0.785 430 024 824 850 447 891 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.785 430 024 824 850 447 891(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
0.785 430 024 824 850 447 891(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: 0.
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;
  • 0 ÷ 2 = 0 + 0;

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.

0(10) =

0(2)


3. Convert to binary (base 2) the fractional part: 0.785 430 024 824 850 447 891.

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) 0.785 430 024 824 850 447 891 × 2 = 1 + 0.570 860 049 649 700 895 782;
  • 2) 0.570 860 049 649 700 895 782 × 2 = 1 + 0.141 720 099 299 401 791 564;
  • 3) 0.141 720 099 299 401 791 564 × 2 = 0 + 0.283 440 198 598 803 583 128;
  • 4) 0.283 440 198 598 803 583 128 × 2 = 0 + 0.566 880 397 197 607 166 256;
  • 5) 0.566 880 397 197 607 166 256 × 2 = 1 + 0.133 760 794 395 214 332 512;
  • 6) 0.133 760 794 395 214 332 512 × 2 = 0 + 0.267 521 588 790 428 665 024;
  • 7) 0.267 521 588 790 428 665 024 × 2 = 0 + 0.535 043 177 580 857 330 048;
  • 8) 0.535 043 177 580 857 330 048 × 2 = 1 + 0.070 086 355 161 714 660 096;
  • 9) 0.070 086 355 161 714 660 096 × 2 = 0 + 0.140 172 710 323 429 320 192;
  • 10) 0.140 172 710 323 429 320 192 × 2 = 0 + 0.280 345 420 646 858 640 384;
  • 11) 0.280 345 420 646 858 640 384 × 2 = 0 + 0.560 690 841 293 717 280 768;
  • 12) 0.560 690 841 293 717 280 768 × 2 = 1 + 0.121 381 682 587 434 561 536;
  • 13) 0.121 381 682 587 434 561 536 × 2 = 0 + 0.242 763 365 174 869 123 072;
  • 14) 0.242 763 365 174 869 123 072 × 2 = 0 + 0.485 526 730 349 738 246 144;
  • 15) 0.485 526 730 349 738 246 144 × 2 = 0 + 0.971 053 460 699 476 492 288;
  • 16) 0.971 053 460 699 476 492 288 × 2 = 1 + 0.942 106 921 398 952 984 576;
  • 17) 0.942 106 921 398 952 984 576 × 2 = 1 + 0.884 213 842 797 905 969 152;
  • 18) 0.884 213 842 797 905 969 152 × 2 = 1 + 0.768 427 685 595 811 938 304;
  • 19) 0.768 427 685 595 811 938 304 × 2 = 1 + 0.536 855 371 191 623 876 608;
  • 20) 0.536 855 371 191 623 876 608 × 2 = 1 + 0.073 710 742 383 247 753 216;
  • 21) 0.073 710 742 383 247 753 216 × 2 = 0 + 0.147 421 484 766 495 506 432;
  • 22) 0.147 421 484 766 495 506 432 × 2 = 0 + 0.294 842 969 532 991 012 864;
  • 23) 0.294 842 969 532 991 012 864 × 2 = 0 + 0.589 685 939 065 982 025 728;
  • 24) 0.589 685 939 065 982 025 728 × 2 = 1 + 0.179 371 878 131 964 051 456;
  • 25) 0.179 371 878 131 964 051 456 × 2 = 0 + 0.358 743 756 263 928 102 912;
  • 26) 0.358 743 756 263 928 102 912 × 2 = 0 + 0.717 487 512 527 856 205 824;
  • 27) 0.717 487 512 527 856 205 824 × 2 = 1 + 0.434 975 025 055 712 411 648;
  • 28) 0.434 975 025 055 712 411 648 × 2 = 0 + 0.869 950 050 111 424 823 296;
  • 29) 0.869 950 050 111 424 823 296 × 2 = 1 + 0.739 900 100 222 849 646 592;
  • 30) 0.739 900 100 222 849 646 592 × 2 = 1 + 0.479 800 200 445 699 293 184;
  • 31) 0.479 800 200 445 699 293 184 × 2 = 0 + 0.959 600 400 891 398 586 368;
  • 32) 0.959 600 400 891 398 586 368 × 2 = 1 + 0.919 200 801 782 797 172 736;
  • 33) 0.919 200 801 782 797 172 736 × 2 = 1 + 0.838 401 603 565 594 345 472;
  • 34) 0.838 401 603 565 594 345 472 × 2 = 1 + 0.676 803 207 131 188 690 944;
  • 35) 0.676 803 207 131 188 690 944 × 2 = 1 + 0.353 606 414 262 377 381 888;
  • 36) 0.353 606 414 262 377 381 888 × 2 = 0 + 0.707 212 828 524 754 763 776;
  • 37) 0.707 212 828 524 754 763 776 × 2 = 1 + 0.414 425 657 049 509 527 552;
  • 38) 0.414 425 657 049 509 527 552 × 2 = 0 + 0.828 851 314 099 019 055 104;
  • 39) 0.828 851 314 099 019 055 104 × 2 = 1 + 0.657 702 628 198 038 110 208;
  • 40) 0.657 702 628 198 038 110 208 × 2 = 1 + 0.315 405 256 396 076 220 416;
  • 41) 0.315 405 256 396 076 220 416 × 2 = 0 + 0.630 810 512 792 152 440 832;
  • 42) 0.630 810 512 792 152 440 832 × 2 = 1 + 0.261 621 025 584 304 881 664;
  • 43) 0.261 621 025 584 304 881 664 × 2 = 0 + 0.523 242 051 168 609 763 328;
  • 44) 0.523 242 051 168 609 763 328 × 2 = 1 + 0.046 484 102 337 219 526 656;
  • 45) 0.046 484 102 337 219 526 656 × 2 = 0 + 0.092 968 204 674 439 053 312;
  • 46) 0.092 968 204 674 439 053 312 × 2 = 0 + 0.185 936 409 348 878 106 624;
  • 47) 0.185 936 409 348 878 106 624 × 2 = 0 + 0.371 872 818 697 756 213 248;
  • 48) 0.371 872 818 697 756 213 248 × 2 = 0 + 0.743 745 637 395 512 426 496;
  • 49) 0.743 745 637 395 512 426 496 × 2 = 1 + 0.487 491 274 791 024 852 992;
  • 50) 0.487 491 274 791 024 852 992 × 2 = 0 + 0.974 982 549 582 049 705 984;
  • 51) 0.974 982 549 582 049 705 984 × 2 = 1 + 0.949 965 099 164 099 411 968;
  • 52) 0.949 965 099 164 099 411 968 × 2 = 1 + 0.899 930 198 328 198 823 936;
  • 53) 0.899 930 198 328 198 823 936 × 2 = 1 + 0.799 860 396 656 397 647 872;

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:


0.785 430 024 824 850 447 891(10) =

0.1100 1001 0001 0001 1111 0001 0010 1101 1110 1011 0101 0000 1011 1(2)

5. Positive number before normalization:

0.785 430 024 824 850 447 891(10) =

0.1100 1001 0001 0001 1111 0001 0010 1101 1110 1011 0101 0000 1011 1(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 1 positions to the right, so that only one non zero digit remains to the left of it:


0.785 430 024 824 850 447 891(10) =

0.1100 1001 0001 0001 1111 0001 0010 1101 1110 1011 0101 0000 1011 1(2) =

0.1100 1001 0001 0001 1111 0001 0010 1101 1110 1011 0101 0000 1011 1(2) × 20 =

1.1001 0010 0010 0011 1110 0010 0101 1011 1101 0110 1010 0001 0111(2) × 2-1


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): -1

Mantissa (not normalized):
1.1001 0010 0010 0011 1110 0010 0101 1011 1101 0110 1010 0001 0111


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

Exponent (unadjusted) + 2(11-1) - 1 =

-1 + 2(11-1) - 1 =

(-1 + 1 023)(10) =

1 022(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;
  • 1 022 ÷ 2 = 511 + 0;
  • 511 ÷ 2 = 255 + 1;
  • 255 ÷ 2 = 127 + 1;
  • 127 ÷ 2 = 63 + 1;
  • 63 ÷ 2 = 31 + 1;
  • 31 ÷ 2 = 15 + 1;
  • 15 ÷ 2 = 7 + 1;
  • 7 ÷ 2 = 3 + 1;
  • 3 ÷ 2 = 1 + 1;
  • 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) =

1022(10) =

011 1111 1110(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, only if necessary (not the case here).


Mantissa (normalized) =

1. 1001 0010 0010 0011 1110 0010 0101 1011 1101 0110 1010 0001 0111 =

1001 0010 0010 0011 1110 0010 0101 1011 1101 0110 1010 0001 0111


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) =
011 1111 1110

Mantissa (52 bits) =
1001 0010 0010 0011 1110 0010 0101 1011 1101 0110 1010 0001 0111


Decimal number 0.785 430 024 824 850 447 891 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1111 1110 - 1001 0010 0010 0011 1110 0010 0101 1011 1101 0110 1010 0001 0111

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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