0.545 253 866 332 628 829 603 504 3 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.545 253 866 332 628 829 603 504 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
0.545 253 866 332 628 829 603 504 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: 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: 0.545 253 866 332 628 829 603 504 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) 0.545 253 866 332 628 829 603 504 3 × 2 = 1 + 0.090 507 732 665 257 659 207 008 6;
  • 2) 0.090 507 732 665 257 659 207 008 6 × 2 = 0 + 0.181 015 465 330 515 318 414 017 2;
  • 3) 0.181 015 465 330 515 318 414 017 2 × 2 = 0 + 0.362 030 930 661 030 636 828 034 4;
  • 4) 0.362 030 930 661 030 636 828 034 4 × 2 = 0 + 0.724 061 861 322 061 273 656 068 8;
  • 5) 0.724 061 861 322 061 273 656 068 8 × 2 = 1 + 0.448 123 722 644 122 547 312 137 6;
  • 6) 0.448 123 722 644 122 547 312 137 6 × 2 = 0 + 0.896 247 445 288 245 094 624 275 2;
  • 7) 0.896 247 445 288 245 094 624 275 2 × 2 = 1 + 0.792 494 890 576 490 189 248 550 4;
  • 8) 0.792 494 890 576 490 189 248 550 4 × 2 = 1 + 0.584 989 781 152 980 378 497 100 8;
  • 9) 0.584 989 781 152 980 378 497 100 8 × 2 = 1 + 0.169 979 562 305 960 756 994 201 6;
  • 10) 0.169 979 562 305 960 756 994 201 6 × 2 = 0 + 0.339 959 124 611 921 513 988 403 2;
  • 11) 0.339 959 124 611 921 513 988 403 2 × 2 = 0 + 0.679 918 249 223 843 027 976 806 4;
  • 12) 0.679 918 249 223 843 027 976 806 4 × 2 = 1 + 0.359 836 498 447 686 055 953 612 8;
  • 13) 0.359 836 498 447 686 055 953 612 8 × 2 = 0 + 0.719 672 996 895 372 111 907 225 6;
  • 14) 0.719 672 996 895 372 111 907 225 6 × 2 = 1 + 0.439 345 993 790 744 223 814 451 2;
  • 15) 0.439 345 993 790 744 223 814 451 2 × 2 = 0 + 0.878 691 987 581 488 447 628 902 4;
  • 16) 0.878 691 987 581 488 447 628 902 4 × 2 = 1 + 0.757 383 975 162 976 895 257 804 8;
  • 17) 0.757 383 975 162 976 895 257 804 8 × 2 = 1 + 0.514 767 950 325 953 790 515 609 6;
  • 18) 0.514 767 950 325 953 790 515 609 6 × 2 = 1 + 0.029 535 900 651 907 581 031 219 2;
  • 19) 0.029 535 900 651 907 581 031 219 2 × 2 = 0 + 0.059 071 801 303 815 162 062 438 4;
  • 20) 0.059 071 801 303 815 162 062 438 4 × 2 = 0 + 0.118 143 602 607 630 324 124 876 8;
  • 21) 0.118 143 602 607 630 324 124 876 8 × 2 = 0 + 0.236 287 205 215 260 648 249 753 6;
  • 22) 0.236 287 205 215 260 648 249 753 6 × 2 = 0 + 0.472 574 410 430 521 296 499 507 2;
  • 23) 0.472 574 410 430 521 296 499 507 2 × 2 = 0 + 0.945 148 820 861 042 592 999 014 4;
  • 24) 0.945 148 820 861 042 592 999 014 4 × 2 = 1 + 0.890 297 641 722 085 185 998 028 8;
  • 25) 0.890 297 641 722 085 185 998 028 8 × 2 = 1 + 0.780 595 283 444 170 371 996 057 6;
  • 26) 0.780 595 283 444 170 371 996 057 6 × 2 = 1 + 0.561 190 566 888 340 743 992 115 2;
  • 27) 0.561 190 566 888 340 743 992 115 2 × 2 = 1 + 0.122 381 133 776 681 487 984 230 4;
  • 28) 0.122 381 133 776 681 487 984 230 4 × 2 = 0 + 0.244 762 267 553 362 975 968 460 8;
  • 29) 0.244 762 267 553 362 975 968 460 8 × 2 = 0 + 0.489 524 535 106 725 951 936 921 6;
  • 30) 0.489 524 535 106 725 951 936 921 6 × 2 = 0 + 0.979 049 070 213 451 903 873 843 2;
  • 31) 0.979 049 070 213 451 903 873 843 2 × 2 = 1 + 0.958 098 140 426 903 807 747 686 4;
  • 32) 0.958 098 140 426 903 807 747 686 4 × 2 = 1 + 0.916 196 280 853 807 615 495 372 8;
  • 33) 0.916 196 280 853 807 615 495 372 8 × 2 = 1 + 0.832 392 561 707 615 230 990 745 6;
  • 34) 0.832 392 561 707 615 230 990 745 6 × 2 = 1 + 0.664 785 123 415 230 461 981 491 2;
  • 35) 0.664 785 123 415 230 461 981 491 2 × 2 = 1 + 0.329 570 246 830 460 923 962 982 4;
  • 36) 0.329 570 246 830 460 923 962 982 4 × 2 = 0 + 0.659 140 493 660 921 847 925 964 8;
  • 37) 0.659 140 493 660 921 847 925 964 8 × 2 = 1 + 0.318 280 987 321 843 695 851 929 6;
  • 38) 0.318 280 987 321 843 695 851 929 6 × 2 = 0 + 0.636 561 974 643 687 391 703 859 2;
  • 39) 0.636 561 974 643 687 391 703 859 2 × 2 = 1 + 0.273 123 949 287 374 783 407 718 4;
  • 40) 0.273 123 949 287 374 783 407 718 4 × 2 = 0 + 0.546 247 898 574 749 566 815 436 8;
  • 41) 0.546 247 898 574 749 566 815 436 8 × 2 = 1 + 0.092 495 797 149 499 133 630 873 6;
  • 42) 0.092 495 797 149 499 133 630 873 6 × 2 = 0 + 0.184 991 594 298 998 267 261 747 2;
  • 43) 0.184 991 594 298 998 267 261 747 2 × 2 = 0 + 0.369 983 188 597 996 534 523 494 4;
  • 44) 0.369 983 188 597 996 534 523 494 4 × 2 = 0 + 0.739 966 377 195 993 069 046 988 8;
  • 45) 0.739 966 377 195 993 069 046 988 8 × 2 = 1 + 0.479 932 754 391 986 138 093 977 6;
  • 46) 0.479 932 754 391 986 138 093 977 6 × 2 = 0 + 0.959 865 508 783 972 276 187 955 2;
  • 47) 0.959 865 508 783 972 276 187 955 2 × 2 = 1 + 0.919 731 017 567 944 552 375 910 4;
  • 48) 0.919 731 017 567 944 552 375 910 4 × 2 = 1 + 0.839 462 035 135 889 104 751 820 8;
  • 49) 0.839 462 035 135 889 104 751 820 8 × 2 = 1 + 0.678 924 070 271 778 209 503 641 6;
  • 50) 0.678 924 070 271 778 209 503 641 6 × 2 = 1 + 0.357 848 140 543 556 419 007 283 2;
  • 51) 0.357 848 140 543 556 419 007 283 2 × 2 = 0 + 0.715 696 281 087 112 838 014 566 4;
  • 52) 0.715 696 281 087 112 838 014 566 4 × 2 = 1 + 0.431 392 562 174 225 676 029 132 8;
  • 53) 0.431 392 562 174 225 676 029 132 8 × 2 = 0 + 0.862 785 124 348 451 352 058 265 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:


0.545 253 866 332 628 829 603 504 3(10) =

0.1000 1011 1001 0101 1100 0001 1110 0011 1110 1010 1000 1011 1101 0(2)

5. Positive number before normalization:

0.545 253 866 332 628 829 603 504 3(10) =

0.1000 1011 1001 0101 1100 0001 1110 0011 1110 1010 1000 1011 1101 0(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:


0.545 253 866 332 628 829 603 504 3(10) =

0.1000 1011 1001 0101 1100 0001 1110 0011 1110 1010 1000 1011 1101 0(2) =

0.1000 1011 1001 0101 1100 0001 1110 0011 1110 1010 1000 1011 1101 0(2) × 20 =

1.0001 0111 0010 1011 1000 0011 1100 0111 1101 0101 0001 0111 1010(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):
1.0001 0111 0010 1011 1000 0011 1100 0111 1101 0101 0001 0111 1010


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. 0001 0111 0010 1011 1000 0011 1100 0111 1101 0101 0001 0111 1010 =

0001 0111 0010 1011 1000 0011 1100 0111 1101 0101 0001 0111 1010


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) =
0001 0111 0010 1011 1000 0011 1100 0111 1101 0101 0001 0111 1010


Decimal number 0.545 253 866 332 628 829 603 504 3 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1111 1110 - 0001 0111 0010 1011 1000 0011 1100 0111 1101 0101 0001 0111 1010

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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