 463 307 468 8;
  • 37) 0.777 777 777 985 463 307 468 8 × 2 = 1 + 0.555 555 555 970 926 614 937 6;
  • 38) 0.555 555 555 970 926 614 937 6 × 2 = 1 + 0.111 111 111 941 853 229 875 2;
  • 39) 0.111 111 111 941 853 229 875 2 × 2 = 0 + 0.222 222 223 883 706 459 750 4;
  • 40) 0.222 222 223 883 706 459 750 4 × 2 = 0 + 0.444 444 447 767 412 919 500 8;
  • 41) 0.444 444 447 767 412 919 500 8 × 2 = 0 + 0.888 888 895 534 825 839 001 6;
  • 42) 0.888 888 895 534 825 839 001 6 × 2 = 1 + 0.777 777 791 069 651 678 003 2;
  • 43) 0.777 777 791 069 651 678 003 2 × 2 = 1 + 0.555 555 582 139 303 356 006 4;
  • 44) 0.555 555 582 139 303 356 006 4 × 2 = 1 + 0.111 111 164 278 606 712 012 8;
  • 45) 0.111 111 164 278 606 712 012 8 × 2 = 0 + 0.222 222 328 557 213 424 025 6;
  • 46) 0.222 222 328 557 213 424 025 6 × 2 = 0 + 0.444 444 657 114 426 848 051 2;
  • 47) 0.444 444 657 114 426 848 051 2 × 2 = 0 + 0.888 889 314 228 853 696 102 4;
  • 48) 0.888 889 314 228 853 696 102 4 × 2 = 1 + 0.777 778 628 457 707 392 204 8;
  • 49) 0.777 778 628 457 707 392 204 8 × 2 = 1 + 0.555 557 256 915 414 784 409 6;
  • 50) 0.555 557 256 915 414 784 409 6 × 2 = 1 + 0.111 114 513 830 829 568 819 2;
  • 51) 0.111 114 513 830 829 568 819 2 × 2 = 0 + 0.222 229 027 661 659 137 638 4;
  • 52) 0.222 229 027 661 659 137 638 4 × 2 = 0 + 0.444 458 055 323 318 275 276 8;
  • 53) 0.444 458 055 323 318 275 276 8 × 2 = 0 + 0.888 916 110 646 636 550 553 6;

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 8(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 8(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 8(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 8 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