0.333 333 333 148 296 162 562 473 909 929 394 715 6 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.333 333 333 148 296 162 562 473 909 929 394 715 6(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
0.333 333 333 148 296 162 562 473 909 929 394 715 6(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: 0.333 333 333 148 296 162 562 473 909 929 394 715 6.

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) 0.333 333 333 148 296 162 562 473 909 929 394 715 6 × 2 = 0 + 0.666 666 666 296 592 325 124 947 819 858 789 431 2;
  • 2) 0.666 666 666 296 592 325 124 947 819 858 789 431 2 × 2 = 1 + 0.333 333 332 593 184 650 249 895 639 717 578 862 4;
  • 3) 0.333 333 332 593 184 650 249 895 639 717 578 862 4 × 2 = 0 + 0.666 666 665 186 369 300 499 791 279 435 157 724 8;
  • 4) 0.666 666 665 186 369 300 499 791 279 435 157 724 8 × 2 = 1 + 0.333 333 330 372 738 600 999 582 558 870 315 449 6;
  • 5) 0.333 333 330 372 738 600 999 582 558 870 315 449 6 × 2 = 0 + 0.666 666 660 745 477 201 999 165 117 740 630 899 2;
  • 6) 0.666 666 660 745 477 201 999 165 117 740 630 899 2 × 2 = 1 + 0.333 333 321 490 954 403 998 330 235 481 261 798 4;
  • 7) 0.333 333 321 490 954 403 998 330 235 481 261 798 4 × 2 = 0 + 0.666 666 642 981 908 807 996 660 470 962 523 596 8;
  • 8) 0.666 666 642 981 908 807 996 660 470 962 523 596 8 × 2 = 1 + 0.333 333 285 963 817 615 993 320 941 925 047 193 6;
  • 9) 0.333 333 285 963 817 615 993 320 941 925 047 193 6 × 2 = 0 + 0.666 666 571 927 635 231 986 641 883 850 094 387 2;
  • 10) 0.666 666 571 927 635 231 986 641 883 850 094 387 2 × 2 = 1 + 0.333 333 143 855 270 463 973 283 767 700 188 774 4;
  • 11) 0.333 333 143 855 270 463 973 283 767 700 188 774 4 × 2 = 0 + 0.666 666 287 710 540 927 946 567 535 400 377 548 8;
  • 12) 0.666 666 287 710 540 927 946 567 535 400 377 548 8 × 2 = 1 + 0.333 332 575 421 081 855 893 135 070 800 755 097 6;
  • 13) 0.333 332 575 421 081 855 893 135 070 800 755 097 6 × 2 = 0 + 0.666 665 150 842 163 711 786 270 141 601 510 195 2;
  • 14) 0.666 665 150 842 163 711 786 270 141 601 510 195 2 × 2 = 1 + 0.333 330 301 684 327 423 572 540 283 203 020 390 4;
  • 15) 0.333 330 301 684 327 423 572 540 283 203 020 390 4 × 2 = 0 + 0.666 660 603 368 654 847 145 080 566 406 040 780 8;
  • 16) 0.666 660 603 368 654 847 145 080 566 406 040 780 8 × 2 = 1 + 0.333 321 206 737 309 694 290 161 132 812 081 561 6;
  • 17) 0.333 321 206 737 309 694 290 161 132 812 081 561 6 × 2 = 0 + 0.666 642 413 474 619 388 580 322 265 624 163 123 2;
  • 18) 0.666 642 413 474 619 388 580 322 265 624 163 123 2 × 2 = 1 + 0.333 284 826 949 238 777 160 644 531 248 326 246 4;
  • 19) 0.333 284 826 949 238 777 160 644 531 248 326 246 4 × 2 = 0 + 0.666 569 653 898 477 554 321 289 062 496 652 492 8;
  • 20) 0.666 569 653 898 477 554 321 289 062 496 652 492 8 × 2 = 1 + 0.333 139 307 796 955 108 642 578 124 993 304 985 6;
  • 21) 0.333 139 307 796 955 108 642 578 124 993 304 985 6 × 2 = 0 + 0.666 278 615 593 910 217 285 156 249 986 609 971 2;
  • 22) 0.666 278 615 593 910 217 285 156 249 986 609 971 2 × 2 = 1 + 0.332 557 231 187 820 434 570 312 499 973 219 942 4;
  • 23) 0.332 557 231 187 820 434 570 312 499 973 219 942 4 × 2 = 0 + 0.665 114 462 375 640 869 140 624 999 946 439 884 8;
  • 24) 0.665 114 462 375 640 869 140 624 999 946 439 884 8 × 2 = 1 + 0.330 228 924 751 281 738 281 249 999 892 879 769 6;
  • 25) 0.330 228 924 751 281 738 281 249 999 892 879 769 6 × 2 = 0 + 0.660 457 849 502 563 476 562 499 999 785 759 539 2;
  • 26) 0.660 457 849 502 563 476 562 499 999 785 759 539 2 × 2 = 1 + 0.320 915 699 005 126 953 124 999 999 571 519 078 4;
  • 27) 0.320 915 699 005 126 953 124 999 999 571 519 078 4 × 2 = 0 + 0.641 831 398 010 253 906 249 999 999 143 038 156 8;
  • 28) 0.641 831 398 010 253 906 249 999 999 143 038 156 8 × 2 = 1 + 0.283 662 796 020 507 812 499 999 998 286 076 313 6;
  • 29) 0.283 662 796 020 507 812 499 999 998 286 076 313 6 × 2 = 0 + 0.567 325 592 041 015 624 999 999 996 572 152 627 2;
  • 30) 0.567 325 592 041 015 624 999 999 996 572 152 627 2 × 2 = 1 + 0.134 651 184 082 031 249 999 999 993 144 305 254 4;
  • 31) 0.134 651 184 082 031 249 999 999 993 144 305 254 4 × 2 = 0 + 0.269 302 368 164 062 499 999 999 986 288 610 508 8;
  • 32) 0.269 302 368 164 062 499 999 999 986 288 610 508 8 × 2 = 0 + 0.538 604 736 328 124 999 999 999 972 577 221 017 6;
  • 33) 0.538 604 736 328 124 999 999 999 972 577 221 017 6 × 2 = 1 + 0.077 209 472 656 249 999 999 999 945 154 442 035 2;
  • 34) 0.077 209 472 656 249 999 999 999 945 154 442 035 2 × 2 = 0 + 0.154 418 945 312 499 999 999 999 890 308 884 070 4;
  • 35) 0.154 418 945 312 499 999 999 999 890 308 884 070 4 × 2 = 0 + 0.308 837 890 624 999 999 999 999 780 617 768 140 8;
  • 36) 0.308 837 890 624 999 999 999 999 780 617 768 140 8 × 2 = 0 + 0.617 675 781 249 999 999 999 999 561 235 536 281 6;
  • 37) 0.617 675 781 249 999 999 999 999 561 235 536 281 6 × 2 = 1 + 0.235 351 562 499 999 999 999 999 122 471 072 563 2;
  • 38) 0.235 351 562 499 999 999 999 999 122 471 072 563 2 × 2 = 0 + 0.470 703 124 999 999 999 999 998 244 942 145 126 4;
  • 39) 0.470 703 124 999 999 999 999 998 244 942 145 126 4 × 2 = 0 + 0.941 406 249 999 999 999 999 996 489 884 290 252 8;
  • 40) 0.941 406 249 999 999 999 999 996 489 884 290 252 8 × 2 = 1 + 0.882 812 499 999 999 999 999 992 979 768 580 505 6;
  • 41) 0.882 812 499 999 999 999 999 992 979 768 580 505 6 × 2 = 1 + 0.765 624 999 999 999 999 999 985 959 537 161 011 2;
  • 42) 0.765 624 999 999 999 999 999 985 959 537 161 011 2 × 2 = 1 + 0.531 249 999 999 999 999 999 971 919 074 322 022 4;
  • 43) 0.531 249 999 999 999 999 999 971 919 074 322 022 4 × 2 = 1 + 0.062 499 999 999 999 999 999 943 838 148 644 044 8;
  • 44) 0.062 499 999 999 999 999 999 943 838 148 644 044 8 × 2 = 0 + 0.124 999 999 999 999 999 999 887 676 297 288 089 6;
  • 45) 0.124 999 999 999 999 999 999 887 676 297 288 089 6 × 2 = 0 + 0.249 999 999 999 999 999 999 775 352 594 576 179 2;
  • 46) 0.249 999 999 999 999 999 999 775 352 594 576 179 2 × 2 = 0 + 0.499 999 999 999 999 999 999 550 705 189 152 358 4;
  • 47) 0.499 999 999 999 999 999 999 550 705 189 152 358 4 × 2 = 0 + 0.999 999 999 999 999 999 999 101 410 378 304 716 8;
  • 48) 0.999 999 999 999 999 999 999 101 410 378 304 716 8 × 2 = 1 + 0.999 999 999 999 999 999 998 202 820 756 609 433 6;
  • 49) 0.999 999 999 999 999 999 998 202 820 756 609 433 6 × 2 = 1 + 0.999 999 999 999 999 999 996 405 641 513 218 867 2;
  • 50) 0.999 999 999 999 999 999 996 405 641 513 218 867 2 × 2 = 1 + 0.999 999 999 999 999 999 992 811 283 026 437 734 4;
  • 51) 0.999 999 999 999 999 999 992 811 283 026 437 734 4 × 2 = 1 + 0.999 999 999 999 999 999 985 622 566 052 875 468 8;
  • 52) 0.999 999 999 999 999 999 985 622 566 052 875 468 8 × 2 = 1 + 0.999 999 999 999 999 999 971 245 132 105 750 937 6;
  • 53) 0.999 999 999 999 999 999 971 245 132 105 750 937 6 × 2 = 1 + 0.999 999 999 999 999 999 942 490 264 211 501 875 2;
  • 54) 0.999 999 999 999 999 999 942 490 264 211 501 875 2 × 2 = 1 + 0.999 999 999 999 999 999 884 980 528 423 003 750 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:


0.333 333 333 148 296 162 562 473 909 929 394 715 6(10) =

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

5. Positive number before normalization:

0.333 333 333 148 296 162 562 473 909 929 394 715 6(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:


0.333 333 333 148 296 162 562 473 909 929 394 715 6(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 0.333 333 333 148 296 162 562 473 909 929 394 715 6 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