24.777 777 777 777 777 781 41 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 24.777 777 777 777 777 781 41(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
24.777 777 777 777 777 781 41(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: 0.777 777 777 777 777 781 41.

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) 0.777 777 777 777 777 781 41 × 2 = 1 + 0.555 555 555 555 555 562 82;
  • 2) 0.555 555 555 555 555 562 82 × 2 = 1 + 0.111 111 111 111 111 125 64;
  • 3) 0.111 111 111 111 111 125 64 × 2 = 0 + 0.222 222 222 222 222 251 28;
  • 4) 0.222 222 222 222 222 251 28 × 2 = 0 + 0.444 444 444 444 444 502 56;
  • 5) 0.444 444 444 444 444 502 56 × 2 = 0 + 0.888 888 888 888 889 005 12;
  • 6) 0.888 888 888 888 889 005 12 × 2 = 1 + 0.777 777 777 777 778 010 24;
  • 7) 0.777 777 777 777 778 010 24 × 2 = 1 + 0.555 555 555 555 556 020 48;
  • 8) 0.555 555 555 555 556 020 48 × 2 = 1 + 0.111 111 111 111 112 040 96;
  • 9) 0.111 111 111 111 112 040 96 × 2 = 0 + 0.222 222 222 222 224 081 92;
  • 10) 0.222 222 222 222 224 081 92 × 2 = 0 + 0.444 444 444 444 448 163 84;
  • 11) 0.444 444 444 444 448 163 84 × 2 = 0 + 0.888 888 888 888 896 327 68;
  • 12) 0.888 888 888 888 896 327 68 × 2 = 1 + 0.777 777 777 777 792 655 36;
  • 13) 0.777 777 777 777 792 655 36 × 2 = 1 + 0.555 555 555 555 585 310 72;
  • 14) 0.555 555 555 555 585 310 72 × 2 = 1 + 0.111 111 111 111 170 621 44;
  • 15) 0.111 111 111 111 170 621 44 × 2 = 0 + 0.222 222 222 222 341 242 88;
  • 16) 0.222 222 222 222 341 242 88 × 2 = 0 + 0.444 444 444 444 682 485 76;
  • 17) 0.444 444 444 444 682 485 76 × 2 = 0 + 0.888 888 888 889 364 971 52;
  • 18) 0.888 888 888 889 364 971 52 × 2 = 1 + 0.777 777 777 778 729 943 04;
  • 19) 0.777 777 777 778 729 943 04 × 2 = 1 + 0.555 555 555 557 459 886 08;
  • 20) 0.555 555 555 557 459 886 08 × 2 = 1 + 0.111 111 111 114 919 772 16;
  • 21) 0.111 111 111 114 919 772 16 × 2 = 0 + 0.222 222 222 229 839 544 32;
  • 22) 0.222 222 222 229 839 544 32 × 2 = 0 + 0.444 444 444 459 679 088 64;
  • 23) 0.444 444 444 459 679 088 64 × 2 = 0 + 0.888 888 888 919 358 177 28;
  • 24) 0.888 888 888 919 358 177 28 × 2 = 1 + 0.777 777 777 838 716 354 56;
  • 25) 0.777 777 777 838 716 354 56 × 2 = 1 + 0.555 555 555 677 432 709 12;
  • 26) 0.555 555 555 677 432 709 12 × 2 = 1 + 0.111 111 111 354 865 418 24;
  • 27) 0.111 111 111 354 865 418 24 × 2 = 0 + 0.222 222 222 709 730 836 48;
  • 28) 0.222 222 222 709 730 836 48 × 2 = 0 + 0.444 444 445 419 461 672 96;
  • 29) 0.444 444 445 419 461 672 96 × 2 = 0 + 0.888 888 890 838 923 345 92;
  • 30) 0.888 888 890 838 923 345 92 × 2 = 1 + 0.777 777 781 677 846 691 84;
  • 31) 0.777 777 781 677 846 691 84 × 2 = 1 + 0.555 555 563 355 693 383 68;
  • 32) 0.555 555 563 355 693 383 68 × 2 = 1 + 0.111 111 126 711 386 767 36;
  • 33) 0.111 111 126 711 386 767 36 × 2 = 0 + 0.222 222 253 422 773 534 72;
  • 34) 0.222 222 253 422 773 534 72 × 2 = 0 + 0.444 444 506 845 547 069 44;
  • 35) 0.444 444 506 845 547 069 44 × 2 = 0 + 0.888 889 013 691 094 138 88;
  • 36) 0.888 889 013 691 094 138 88 × 2 = 1 + 0.777 778 027 382 188 277 76;
  • 37) 0.777 778 027 382 188 277 76 × 2 = 1 + 0.555 556 054 764 376 555 52;
  • 38) 0.555 556 054 764 376 555 52 × 2 = 1 + 0.111 112 109 528 753 111 04;
  • 39) 0.111 112 109 528 753 111 04 × 2 = 0 + 0.222 224 219 057 506 222 08;
  • 40) 0.222 224 219 057 506 222 08 × 2 = 0 + 0.444 448 438 115 012 444 16;
  • 41) 0.444 448 438 115 012 444 16 × 2 = 0 + 0.888 896 876 230 024 888 32;
  • 42) 0.888 896 876 230 024 888 32 × 2 = 1 + 0.777 793 752 460 049 776 64;
  • 43) 0.777 793 752 460 049 776 64 × 2 = 1 + 0.555 587 504 920 099 553 28;
  • 44) 0.555 587 504 920 099 553 28 × 2 = 1 + 0.111 175 009 840 199 106 56;
  • 45) 0.111 175 009 840 199 106 56 × 2 = 0 + 0.222 350 019 680 398 213 12;
  • 46) 0.222 350 019 680 398 213 12 × 2 = 0 + 0.444 700 039 360 796 426 24;
  • 47) 0.444 700 039 360 796 426 24 × 2 = 0 + 0.889 400 078 721 592 852 48;
  • 48) 0.889 400 078 721 592 852 48 × 2 = 1 + 0.778 800 157 443 185 704 96;
  • 49) 0.778 800 157 443 185 704 96 × 2 = 1 + 0.557 600 314 886 371 409 92;
  • 50) 0.557 600 314 886 371 409 92 × 2 = 1 + 0.115 200 629 772 742 819 84;
  • 51) 0.115 200 629 772 742 819 84 × 2 = 0 + 0.230 401 259 545 485 639 68;
  • 52) 0.230 401 259 545 485 639 68 × 2 = 0 + 0.460 802 519 090 971 279 36;
  • 53) 0.460 802 519 090 971 279 36 × 2 = 0 + 0.921 605 038 181 942 558 72;

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:


0.777 777 777 777 777 781 41(10) =

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

5. Positive number before normalization:

24.777 777 777 777 777 781 41(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:


24.777 777 777 777 777 781 41(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 24.777 777 777 777 777 781 41 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