24.777 777 777 777 777 772 2 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 24.777 777 777 777 777 772 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
24.777 777 777 777 777 772 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: 0.777 777 777 777 777 772 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) 0.777 777 777 777 777 772 2 × 2 = 1 + 0.555 555 555 555 555 544 4;
  • 2) 0.555 555 555 555 555 544 4 × 2 = 1 + 0.111 111 111 111 111 088 8;
  • 3) 0.111 111 111 111 111 088 8 × 2 = 0 + 0.222 222 222 222 222 177 6;
  • 4) 0.222 222 222 222 222 177 6 × 2 = 0 + 0.444 444 444 444 444 355 2;
  • 5) 0.444 444 444 444 444 355 2 × 2 = 0 + 0.888 888 888 888 888 710 4;
  • 6) 0.888 888 888 888 888 710 4 × 2 = 1 + 0.777 777 777 777 777 420 8;
  • 7) 0.777 777 777 777 777 420 8 × 2 = 1 + 0.555 555 555 555 554 841 6;
  • 8) 0.555 555 555 555 554 841 6 × 2 = 1 + 0.111 111 111 111 109 683 2;
  • 9) 0.111 111 111 111 109 683 2 × 2 = 0 + 0.222 222 222 222 219 366 4;
  • 10) 0.222 222 222 222 219 366 4 × 2 = 0 + 0.444 444 444 444 438 732 8;
  • 11) 0.444 444 444 444 438 732 8 × 2 = 0 + 0.888 888 888 888 877 465 6;
  • 12) 0.888 888 888 888 877 465 6 × 2 = 1 + 0.777 777 777 777 754 931 2;
  • 13) 0.777 777 777 777 754 931 2 × 2 = 1 + 0.555 555 555 555 509 862 4;
  • 14) 0.555 555 555 555 509 862 4 × 2 = 1 + 0.111 111 111 111 019 724 8;
  • 15) 0.111 111 111 111 019 724 8 × 2 = 0 + 0.222 222 222 222 039 449 6;
  • 16) 0.222 222 222 222 039 449 6 × 2 = 0 + 0.444 444 444 444 078 899 2;
  • 17) 0.444 444 444 444 078 899 2 × 2 = 0 + 0.888 888 888 888 157 798 4;
  • 18) 0.888 888 888 888 157 798 4 × 2 = 1 + 0.777 777 777 776 315 596 8;
  • 19) 0.777 777 777 776 315 596 8 × 2 = 1 + 0.555 555 555 552 631 193 6;
  • 20) 0.555 555 555 552 631 193 6 × 2 = 1 + 0.111 111 111 105 262 387 2;
  • 21) 0.111 111 111 105 262 387 2 × 2 = 0 + 0.222 222 222 210 524 774 4;
  • 22) 0.222 222 222 210 524 774 4 × 2 = 0 + 0.444 444 444 421 049 548 8;
  • 23) 0.444 444 444 421 049 548 8 × 2 = 0 + 0.888 888 888 842 099 097 6;
  • 24) 0.888 888 888 842 099 097 6 × 2 = 1 + 0.777 777 777 684 198 195 2;
  • 25) 0.777 777 777 684 198 195 2 × 2 = 1 + 0.555 555 555 368 396 390 4;
  • 26) 0.555 555 555 368 396 390 4 × 2 = 1 + 0.111 111 110 736 792 780 8;
  • 27) 0.111 111 110 736 792 780 8 × 2 = 0 + 0.222 222 221 473 585 561 6;
  • 28) 0.222 222 221 473 585 561 6 × 2 = 0 + 0.444 444 442 947 171 123 2;
  • 29) 0.444 444 442 947 171 123 2 × 2 = 0 + 0.888 888 885 894 342 246 4;
  • 30) 0.888 888 885 894 342 246 4 × 2 = 1 + 0.777 777 771 788 684 492 8;
  • 31) 0.777 777 771 788 684 492 8 × 2 = 1 + 0.555 555 543 577 368 985 6;
  • 32) 0.555 555 543 577 368 985 6 × 2 = 1 + 0.111 111 087 154 737 971 2;
  • 33) 0.111 111 087 154 737 971 2 × 2 = 0 + 0.222 222 174 309 475 942 4;
  • 34) 0.222 222 174 309 475 942 4 × 2 = 0 + 0.444 444 348 618 951 884 8;
  • 35) 0.444 444 348 618 951 884 8 × 2 = 0 + 0.888 888 697 237 903 769 6;
  • 36) 0.888 888 697 237 903 769 6 × 2 = 1 + 0.777 777 394 475 807 539 2;
  • 37) 0.777 777 394 475 807 539 2 × 2 = 1 + 0.555 554 788 951 615 078 4;
  • 38) 0.555 554 788 951 615 078 4 × 2 = 1 + 0.111 109 577 903 230 156 8;
  • 39) 0.111 109 577 903 230 156 8 × 2 = 0 + 0.222 219 155 806 460 313 6;
  • 40) 0.222 219 155 806 460 313 6 × 2 = 0 + 0.444 438 311 612 920 627 2;
  • 41) 0.444 438 311 612 920 627 2 × 2 = 0 + 0.888 876 623 225 841 254 4;
  • 42) 0.888 876 623 225 841 254 4 × 2 = 1 + 0.777 753 246 451 682 508 8;
  • 43) 0.777 753 246 451 682 508 8 × 2 = 1 + 0.555 506 492 903 365 017 6;
  • 44) 0.555 506 492 903 365 017 6 × 2 = 1 + 0.111 012 985 806 730 035 2;
  • 45) 0.111 012 985 806 730 035 2 × 2 = 0 + 0.222 025 971 613 460 070 4;
  • 46) 0.222 025 971 613 460 070 4 × 2 = 0 + 0.444 051 943 226 920 140 8;
  • 47) 0.444 051 943 226 920 140 8 × 2 = 0 + 0.888 103 886 453 840 281 6;
  • 48) 0.888 103 886 453 840 281 6 × 2 = 1 + 0.776 207 772 907 680 563 2;
  • 49) 0.776 207 772 907 680 563 2 × 2 = 1 + 0.552 415 545 815 361 126 4;
  • 50) 0.552 415 545 815 361 126 4 × 2 = 1 + 0.104 831 091 630 722 252 8;
  • 51) 0.104 831 091 630 722 252 8 × 2 = 0 + 0.209 662 183 261 444 505 6;
  • 52) 0.209 662 183 261 444 505 6 × 2 = 0 + 0.419 324 366 522 889 011 2;
  • 53) 0.419 324 366 522 889 011 2 × 2 = 0 + 0.838 648 733 045 778 022 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:


0.777 777 777 777 777 772 2(10) =

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

5. Positive number before normalization:

24.777 777 777 777 777 772 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:


24.777 777 777 777 777 772 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 24.777 777 777 777 777 772 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