0.333 333 333 148 296 162 562 473 909 929 394 721 945 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.333 333 333 148 296 162 562 473 909 929 394 721 945(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
0.333 333 333 148 296 162 562 473 909 929 394 721 945(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: 0.333 333 333 148 296 162 562 473 909 929 394 721 945.

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) 0.333 333 333 148 296 162 562 473 909 929 394 721 945 × 2 = 0 + 0.666 666 666 296 592 325 124 947 819 858 789 443 89;
  • 2) 0.666 666 666 296 592 325 124 947 819 858 789 443 89 × 2 = 1 + 0.333 333 332 593 184 650 249 895 639 717 578 887 78;
  • 3) 0.333 333 332 593 184 650 249 895 639 717 578 887 78 × 2 = 0 + 0.666 666 665 186 369 300 499 791 279 435 157 775 56;
  • 4) 0.666 666 665 186 369 300 499 791 279 435 157 775 56 × 2 = 1 + 0.333 333 330 372 738 600 999 582 558 870 315 551 12;
  • 5) 0.333 333 330 372 738 600 999 582 558 870 315 551 12 × 2 = 0 + 0.666 666 660 745 477 201 999 165 117 740 631 102 24;
  • 6) 0.666 666 660 745 477 201 999 165 117 740 631 102 24 × 2 = 1 + 0.333 333 321 490 954 403 998 330 235 481 262 204 48;
  • 7) 0.333 333 321 490 954 403 998 330 235 481 262 204 48 × 2 = 0 + 0.666 666 642 981 908 807 996 660 470 962 524 408 96;
  • 8) 0.666 666 642 981 908 807 996 660 470 962 524 408 96 × 2 = 1 + 0.333 333 285 963 817 615 993 320 941 925 048 817 92;
  • 9) 0.333 333 285 963 817 615 993 320 941 925 048 817 92 × 2 = 0 + 0.666 666 571 927 635 231 986 641 883 850 097 635 84;
  • 10) 0.666 666 571 927 635 231 986 641 883 850 097 635 84 × 2 = 1 + 0.333 333 143 855 270 463 973 283 767 700 195 271 68;
  • 11) 0.333 333 143 855 270 463 973 283 767 700 195 271 68 × 2 = 0 + 0.666 666 287 710 540 927 946 567 535 400 390 543 36;
  • 12) 0.666 666 287 710 540 927 946 567 535 400 390 543 36 × 2 = 1 + 0.333 332 575 421 081 855 893 135 070 800 781 086 72;
  • 13) 0.333 332 575 421 081 855 893 135 070 800 781 086 72 × 2 = 0 + 0.666 665 150 842 163 711 786 270 141 601 562 173 44;
  • 14) 0.666 665 150 842 163 711 786 270 141 601 562 173 44 × 2 = 1 + 0.333 330 301 684 327 423 572 540 283 203 124 346 88;
  • 15) 0.333 330 301 684 327 423 572 540 283 203 124 346 88 × 2 = 0 + 0.666 660 603 368 654 847 145 080 566 406 248 693 76;
  • 16) 0.666 660 603 368 654 847 145 080 566 406 248 693 76 × 2 = 1 + 0.333 321 206 737 309 694 290 161 132 812 497 387 52;
  • 17) 0.333 321 206 737 309 694 290 161 132 812 497 387 52 × 2 = 0 + 0.666 642 413 474 619 388 580 322 265 624 994 775 04;
  • 18) 0.666 642 413 474 619 388 580 322 265 624 994 775 04 × 2 = 1 + 0.333 284 826 949 238 777 160 644 531 249 989 550 08;
  • 19) 0.333 284 826 949 238 777 160 644 531 249 989 550 08 × 2 = 0 + 0.666 569 653 898 477 554 321 289 062 499 979 100 16;
  • 20) 0.666 569 653 898 477 554 321 289 062 499 979 100 16 × 2 = 1 + 0.333 139 307 796 955 108 642 578 124 999 958 200 32;
  • 21) 0.333 139 307 796 955 108 642 578 124 999 958 200 32 × 2 = 0 + 0.666 278 615 593 910 217 285 156 249 999 916 400 64;
  • 22) 0.666 278 615 593 910 217 285 156 249 999 916 400 64 × 2 = 1 + 0.332 557 231 187 820 434 570 312 499 999 832 801 28;
  • 23) 0.332 557 231 187 820 434 570 312 499 999 832 801 28 × 2 = 0 + 0.665 114 462 375 640 869 140 624 999 999 665 602 56;
  • 24) 0.665 114 462 375 640 869 140 624 999 999 665 602 56 × 2 = 1 + 0.330 228 924 751 281 738 281 249 999 999 331 205 12;
  • 25) 0.330 228 924 751 281 738 281 249 999 999 331 205 12 × 2 = 0 + 0.660 457 849 502 563 476 562 499 999 998 662 410 24;
  • 26) 0.660 457 849 502 563 476 562 499 999 998 662 410 24 × 2 = 1 + 0.320 915 699 005 126 953 124 999 999 997 324 820 48;
  • 27) 0.320 915 699 005 126 953 124 999 999 997 324 820 48 × 2 = 0 + 0.641 831 398 010 253 906 249 999 999 994 649 640 96;
  • 28) 0.641 831 398 010 253 906 249 999 999 994 649 640 96 × 2 = 1 + 0.283 662 796 020 507 812 499 999 999 989 299 281 92;
  • 29) 0.283 662 796 020 507 812 499 999 999 989 299 281 92 × 2 = 0 + 0.567 325 592 041 015 624 999 999 999 978 598 563 84;
  • 30) 0.567 325 592 041 015 624 999 999 999 978 598 563 84 × 2 = 1 + 0.134 651 184 082 031 249 999 999 999 957 197 127 68;
  • 31) 0.134 651 184 082 031 249 999 999 999 957 197 127 68 × 2 = 0 + 0.269 302 368 164 062 499 999 999 999 914 394 255 36;
  • 32) 0.269 302 368 164 062 499 999 999 999 914 394 255 36 × 2 = 0 + 0.538 604 736 328 124 999 999 999 999 828 788 510 72;
  • 33) 0.538 604 736 328 124 999 999 999 999 828 788 510 72 × 2 = 1 + 0.077 209 472 656 249 999 999 999 999 657 577 021 44;
  • 34) 0.077 209 472 656 249 999 999 999 999 657 577 021 44 × 2 = 0 + 0.154 418 945 312 499 999 999 999 999 315 154 042 88;
  • 35) 0.154 418 945 312 499 999 999 999 999 315 154 042 88 × 2 = 0 + 0.308 837 890 624 999 999 999 999 998 630 308 085 76;
  • 36) 0.308 837 890 624 999 999 999 999 998 630 308 085 76 × 2 = 0 + 0.617 675 781 249 999 999 999 999 997 260 616 171 52;
  • 37) 0.617 675 781 249 999 999 999 999 997 260 616 171 52 × 2 = 1 + 0.235 351 562 499 999 999 999 999 994 521 232 343 04;
  • 38) 0.235 351 562 499 999 999 999 999 994 521 232 343 04 × 2 = 0 + 0.470 703 124 999 999 999 999 999 989 042 464 686 08;
  • 39) 0.470 703 124 999 999 999 999 999 989 042 464 686 08 × 2 = 0 + 0.941 406 249 999 999 999 999 999 978 084 929 372 16;
  • 40) 0.941 406 249 999 999 999 999 999 978 084 929 372 16 × 2 = 1 + 0.882 812 499 999 999 999 999 999 956 169 858 744 32;
  • 41) 0.882 812 499 999 999 999 999 999 956 169 858 744 32 × 2 = 1 + 0.765 624 999 999 999 999 999 999 912 339 717 488 64;
  • 42) 0.765 624 999 999 999 999 999 999 912 339 717 488 64 × 2 = 1 + 0.531 249 999 999 999 999 999 999 824 679 434 977 28;
  • 43) 0.531 249 999 999 999 999 999 999 824 679 434 977 28 × 2 = 1 + 0.062 499 999 999 999 999 999 999 649 358 869 954 56;
  • 44) 0.062 499 999 999 999 999 999 999 649 358 869 954 56 × 2 = 0 + 0.124 999 999 999 999 999 999 999 298 717 739 909 12;
  • 45) 0.124 999 999 999 999 999 999 999 298 717 739 909 12 × 2 = 0 + 0.249 999 999 999 999 999 999 998 597 435 479 818 24;
  • 46) 0.249 999 999 999 999 999 999 998 597 435 479 818 24 × 2 = 0 + 0.499 999 999 999 999 999 999 997 194 870 959 636 48;
  • 47) 0.499 999 999 999 999 999 999 997 194 870 959 636 48 × 2 = 0 + 0.999 999 999 999 999 999 999 994 389 741 919 272 96;
  • 48) 0.999 999 999 999 999 999 999 994 389 741 919 272 96 × 2 = 1 + 0.999 999 999 999 999 999 999 988 779 483 838 545 92;
  • 49) 0.999 999 999 999 999 999 999 988 779 483 838 545 92 × 2 = 1 + 0.999 999 999 999 999 999 999 977 558 967 677 091 84;
  • 50) 0.999 999 999 999 999 999 999 977 558 967 677 091 84 × 2 = 1 + 0.999 999 999 999 999 999 999 955 117 935 354 183 68;
  • 51) 0.999 999 999 999 999 999 999 955 117 935 354 183 68 × 2 = 1 + 0.999 999 999 999 999 999 999 910 235 870 708 367 36;
  • 52) 0.999 999 999 999 999 999 999 910 235 870 708 367 36 × 2 = 1 + 0.999 999 999 999 999 999 999 820 471 741 416 734 72;
  • 53) 0.999 999 999 999 999 999 999 820 471 741 416 734 72 × 2 = 1 + 0.999 999 999 999 999 999 999 640 943 482 833 469 44;
  • 54) 0.999 999 999 999 999 999 999 640 943 482 833 469 44 × 2 = 1 + 0.999 999 999 999 999 999 999 281 886 965 666 938 88;

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:


0.333 333 333 148 296 162 562 473 909 929 394 721 945(10) =

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

5. Positive number before normalization:

0.333 333 333 148 296 162 562 473 909 929 394 721 945(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:


0.333 333 333 148 296 162 562 473 909 929 394 721 945(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 0.333 333 333 148 296 162 562 473 909 929 394 721 945 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