24.777 777 777 777 777 767 9 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 24.777 777 777 777 777 767 9(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
24.777 777 777 777 777 767 9(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: 0.777 777 777 777 777 767 9.

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) 0.777 777 777 777 777 767 9 × 2 = 1 + 0.555 555 555 555 555 535 8;
  • 2) 0.555 555 555 555 555 535 8 × 2 = 1 + 0.111 111 111 111 111 071 6;
  • 3) 0.111 111 111 111 111 071 6 × 2 = 0 + 0.222 222 222 222 222 143 2;
  • 4) 0.222 222 222 222 222 143 2 × 2 = 0 + 0.444 444 444 444 444 286 4;
  • 5) 0.444 444 444 444 444 286 4 × 2 = 0 + 0.888 888 888 888 888 572 8;
  • 6) 0.888 888 888 888 888 572 8 × 2 = 1 + 0.777 777 777 777 777 145 6;
  • 7) 0.777 777 777 777 777 145 6 × 2 = 1 + 0.555 555 555 555 554 291 2;
  • 8) 0.555 555 555 555 554 291 2 × 2 = 1 + 0.111 111 111 111 108 582 4;
  • 9) 0.111 111 111 111 108 582 4 × 2 = 0 + 0.222 222 222 222 217 164 8;
  • 10) 0.222 222 222 222 217 164 8 × 2 = 0 + 0.444 444 444 444 434 329 6;
  • 11) 0.444 444 444 444 434 329 6 × 2 = 0 + 0.888 888 888 888 868 659 2;
  • 12) 0.888 888 888 888 868 659 2 × 2 = 1 + 0.777 777 777 777 737 318 4;
  • 13) 0.777 777 777 777 737 318 4 × 2 = 1 + 0.555 555 555 555 474 636 8;
  • 14) 0.555 555 555 555 474 636 8 × 2 = 1 + 0.111 111 111 110 949 273 6;
  • 15) 0.111 111 111 110 949 273 6 × 2 = 0 + 0.222 222 222 221 898 547 2;
  • 16) 0.222 222 222 221 898 547 2 × 2 = 0 + 0.444 444 444 443 797 094 4;
  • 17) 0.444 444 444 443 797 094 4 × 2 = 0 + 0.888 888 888 887 594 188 8;
  • 18) 0.888 888 888 887 594 188 8 × 2 = 1 + 0.777 777 777 775 188 377 6;
  • 19) 0.777 777 777 775 188 377 6 × 2 = 1 + 0.555 555 555 550 376 755 2;
  • 20) 0.555 555 555 550 376 755 2 × 2 = 1 + 0.111 111 111 100 753 510 4;
  • 21) 0.111 111 111 100 753 510 4 × 2 = 0 + 0.222 222 222 201 507 020 8;
  • 22) 0.222 222 222 201 507 020 8 × 2 = 0 + 0.444 444 444 403 014 041 6;
  • 23) 0.444 444 444 403 014 041 6 × 2 = 0 + 0.888 888 888 806 028 083 2;
  • 24) 0.888 888 888 806 028 083 2 × 2 = 1 + 0.777 777 777 612 056 166 4;
  • 25) 0.777 777 777 612 056 166 4 × 2 = 1 + 0.555 555 555 224 112 332 8;
  • 26) 0.555 555 555 224 112 332 8 × 2 = 1 + 0.111 111 110 448 224 665 6;
  • 27) 0.111 111 110 448 224 665 6 × 2 = 0 + 0.222 222 220 896 449 331 2;
  • 28) 0.222 222 220 896 449 331 2 × 2 = 0 + 0.444 444 441 792 898 662 4;
  • 29) 0.444 444 441 792 898 662 4 × 2 = 0 + 0.888 888 883 585 797 324 8;
  • 30) 0.888 888 883 585 797 324 8 × 2 = 1 + 0.777 777 767 171 594 649 6;
  • 31) 0.777 777 767 171 594 649 6 × 2 = 1 + 0.555 555 534 343 189 299 2;
  • 32) 0.555 555 534 343 189 299 2 × 2 = 1 + 0.111 111 068 686 378 598 4;
  • 33) 0.111 111 068 686 378 598 4 × 2 = 0 + 0.222 222 137 372 757 196 8;
  • 34) 0.222 222 137 372 757 196 8 × 2 = 0 + 0.444 444 274 745 514 393 6;
  • 35) 0.444 444 274 745 514 393 6 × 2 = 0 + 0.888 888 549 491 028 787 2;
  • 36) 0.888 888 549 491 028 787 2 × 2 = 1 + 0.777 777 098 982 057 574 4;
  • 37) 0.777 777 098 982 057 574 4 × 2 = 1 + 0.555 554 197 964 115 148 8;
  • 38) 0.555 554 197 964 115 148 8 × 2 = 1 + 0.111 108 395 928 230 297 6;
  • 39) 0.111 108 395 928 230 297 6 × 2 = 0 + 0.222 216 791 856 460 595 2;
  • 40) 0.222 216 791 856 460 595 2 × 2 = 0 + 0.444 433 583 712 921 190 4;
  • 41) 0.444 433 583 712 921 190 4 × 2 = 0 + 0.888 867 167 425 842 380 8;
  • 42) 0.888 867 167 425 842 380 8 × 2 = 1 + 0.777 734 334 851 684 761 6;
  • 43) 0.777 734 334 851 684 761 6 × 2 = 1 + 0.555 468 669 703 369 523 2;
  • 44) 0.555 468 669 703 369 523 2 × 2 = 1 + 0.110 937 339 406 739 046 4;
  • 45) 0.110 937 339 406 739 046 4 × 2 = 0 + 0.221 874 678 813 478 092 8;
  • 46) 0.221 874 678 813 478 092 8 × 2 = 0 + 0.443 749 357 626 956 185 6;
  • 47) 0.443 749 357 626 956 185 6 × 2 = 0 + 0.887 498 715 253 912 371 2;
  • 48) 0.887 498 715 253 912 371 2 × 2 = 1 + 0.774 997 430 507 824 742 4;
  • 49) 0.774 997 430 507 824 742 4 × 2 = 1 + 0.549 994 861 015 649 484 8;
  • 50) 0.549 994 861 015 649 484 8 × 2 = 1 + 0.099 989 722 031 298 969 6;
  • 51) 0.099 989 722 031 298 969 6 × 2 = 0 + 0.199 979 444 062 597 939 2;
  • 52) 0.199 979 444 062 597 939 2 × 2 = 0 + 0.399 958 888 125 195 878 4;
  • 53) 0.399 958 888 125 195 878 4 × 2 = 0 + 0.799 917 776 250 391 756 8;

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:


0.777 777 777 777 777 767 9(10) =

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

5. Positive number before normalization:

24.777 777 777 777 777 767 9(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:


24.777 777 777 777 777 767 9(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 24.777 777 777 777 777 767 9 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