24.777 777 777 777 777 787 2 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 24.777 777 777 777 777 787 2(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
24.777 777 777 777 777 787 2(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: 0.777 777 777 777 777 787 2.

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) 0.777 777 777 777 777 787 2 × 2 = 1 + 0.555 555 555 555 555 574 4;
  • 2) 0.555 555 555 555 555 574 4 × 2 = 1 + 0.111 111 111 111 111 148 8;
  • 3) 0.111 111 111 111 111 148 8 × 2 = 0 + 0.222 222 222 222 222 297 6;
  • 4) 0.222 222 222 222 222 297 6 × 2 = 0 + 0.444 444 444 444 444 595 2;
  • 5) 0.444 444 444 444 444 595 2 × 2 = 0 + 0.888 888 888 888 889 190 4;
  • 6) 0.888 888 888 888 889 190 4 × 2 = 1 + 0.777 777 777 777 778 380 8;
  • 7) 0.777 777 777 777 778 380 8 × 2 = 1 + 0.555 555 555 555 556 761 6;
  • 8) 0.555 555 555 555 556 761 6 × 2 = 1 + 0.111 111 111 111 113 523 2;
  • 9) 0.111 111 111 111 113 523 2 × 2 = 0 + 0.222 222 222 222 227 046 4;
  • 10) 0.222 222 222 222 227 046 4 × 2 = 0 + 0.444 444 444 444 454 092 8;
  • 11) 0.444 444 444 444 454 092 8 × 2 = 0 + 0.888 888 888 888 908 185 6;
  • 12) 0.888 888 888 888 908 185 6 × 2 = 1 + 0.777 777 777 777 816 371 2;
  • 13) 0.777 777 777 777 816 371 2 × 2 = 1 + 0.555 555 555 555 632 742 4;
  • 14) 0.555 555 555 555 632 742 4 × 2 = 1 + 0.111 111 111 111 265 484 8;
  • 15) 0.111 111 111 111 265 484 8 × 2 = 0 + 0.222 222 222 222 530 969 6;
  • 16) 0.222 222 222 222 530 969 6 × 2 = 0 + 0.444 444 444 445 061 939 2;
  • 17) 0.444 444 444 445 061 939 2 × 2 = 0 + 0.888 888 888 890 123 878 4;
  • 18) 0.888 888 888 890 123 878 4 × 2 = 1 + 0.777 777 777 780 247 756 8;
  • 19) 0.777 777 777 780 247 756 8 × 2 = 1 + 0.555 555 555 560 495 513 6;
  • 20) 0.555 555 555 560 495 513 6 × 2 = 1 + 0.111 111 111 120 991 027 2;
  • 21) 0.111 111 111 120 991 027 2 × 2 = 0 + 0.222 222 222 241 982 054 4;
  • 22) 0.222 222 222 241 982 054 4 × 2 = 0 + 0.444 444 444 483 964 108 8;
  • 23) 0.444 444 444 483 964 108 8 × 2 = 0 + 0.888 888 888 967 928 217 6;
  • 24) 0.888 888 888 967 928 217 6 × 2 = 1 + 0.777 777 777 935 856 435 2;
  • 25) 0.777 777 777 935 856 435 2 × 2 = 1 + 0.555 555 555 871 712 870 4;
  • 26) 0.555 555 555 871 712 870 4 × 2 = 1 + 0.111 111 111 743 425 740 8;
  • 27) 0.111 111 111 743 425 740 8 × 2 = 0 + 0.222 222 223 486 851 481 6;
  • 28) 0.222 222 223 486 851 481 6 × 2 = 0 + 0.444 444 446 973 702 963 2;
  • 29) 0.444 444 446 973 702 963 2 × 2 = 0 + 0.888 888 893 947 405 926 4;
  • 30) 0.888 888 893 947 405 926 4 × 2 = 1 + 0.777 777 787 894 811 852 8;
  • 31) 0.777 777 787 894 811 852 8 × 2 = 1 + 0.555 555 575 789 623 705 6;
  • 32) 0.555 555 575 789 623 705 6 × 2 = 1 + 0.111 111 151 579 247 411 2;
  • 33) 0.111 111 151 579 247 411 2 × 2 = 0 + 0.222 222 303 158 494 822 4;
  • 34) 0.222 222 303 158 494 822 4 × 2 = 0 + 0.444 444 606 316 989 644 8;
  • 35) 0.444 444 606 316 989 644 8 × 2 = 0 + 0.888 889 212 633 979 289 6;
  • 36) 0.888 889 212 633 979 289 6 × 2 = 1 + 0.777 778 425 267 958 579 2;
  • 37) 0.777 778 425 267 958 579 2 × 2 = 1 + 0.555 556 850 535 917 158 4;
  • 38) 0.555 556 850 535 917 158 4 × 2 = 1 + 0.111 113 701 071 834 316 8;
  • 39) 0.111 113 701 071 834 316 8 × 2 = 0 + 0.222 227 402 143 668 633 6;
  • 40) 0.222 227 402 143 668 633 6 × 2 = 0 + 0.444 454 804 287 337 267 2;
  • 41) 0.444 454 804 287 337 267 2 × 2 = 0 + 0.888 909 608 574 674 534 4;
  • 42) 0.888 909 608 574 674 534 4 × 2 = 1 + 0.777 819 217 149 349 068 8;
  • 43) 0.777 819 217 149 349 068 8 × 2 = 1 + 0.555 638 434 298 698 137 6;
  • 44) 0.555 638 434 298 698 137 6 × 2 = 1 + 0.111 276 868 597 396 275 2;
  • 45) 0.111 276 868 597 396 275 2 × 2 = 0 + 0.222 553 737 194 792 550 4;
  • 46) 0.222 553 737 194 792 550 4 × 2 = 0 + 0.445 107 474 389 585 100 8;
  • 47) 0.445 107 474 389 585 100 8 × 2 = 0 + 0.890 214 948 779 170 201 6;
  • 48) 0.890 214 948 779 170 201 6 × 2 = 1 + 0.780 429 897 558 340 403 2;
  • 49) 0.780 429 897 558 340 403 2 × 2 = 1 + 0.560 859 795 116 680 806 4;
  • 50) 0.560 859 795 116 680 806 4 × 2 = 1 + 0.121 719 590 233 361 612 8;
  • 51) 0.121 719 590 233 361 612 8 × 2 = 0 + 0.243 439 180 466 723 225 6;
  • 52) 0.243 439 180 466 723 225 6 × 2 = 0 + 0.486 878 360 933 446 451 2;
  • 53) 0.486 878 360 933 446 451 2 × 2 = 0 + 0.973 756 721 866 892 902 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:


0.777 777 777 777 777 787 2(10) =

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

5. Positive number before normalization:

24.777 777 777 777 777 787 2(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:


24.777 777 777 777 777 787 2(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 24.777 777 777 777 777 787 2 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