24.777 777 777 777 59 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 24.777 777 777 777 59(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 59(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 59.

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 59 × 2 = 1 + 0.555 555 555 555 18;
  • 2) 0.555 555 555 555 18 × 2 = 1 + 0.111 111 111 110 36;
  • 3) 0.111 111 111 110 36 × 2 = 0 + 0.222 222 222 220 72;
  • 4) 0.222 222 222 220 72 × 2 = 0 + 0.444 444 444 441 44;
  • 5) 0.444 444 444 441 44 × 2 = 0 + 0.888 888 888 882 88;
  • 6) 0.888 888 888 882 88 × 2 = 1 + 0.777 777 777 765 76;
  • 7) 0.777 777 777 765 76 × 2 = 1 + 0.555 555 555 531 52;
  • 8) 0.555 555 555 531 52 × 2 = 1 + 0.111 111 111 063 04;
  • 9) 0.111 111 111 063 04 × 2 = 0 + 0.222 222 222 126 08;
  • 10) 0.222 222 222 126 08 × 2 = 0 + 0.444 444 444 252 16;
  • 11) 0.444 444 444 252 16 × 2 = 0 + 0.888 888 888 504 32;
  • 12) 0.888 888 888 504 32 × 2 = 1 + 0.777 777 777 008 64;
  • 13) 0.777 777 777 008 64 × 2 = 1 + 0.555 555 554 017 28;
  • 14) 0.555 555 554 017 28 × 2 = 1 + 0.111 111 108 034 56;
  • 15) 0.111 111 108 034 56 × 2 = 0 + 0.222 222 216 069 12;
  • 16) 0.222 222 216 069 12 × 2 = 0 + 0.444 444 432 138 24;
  • 17) 0.444 444 432 138 24 × 2 = 0 + 0.888 888 864 276 48;
  • 18) 0.888 888 864 276 48 × 2 = 1 + 0.777 777 728 552 96;
  • 19) 0.777 777 728 552 96 × 2 = 1 + 0.555 555 457 105 92;
  • 20) 0.555 555 457 105 92 × 2 = 1 + 0.111 110 914 211 84;
  • 21) 0.111 110 914 211 84 × 2 = 0 + 0.222 221 828 423 68;
  • 22) 0.222 221 828 423 68 × 2 = 0 + 0.444 443 656 847 36;
  • 23) 0.444 443 656 847 36 × 2 = 0 + 0.888 887 313 694 72;
  • 24) 0.888 887 313 694 72 × 2 = 1 + 0.777 774 627 389 44;
  • 25) 0.777 774 627 389 44 × 2 = 1 + 0.555 549 254 778 88;
  • 26) 0.555 549 254 778 88 × 2 = 1 + 0.111 098 509 557 76;
  • 27) 0.111 098 509 557 76 × 2 = 0 + 0.222 197 019 115 52;
  • 28) 0.222 197 019 115 52 × 2 = 0 + 0.444 394 038 231 04;
  • 29) 0.444 394 038 231 04 × 2 = 0 + 0.888 788 076 462 08;
  • 30) 0.888 788 076 462 08 × 2 = 1 + 0.777 576 152 924 16;
  • 31) 0.777 576 152 924 16 × 2 = 1 + 0.555 152 305 848 32;
  • 32) 0.555 152 305 848 32 × 2 = 1 + 0.110 304 611 696 64;
  • 33) 0.110 304 611 696 64 × 2 = 0 + 0.220 609 223 393 28;
  • 34) 0.220 609 223 393 28 × 2 = 0 + 0.441 218 446 786 56;
  • 35) 0.441 218 446 786 56 × 2 = 0 + 0.882 436 893 573 12;
  • 36) 0.882 436 893 573 12 × 2 = 1 + 0.764 873 787 146 24;
  • 37) 0.764 873 787 146 24 × 2 = 1 + 0.529 747 574 292 48;
  • 38) 0.529 747 574 292 48 × 2 = 1 + 0.059 495 148 584 96;
  • 39) 0.059 495 148 584 96 × 2 = 0 + 0.118 990 297 169 92;
  • 40) 0.118 990 297 169 92 × 2 = 0 + 0.237 980 594 339 84;
  • 41) 0.237 980 594 339 84 × 2 = 0 + 0.475 961 188 679 68;
  • 42) 0.475 961 188 679 68 × 2 = 0 + 0.951 922 377 359 36;
  • 43) 0.951 922 377 359 36 × 2 = 1 + 0.903 844 754 718 72;
  • 44) 0.903 844 754 718 72 × 2 = 1 + 0.807 689 509 437 44;
  • 45) 0.807 689 509 437 44 × 2 = 1 + 0.615 379 018 874 88;
  • 46) 0.615 379 018 874 88 × 2 = 1 + 0.230 758 037 749 76;
  • 47) 0.230 758 037 749 76 × 2 = 0 + 0.461 516 075 499 52;
  • 48) 0.461 516 075 499 52 × 2 = 0 + 0.923 032 150 999 04;
  • 49) 0.923 032 150 999 04 × 2 = 1 + 0.846 064 301 998 08;
  • 50) 0.846 064 301 998 08 × 2 = 1 + 0.692 128 603 996 16;
  • 51) 0.692 128 603 996 16 × 2 = 1 + 0.384 257 207 992 32;
  • 52) 0.384 257 207 992 32 × 2 = 0 + 0.768 514 415 984 64;
  • 53) 0.768 514 415 984 64 × 2 = 1 + 0.537 028 831 969 28;

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 59(10) =

0.1100 0111 0001 1100 0111 0001 1100 0111 0001 1100 0011 1100 1110 1(2)

5. Positive number before normalization:

24.777 777 777 777 59(10) =

1 1000.1100 0111 0001 1100 0111 0001 1100 0111 0001 1100 0011 1100 1110 1(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 59(10) =

1 1000.1100 0111 0001 1100 0111 0001 1100 0111 0001 1100 0011 1100 1110 1(2) =

1 1000.1100 0111 0001 1100 0111 0001 1100 0111 0001 1100 0011 1100 1110 1(2) × 20 =

1.1000 1100 0111 0001 1100 0111 0001 1100 0111 0001 1100 0011 1100 1110 1(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 0011 1100 1110 1


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 0011 1100 1 1101 =

1000 1100 0111 0001 1100 0111 0001 1100 0111 0001 1100 0011 1100


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 0011 1100


Decimal number 24.777 777 777 777 59 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 0011 1100

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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