0.333 333 333 148 296 162 562 473 909 929 394 721 868 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.333 333 333 148 296 162 562 473 909 929 394 721 868(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
0.333 333 333 148 296 162 562 473 909 929 394 721 868(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: 0.333 333 333 148 296 162 562 473 909 929 394 721 868.

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) 0.333 333 333 148 296 162 562 473 909 929 394 721 868 × 2 = 0 + 0.666 666 666 296 592 325 124 947 819 858 789 443 736;
  • 2) 0.666 666 666 296 592 325 124 947 819 858 789 443 736 × 2 = 1 + 0.333 333 332 593 184 650 249 895 639 717 578 887 472;
  • 3) 0.333 333 332 593 184 650 249 895 639 717 578 887 472 × 2 = 0 + 0.666 666 665 186 369 300 499 791 279 435 157 774 944;
  • 4) 0.666 666 665 186 369 300 499 791 279 435 157 774 944 × 2 = 1 + 0.333 333 330 372 738 600 999 582 558 870 315 549 888;
  • 5) 0.333 333 330 372 738 600 999 582 558 870 315 549 888 × 2 = 0 + 0.666 666 660 745 477 201 999 165 117 740 631 099 776;
  • 6) 0.666 666 660 745 477 201 999 165 117 740 631 099 776 × 2 = 1 + 0.333 333 321 490 954 403 998 330 235 481 262 199 552;
  • 7) 0.333 333 321 490 954 403 998 330 235 481 262 199 552 × 2 = 0 + 0.666 666 642 981 908 807 996 660 470 962 524 399 104;
  • 8) 0.666 666 642 981 908 807 996 660 470 962 524 399 104 × 2 = 1 + 0.333 333 285 963 817 615 993 320 941 925 048 798 208;
  • 9) 0.333 333 285 963 817 615 993 320 941 925 048 798 208 × 2 = 0 + 0.666 666 571 927 635 231 986 641 883 850 097 596 416;
  • 10) 0.666 666 571 927 635 231 986 641 883 850 097 596 416 × 2 = 1 + 0.333 333 143 855 270 463 973 283 767 700 195 192 832;
  • 11) 0.333 333 143 855 270 463 973 283 767 700 195 192 832 × 2 = 0 + 0.666 666 287 710 540 927 946 567 535 400 390 385 664;
  • 12) 0.666 666 287 710 540 927 946 567 535 400 390 385 664 × 2 = 1 + 0.333 332 575 421 081 855 893 135 070 800 780 771 328;
  • 13) 0.333 332 575 421 081 855 893 135 070 800 780 771 328 × 2 = 0 + 0.666 665 150 842 163 711 786 270 141 601 561 542 656;
  • 14) 0.666 665 150 842 163 711 786 270 141 601 561 542 656 × 2 = 1 + 0.333 330 301 684 327 423 572 540 283 203 123 085 312;
  • 15) 0.333 330 301 684 327 423 572 540 283 203 123 085 312 × 2 = 0 + 0.666 660 603 368 654 847 145 080 566 406 246 170 624;
  • 16) 0.666 660 603 368 654 847 145 080 566 406 246 170 624 × 2 = 1 + 0.333 321 206 737 309 694 290 161 132 812 492 341 248;
  • 17) 0.333 321 206 737 309 694 290 161 132 812 492 341 248 × 2 = 0 + 0.666 642 413 474 619 388 580 322 265 624 984 682 496;
  • 18) 0.666 642 413 474 619 388 580 322 265 624 984 682 496 × 2 = 1 + 0.333 284 826 949 238 777 160 644 531 249 969 364 992;
  • 19) 0.333 284 826 949 238 777 160 644 531 249 969 364 992 × 2 = 0 + 0.666 569 653 898 477 554 321 289 062 499 938 729 984;
  • 20) 0.666 569 653 898 477 554 321 289 062 499 938 729 984 × 2 = 1 + 0.333 139 307 796 955 108 642 578 124 999 877 459 968;
  • 21) 0.333 139 307 796 955 108 642 578 124 999 877 459 968 × 2 = 0 + 0.666 278 615 593 910 217 285 156 249 999 754 919 936;
  • 22) 0.666 278 615 593 910 217 285 156 249 999 754 919 936 × 2 = 1 + 0.332 557 231 187 820 434 570 312 499 999 509 839 872;
  • 23) 0.332 557 231 187 820 434 570 312 499 999 509 839 872 × 2 = 0 + 0.665 114 462 375 640 869 140 624 999 999 019 679 744;
  • 24) 0.665 114 462 375 640 869 140 624 999 999 019 679 744 × 2 = 1 + 0.330 228 924 751 281 738 281 249 999 998 039 359 488;
  • 25) 0.330 228 924 751 281 738 281 249 999 998 039 359 488 × 2 = 0 + 0.660 457 849 502 563 476 562 499 999 996 078 718 976;
  • 26) 0.660 457 849 502 563 476 562 499 999 996 078 718 976 × 2 = 1 + 0.320 915 699 005 126 953 124 999 999 992 157 437 952;
  • 27) 0.320 915 699 005 126 953 124 999 999 992 157 437 952 × 2 = 0 + 0.641 831 398 010 253 906 249 999 999 984 314 875 904;
  • 28) 0.641 831 398 010 253 906 249 999 999 984 314 875 904 × 2 = 1 + 0.283 662 796 020 507 812 499 999 999 968 629 751 808;
  • 29) 0.283 662 796 020 507 812 499 999 999 968 629 751 808 × 2 = 0 + 0.567 325 592 041 015 624 999 999 999 937 259 503 616;
  • 30) 0.567 325 592 041 015 624 999 999 999 937 259 503 616 × 2 = 1 + 0.134 651 184 082 031 249 999 999 999 874 519 007 232;
  • 31) 0.134 651 184 082 031 249 999 999 999 874 519 007 232 × 2 = 0 + 0.269 302 368 164 062 499 999 999 999 749 038 014 464;
  • 32) 0.269 302 368 164 062 499 999 999 999 749 038 014 464 × 2 = 0 + 0.538 604 736 328 124 999 999 999 999 498 076 028 928;
  • 33) 0.538 604 736 328 124 999 999 999 999 498 076 028 928 × 2 = 1 + 0.077 209 472 656 249 999 999 999 998 996 152 057 856;
  • 34) 0.077 209 472 656 249 999 999 999 998 996 152 057 856 × 2 = 0 + 0.154 418 945 312 499 999 999 999 997 992 304 115 712;
  • 35) 0.154 418 945 312 499 999 999 999 997 992 304 115 712 × 2 = 0 + 0.308 837 890 624 999 999 999 999 995 984 608 231 424;
  • 36) 0.308 837 890 624 999 999 999 999 995 984 608 231 424 × 2 = 0 + 0.617 675 781 249 999 999 999 999 991 969 216 462 848;
  • 37) 0.617 675 781 249 999 999 999 999 991 969 216 462 848 × 2 = 1 + 0.235 351 562 499 999 999 999 999 983 938 432 925 696;
  • 38) 0.235 351 562 499 999 999 999 999 983 938 432 925 696 × 2 = 0 + 0.470 703 124 999 999 999 999 999 967 876 865 851 392;
  • 39) 0.470 703 124 999 999 999 999 999 967 876 865 851 392 × 2 = 0 + 0.941 406 249 999 999 999 999 999 935 753 731 702 784;
  • 40) 0.941 406 249 999 999 999 999 999 935 753 731 702 784 × 2 = 1 + 0.882 812 499 999 999 999 999 999 871 507 463 405 568;
  • 41) 0.882 812 499 999 999 999 999 999 871 507 463 405 568 × 2 = 1 + 0.765 624 999 999 999 999 999 999 743 014 926 811 136;
  • 42) 0.765 624 999 999 999 999 999 999 743 014 926 811 136 × 2 = 1 + 0.531 249 999 999 999 999 999 999 486 029 853 622 272;
  • 43) 0.531 249 999 999 999 999 999 999 486 029 853 622 272 × 2 = 1 + 0.062 499 999 999 999 999 999 998 972 059 707 244 544;
  • 44) 0.062 499 999 999 999 999 999 998 972 059 707 244 544 × 2 = 0 + 0.124 999 999 999 999 999 999 997 944 119 414 489 088;
  • 45) 0.124 999 999 999 999 999 999 997 944 119 414 489 088 × 2 = 0 + 0.249 999 999 999 999 999 999 995 888 238 828 978 176;
  • 46) 0.249 999 999 999 999 999 999 995 888 238 828 978 176 × 2 = 0 + 0.499 999 999 999 999 999 999 991 776 477 657 956 352;
  • 47) 0.499 999 999 999 999 999 999 991 776 477 657 956 352 × 2 = 0 + 0.999 999 999 999 999 999 999 983 552 955 315 912 704;
  • 48) 0.999 999 999 999 999 999 999 983 552 955 315 912 704 × 2 = 1 + 0.999 999 999 999 999 999 999 967 105 910 631 825 408;
  • 49) 0.999 999 999 999 999 999 999 967 105 910 631 825 408 × 2 = 1 + 0.999 999 999 999 999 999 999 934 211 821 263 650 816;
  • 50) 0.999 999 999 999 999 999 999 934 211 821 263 650 816 × 2 = 1 + 0.999 999 999 999 999 999 999 868 423 642 527 301 632;
  • 51) 0.999 999 999 999 999 999 999 868 423 642 527 301 632 × 2 = 1 + 0.999 999 999 999 999 999 999 736 847 285 054 603 264;
  • 52) 0.999 999 999 999 999 999 999 736 847 285 054 603 264 × 2 = 1 + 0.999 999 999 999 999 999 999 473 694 570 109 206 528;
  • 53) 0.999 999 999 999 999 999 999 473 694 570 109 206 528 × 2 = 1 + 0.999 999 999 999 999 999 998 947 389 140 218 413 056;
  • 54) 0.999 999 999 999 999 999 998 947 389 140 218 413 056 × 2 = 1 + 0.999 999 999 999 999 999 997 894 778 280 436 826 112;

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:


0.333 333 333 148 296 162 562 473 909 929 394 721 868(10) =

0.0101 0101 0101 0101 0101 0101 0101 0100 1000 1001 1110 0001 1111 11(2)

5. Positive number before normalization:

0.333 333 333 148 296 162 562 473 909 929 394 721 868(10) =

0.0101 0101 0101 0101 0101 0101 0101 0100 1000 1001 1110 0001 1111 11(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 2 positions to the right, so that only one non zero digit remains to the left of it:


0.333 333 333 148 296 162 562 473 909 929 394 721 868(10) =

0.0101 0101 0101 0101 0101 0101 0101 0100 1000 1001 1110 0001 1111 11(2) =

0.0101 0101 0101 0101 0101 0101 0101 0100 1000 1001 1110 0001 1111 11(2) × 20 =

1.0101 0101 0101 0101 0101 0101 0101 0010 0010 0111 1000 0111 1111(2) × 2-2


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): -2

Mantissa (not normalized):
1.0101 0101 0101 0101 0101 0101 0101 0010 0010 0111 1000 0111 1111


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

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

(-2 + 1 023)(10) =

1 021(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;
  • 1 021 ÷ 2 = 510 + 1;
  • 510 ÷ 2 = 255 + 0;
  • 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) =

1021(10) =

011 1111 1101(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. 0101 0101 0101 0101 0101 0101 0101 0010 0010 0111 1000 0111 1111 =

0101 0101 0101 0101 0101 0101 0101 0010 0010 0111 1000 0111 1111


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 1101

Mantissa (52 bits) =
0101 0101 0101 0101 0101 0101 0101 0010 0010 0111 1000 0111 1111


Decimal number 0.333 333 333 148 296 162 562 473 909 929 394 721 868 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1111 1101 - 0101 0101 0101 0101 0101 0101 0101 0010 0010 0111 1000 0111 1111

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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