24.777 777 777 777 777 789 3 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 24.777 777 777 777 777 789 3(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
24.777 777 777 777 777 789 3(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: 0.777 777 777 777 777 789 3.

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) 0.777 777 777 777 777 789 3 × 2 = 1 + 0.555 555 555 555 555 578 6;
  • 2) 0.555 555 555 555 555 578 6 × 2 = 1 + 0.111 111 111 111 111 157 2;
  • 3) 0.111 111 111 111 111 157 2 × 2 = 0 + 0.222 222 222 222 222 314 4;
  • 4) 0.222 222 222 222 222 314 4 × 2 = 0 + 0.444 444 444 444 444 628 8;
  • 5) 0.444 444 444 444 444 628 8 × 2 = 0 + 0.888 888 888 888 889 257 6;
  • 6) 0.888 888 888 888 889 257 6 × 2 = 1 + 0.777 777 777 777 778 515 2;
  • 7) 0.777 777 777 777 778 515 2 × 2 = 1 + 0.555 555 555 555 557 030 4;
  • 8) 0.555 555 555 555 557 030 4 × 2 = 1 + 0.111 111 111 111 114 060 8;
  • 9) 0.111 111 111 111 114 060 8 × 2 = 0 + 0.222 222 222 222 228 121 6;
  • 10) 0.222 222 222 222 228 121 6 × 2 = 0 + 0.444 444 444 444 456 243 2;
  • 11) 0.444 444 444 444 456 243 2 × 2 = 0 + 0.888 888 888 888 912 486 4;
  • 12) 0.888 888 888 888 912 486 4 × 2 = 1 + 0.777 777 777 777 824 972 8;
  • 13) 0.777 777 777 777 824 972 8 × 2 = 1 + 0.555 555 555 555 649 945 6;
  • 14) 0.555 555 555 555 649 945 6 × 2 = 1 + 0.111 111 111 111 299 891 2;
  • 15) 0.111 111 111 111 299 891 2 × 2 = 0 + 0.222 222 222 222 599 782 4;
  • 16) 0.222 222 222 222 599 782 4 × 2 = 0 + 0.444 444 444 445 199 564 8;
  • 17) 0.444 444 444 445 199 564 8 × 2 = 0 + 0.888 888 888 890 399 129 6;
  • 18) 0.888 888 888 890 399 129 6 × 2 = 1 + 0.777 777 777 780 798 259 2;
  • 19) 0.777 777 777 780 798 259 2 × 2 = 1 + 0.555 555 555 561 596 518 4;
  • 20) 0.555 555 555 561 596 518 4 × 2 = 1 + 0.111 111 111 123 193 036 8;
  • 21) 0.111 111 111 123 193 036 8 × 2 = 0 + 0.222 222 222 246 386 073 6;
  • 22) 0.222 222 222 246 386 073 6 × 2 = 0 + 0.444 444 444 492 772 147 2;
  • 23) 0.444 444 444 492 772 147 2 × 2 = 0 + 0.888 888 888 985 544 294 4;
  • 24) 0.888 888 888 985 544 294 4 × 2 = 1 + 0.777 777 777 971 088 588 8;
  • 25) 0.777 777 777 971 088 588 8 × 2 = 1 + 0.555 555 555 942 177 177 6;
  • 26) 0.555 555 555 942 177 177 6 × 2 = 1 + 0.111 111 111 884 354 355 2;
  • 27) 0.111 111 111 884 354 355 2 × 2 = 0 + 0.222 222 223 768 708 710 4;
  • 28) 0.222 222 223 768 708 710 4 × 2 = 0 + 0.444 444 447 537 417 420 8;
  • 29) 0.444 444 447 537 417 420 8 × 2 = 0 + 0.888 888 895 074 834 841 6;
  • 30) 0.888 888 895 074 834 841 6 × 2 = 1 + 0.777 777 790 149 669 683 2;
  • 31) 0.777 777 790 149 669 683 2 × 2 = 1 + 0.555 555 580 299 339 366 4;
  • 32) 0.555 555 580 299 339 366 4 × 2 = 1 + 0.111 111 160 598 678 732 8;
  • 33) 0.111 111 160 598 678 732 8 × 2 = 0 + 0.222 222 321 197 357 465 6;
  • 34) 0.222 222 321 197 357 465 6 × 2 = 0 + 0.444 444 642 394 714 931 2;
  • 35) 0.444 444 642 394 714 931 2 × 2 = 0 + 0.888 889 284 789 429 862 4;
  • 36) 0.888 889 284 789 429 862 4 × 2 = 1 + 0.777 778 569 578 859 724 8;
  • 37) 0.777 778 569 578 859 724 8 × 2 = 1 + 0.555 557 139 157 719 449 6;
  • 38) 0.555 557 139 157 719 449 6 × 2 = 1 + 0.111 114 278 315 438 899 2;
  • 39) 0.111 114 278 315 438 899 2 × 2 = 0 + 0.222 228 556 630 877 798 4;
  • 40) 0.222 228 556 630 877 798 4 × 2 = 0 + 0.444 457 113 261 755 596 8;
  • 41) 0.444 457 113 261 755 596 8 × 2 = 0 + 0.888 914 226 523 511 193 6;
  • 42) 0.888 914 226 523 511 193 6 × 2 = 1 + 0.777 828 453 047 022 387 2;
  • 43) 0.777 828 453 047 022 387 2 × 2 = 1 + 0.555 656 906 094 044 774 4;
  • 44) 0.555 656 906 094 044 774 4 × 2 = 1 + 0.111 313 812 188 089 548 8;
  • 45) 0.111 313 812 188 089 548 8 × 2 = 0 + 0.222 627 624 376 179 097 6;
  • 46) 0.222 627 624 376 179 097 6 × 2 = 0 + 0.445 255 248 752 358 195 2;
  • 47) 0.445 255 248 752 358 195 2 × 2 = 0 + 0.890 510 497 504 716 390 4;
  • 48) 0.890 510 497 504 716 390 4 × 2 = 1 + 0.781 020 995 009 432 780 8;
  • 49) 0.781 020 995 009 432 780 8 × 2 = 1 + 0.562 041 990 018 865 561 6;
  • 50) 0.562 041 990 018 865 561 6 × 2 = 1 + 0.124 083 980 037 731 123 2;
  • 51) 0.124 083 980 037 731 123 2 × 2 = 0 + 0.248 167 960 075 462 246 4;
  • 52) 0.248 167 960 075 462 246 4 × 2 = 0 + 0.496 335 920 150 924 492 8;
  • 53) 0.496 335 920 150 924 492 8 × 2 = 0 + 0.992 671 840 301 848 985 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:


0.777 777 777 777 777 789 3(10) =

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

5. Positive number before normalization:

24.777 777 777 777 777 789 3(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:


24.777 777 777 777 777 789 3(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 24.777 777 777 777 777 789 3 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