0.000 000 000 000 002 220 446 049 247 5 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 002 220 446 049 247 5(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
0.000 000 000 000 002 220 446 049 247 5(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: 0.000 000 000 000 002 220 446 049 247 5.

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) 0.000 000 000 000 002 220 446 049 247 5 × 2 = 0 + 0.000 000 000 000 004 440 892 098 495;
  • 2) 0.000 000 000 000 004 440 892 098 495 × 2 = 0 + 0.000 000 000 000 008 881 784 196 99;
  • 3) 0.000 000 000 000 008 881 784 196 99 × 2 = 0 + 0.000 000 000 000 017 763 568 393 98;
  • 4) 0.000 000 000 000 017 763 568 393 98 × 2 = 0 + 0.000 000 000 000 035 527 136 787 96;
  • 5) 0.000 000 000 000 035 527 136 787 96 × 2 = 0 + 0.000 000 000 000 071 054 273 575 92;
  • 6) 0.000 000 000 000 071 054 273 575 92 × 2 = 0 + 0.000 000 000 000 142 108 547 151 84;
  • 7) 0.000 000 000 000 142 108 547 151 84 × 2 = 0 + 0.000 000 000 000 284 217 094 303 68;
  • 8) 0.000 000 000 000 284 217 094 303 68 × 2 = 0 + 0.000 000 000 000 568 434 188 607 36;
  • 9) 0.000 000 000 000 568 434 188 607 36 × 2 = 0 + 0.000 000 000 001 136 868 377 214 72;
  • 10) 0.000 000 000 001 136 868 377 214 72 × 2 = 0 + 0.000 000 000 002 273 736 754 429 44;
  • 11) 0.000 000 000 002 273 736 754 429 44 × 2 = 0 + 0.000 000 000 004 547 473 508 858 88;
  • 12) 0.000 000 000 004 547 473 508 858 88 × 2 = 0 + 0.000 000 000 009 094 947 017 717 76;
  • 13) 0.000 000 000 009 094 947 017 717 76 × 2 = 0 + 0.000 000 000 018 189 894 035 435 52;
  • 14) 0.000 000 000 018 189 894 035 435 52 × 2 = 0 + 0.000 000 000 036 379 788 070 871 04;
  • 15) 0.000 000 000 036 379 788 070 871 04 × 2 = 0 + 0.000 000 000 072 759 576 141 742 08;
  • 16) 0.000 000 000 072 759 576 141 742 08 × 2 = 0 + 0.000 000 000 145 519 152 283 484 16;
  • 17) 0.000 000 000 145 519 152 283 484 16 × 2 = 0 + 0.000 000 000 291 038 304 566 968 32;
  • 18) 0.000 000 000 291 038 304 566 968 32 × 2 = 0 + 0.000 000 000 582 076 609 133 936 64;
  • 19) 0.000 000 000 582 076 609 133 936 64 × 2 = 0 + 0.000 000 001 164 153 218 267 873 28;
  • 20) 0.000 000 001 164 153 218 267 873 28 × 2 = 0 + 0.000 000 002 328 306 436 535 746 56;
  • 21) 0.000 000 002 328 306 436 535 746 56 × 2 = 0 + 0.000 000 004 656 612 873 071 493 12;
  • 22) 0.000 000 004 656 612 873 071 493 12 × 2 = 0 + 0.000 000 009 313 225 746 142 986 24;
  • 23) 0.000 000 009 313 225 746 142 986 24 × 2 = 0 + 0.000 000 018 626 451 492 285 972 48;
  • 24) 0.000 000 018 626 451 492 285 972 48 × 2 = 0 + 0.000 000 037 252 902 984 571 944 96;
  • 25) 0.000 000 037 252 902 984 571 944 96 × 2 = 0 + 0.000 000 074 505 805 969 143 889 92;
  • 26) 0.000 000 074 505 805 969 143 889 92 × 2 = 0 + 0.000 000 149 011 611 938 287 779 84;
  • 27) 0.000 000 149 011 611 938 287 779 84 × 2 = 0 + 0.000 000 298 023 223 876 575 559 68;
  • 28) 0.000 000 298 023 223 876 575 559 68 × 2 = 0 + 0.000 000 596 046 447 753 151 119 36;
  • 29) 0.000 000 596 046 447 753 151 119 36 × 2 = 0 + 0.000 001 192 092 895 506 302 238 72;
  • 30) 0.000 001 192 092 895 506 302 238 72 × 2 = 0 + 0.000 002 384 185 791 012 604 477 44;
  • 31) 0.000 002 384 185 791 012 604 477 44 × 2 = 0 + 0.000 004 768 371 582 025 208 954 88;
  • 32) 0.000 004 768 371 582 025 208 954 88 × 2 = 0 + 0.000 009 536 743 164 050 417 909 76;
  • 33) 0.000 009 536 743 164 050 417 909 76 × 2 = 0 + 0.000 019 073 486 328 100 835 819 52;
  • 34) 0.000 019 073 486 328 100 835 819 52 × 2 = 0 + 0.000 038 146 972 656 201 671 639 04;
  • 35) 0.000 038 146 972 656 201 671 639 04 × 2 = 0 + 0.000 076 293 945 312 403 343 278 08;
  • 36) 0.000 076 293 945 312 403 343 278 08 × 2 = 0 + 0.000 152 587 890 624 806 686 556 16;
  • 37) 0.000 152 587 890 624 806 686 556 16 × 2 = 0 + 0.000 305 175 781 249 613 373 112 32;
  • 38) 0.000 305 175 781 249 613 373 112 32 × 2 = 0 + 0.000 610 351 562 499 226 746 224 64;
  • 39) 0.000 610 351 562 499 226 746 224 64 × 2 = 0 + 0.001 220 703 124 998 453 492 449 28;
  • 40) 0.001 220 703 124 998 453 492 449 28 × 2 = 0 + 0.002 441 406 249 996 906 984 898 56;
  • 41) 0.002 441 406 249 996 906 984 898 56 × 2 = 0 + 0.004 882 812 499 993 813 969 797 12;
  • 42) 0.004 882 812 499 993 813 969 797 12 × 2 = 0 + 0.009 765 624 999 987 627 939 594 24;
  • 43) 0.009 765 624 999 987 627 939 594 24 × 2 = 0 + 0.019 531 249 999 975 255 879 188 48;
  • 44) 0.019 531 249 999 975 255 879 188 48 × 2 = 0 + 0.039 062 499 999 950 511 758 376 96;
  • 45) 0.039 062 499 999 950 511 758 376 96 × 2 = 0 + 0.078 124 999 999 901 023 516 753 92;
  • 46) 0.078 124 999 999 901 023 516 753 92 × 2 = 0 + 0.156 249 999 999 802 047 033 507 84;
  • 47) 0.156 249 999 999 802 047 033 507 84 × 2 = 0 + 0.312 499 999 999 604 094 067 015 68;
  • 48) 0.312 499 999 999 604 094 067 015 68 × 2 = 0 + 0.624 999 999 999 208 188 134 031 36;
  • 49) 0.624 999 999 999 208 188 134 031 36 × 2 = 1 + 0.249 999 999 998 416 376 268 062 72;
  • 50) 0.249 999 999 998 416 376 268 062 72 × 2 = 0 + 0.499 999 999 996 832 752 536 125 44;
  • 51) 0.499 999 999 996 832 752 536 125 44 × 2 = 0 + 0.999 999 999 993 665 505 072 250 88;
  • 52) 0.999 999 999 993 665 505 072 250 88 × 2 = 1 + 0.999 999 999 987 331 010 144 501 76;
  • 53) 0.999 999 999 987 331 010 144 501 76 × 2 = 1 + 0.999 999 999 974 662 020 289 003 52;
  • 54) 0.999 999 999 974 662 020 289 003 52 × 2 = 1 + 0.999 999 999 949 324 040 578 007 04;
  • 55) 0.999 999 999 949 324 040 578 007 04 × 2 = 1 + 0.999 999 999 898 648 081 156 014 08;
  • 56) 0.999 999 999 898 648 081 156 014 08 × 2 = 1 + 0.999 999 999 797 296 162 312 028 16;
  • 57) 0.999 999 999 797 296 162 312 028 16 × 2 = 1 + 0.999 999 999 594 592 324 624 056 32;
  • 58) 0.999 999 999 594 592 324 624 056 32 × 2 = 1 + 0.999 999 999 189 184 649 248 112 64;
  • 59) 0.999 999 999 189 184 649 248 112 64 × 2 = 1 + 0.999 999 998 378 369 298 496 225 28;
  • 60) 0.999 999 998 378 369 298 496 225 28 × 2 = 1 + 0.999 999 996 756 738 596 992 450 56;
  • 61) 0.999 999 996 756 738 596 992 450 56 × 2 = 1 + 0.999 999 993 513 477 193 984 901 12;
  • 62) 0.999 999 993 513 477 193 984 901 12 × 2 = 1 + 0.999 999 987 026 954 387 969 802 24;
  • 63) 0.999 999 987 026 954 387 969 802 24 × 2 = 1 + 0.999 999 974 053 908 775 939 604 48;
  • 64) 0.999 999 974 053 908 775 939 604 48 × 2 = 1 + 0.999 999 948 107 817 551 879 208 96;
  • 65) 0.999 999 948 107 817 551 879 208 96 × 2 = 1 + 0.999 999 896 215 635 103 758 417 92;
  • 66) 0.999 999 896 215 635 103 758 417 92 × 2 = 1 + 0.999 999 792 431 270 207 516 835 84;
  • 67) 0.999 999 792 431 270 207 516 835 84 × 2 = 1 + 0.999 999 584 862 540 415 033 671 68;
  • 68) 0.999 999 584 862 540 415 033 671 68 × 2 = 1 + 0.999 999 169 725 080 830 067 343 36;
  • 69) 0.999 999 169 725 080 830 067 343 36 × 2 = 1 + 0.999 998 339 450 161 660 134 686 72;
  • 70) 0.999 998 339 450 161 660 134 686 72 × 2 = 1 + 0.999 996 678 900 323 320 269 373 44;
  • 71) 0.999 996 678 900 323 320 269 373 44 × 2 = 1 + 0.999 993 357 800 646 640 538 746 88;
  • 72) 0.999 993 357 800 646 640 538 746 88 × 2 = 1 + 0.999 986 715 601 293 281 077 493 76;
  • 73) 0.999 986 715 601 293 281 077 493 76 × 2 = 1 + 0.999 973 431 202 586 562 154 987 52;
  • 74) 0.999 973 431 202 586 562 154 987 52 × 2 = 1 + 0.999 946 862 405 173 124 309 975 04;
  • 75) 0.999 946 862 405 173 124 309 975 04 × 2 = 1 + 0.999 893 724 810 346 248 619 950 08;
  • 76) 0.999 893 724 810 346 248 619 950 08 × 2 = 1 + 0.999 787 449 620 692 497 239 900 16;
  • 77) 0.999 787 449 620 692 497 239 900 16 × 2 = 1 + 0.999 574 899 241 384 994 479 800 32;
  • 78) 0.999 574 899 241 384 994 479 800 32 × 2 = 1 + 0.999 149 798 482 769 988 959 600 64;
  • 79) 0.999 149 798 482 769 988 959 600 64 × 2 = 1 + 0.998 299 596 965 539 977 919 201 28;
  • 80) 0.998 299 596 965 539 977 919 201 28 × 2 = 1 + 0.996 599 193 931 079 955 838 402 56;
  • 81) 0.996 599 193 931 079 955 838 402 56 × 2 = 1 + 0.993 198 387 862 159 911 676 805 12;
  • 82) 0.993 198 387 862 159 911 676 805 12 × 2 = 1 + 0.986 396 775 724 319 823 353 610 24;
  • 83) 0.986 396 775 724 319 823 353 610 24 × 2 = 1 + 0.972 793 551 448 639 646 707 220 48;
  • 84) 0.972 793 551 448 639 646 707 220 48 × 2 = 1 + 0.945 587 102 897 279 293 414 440 96;
  • 85) 0.945 587 102 897 279 293 414 440 96 × 2 = 1 + 0.891 174 205 794 558 586 828 881 92;
  • 86) 0.891 174 205 794 558 586 828 881 92 × 2 = 1 + 0.782 348 411 589 117 173 657 763 84;
  • 87) 0.782 348 411 589 117 173 657 763 84 × 2 = 1 + 0.564 696 823 178 234 347 315 527 68;
  • 88) 0.564 696 823 178 234 347 315 527 68 × 2 = 1 + 0.129 393 646 356 468 694 631 055 36;
  • 89) 0.129 393 646 356 468 694 631 055 36 × 2 = 0 + 0.258 787 292 712 937 389 262 110 72;
  • 90) 0.258 787 292 712 937 389 262 110 72 × 2 = 0 + 0.517 574 585 425 874 778 524 221 44;
  • 91) 0.517 574 585 425 874 778 524 221 44 × 2 = 1 + 0.035 149 170 851 749 557 048 442 88;
  • 92) 0.035 149 170 851 749 557 048 442 88 × 2 = 0 + 0.070 298 341 703 499 114 096 885 76;
  • 93) 0.070 298 341 703 499 114 096 885 76 × 2 = 0 + 0.140 596 683 406 998 228 193 771 52;
  • 94) 0.140 596 683 406 998 228 193 771 52 × 2 = 0 + 0.281 193 366 813 996 456 387 543 04;
  • 95) 0.281 193 366 813 996 456 387 543 04 × 2 = 0 + 0.562 386 733 627 992 912 775 086 08;
  • 96) 0.562 386 733 627 992 912 775 086 08 × 2 = 1 + 0.124 773 467 255 985 825 550 172 16;
  • 97) 0.124 773 467 255 985 825 550 172 16 × 2 = 0 + 0.249 546 934 511 971 651 100 344 32;
  • 98) 0.249 546 934 511 971 651 100 344 32 × 2 = 0 + 0.499 093 869 023 943 302 200 688 64;
  • 99) 0.499 093 869 023 943 302 200 688 64 × 2 = 0 + 0.998 187 738 047 886 604 401 377 28;
  • 100) 0.998 187 738 047 886 604 401 377 28 × 2 = 1 + 0.996 375 476 095 773 208 802 754 56;
  • 101) 0.996 375 476 095 773 208 802 754 56 × 2 = 1 + 0.992 750 952 191 546 417 605 509 12;

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:


0.000 000 000 000 002 220 446 049 247 5(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1001 1111 1111 1111 1111 1111 1111 1111 1111 1111 0010 0001 0001 1(2)

5. Positive number before normalization:

0.000 000 000 000 002 220 446 049 247 5(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1001 1111 1111 1111 1111 1111 1111 1111 1111 1111 0010 0001 0001 1(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 49 positions to the right, so that only one non zero digit remains to the left of it:


0.000 000 000 000 002 220 446 049 247 5(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1001 1111 1111 1111 1111 1111 1111 1111 1111 1111 0010 0001 0001 1(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1001 1111 1111 1111 1111 1111 1111 1111 1111 1111 0010 0001 0001 1(2) × 20 =

1.0011 1111 1111 1111 1111 1111 1111 1111 1111 1110 0100 0010 0011(2) × 2-49


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): -49

Mantissa (not normalized):
1.0011 1111 1111 1111 1111 1111 1111 1111 1111 1110 0100 0010 0011


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

-49 + 2(11-1) - 1 =

(-49 + 1 023)(10) =

974(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;
  • 974 ÷ 2 = 487 + 0;
  • 487 ÷ 2 = 243 + 1;
  • 243 ÷ 2 = 121 + 1;
  • 121 ÷ 2 = 60 + 1;
  • 60 ÷ 2 = 30 + 0;
  • 30 ÷ 2 = 15 + 0;
  • 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) =

974(10) =

011 1100 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. 0011 1111 1111 1111 1111 1111 1111 1111 1111 1110 0100 0010 0011 =

0011 1111 1111 1111 1111 1111 1111 1111 1111 1110 0100 0010 0011


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 1100 1110

Mantissa (52 bits) =
0011 1111 1111 1111 1111 1111 1111 1111 1111 1110 0100 0010 0011


Decimal number 0.000 000 000 000 002 220 446 049 247 5 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1100 1110 - 0011 1111 1111 1111 1111 1111 1111 1111 1111 1110 0100 0010 0011

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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