0.000 000 000 000 000 000 008 73 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 000 000 008 73(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
0.000 000 000 000 000 000 008 73(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: 0.000 000 000 000 000 000 008 73.

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) 0.000 000 000 000 000 000 008 73 × 2 = 0 + 0.000 000 000 000 000 000 017 46;
  • 2) 0.000 000 000 000 000 000 017 46 × 2 = 0 + 0.000 000 000 000 000 000 034 92;
  • 3) 0.000 000 000 000 000 000 034 92 × 2 = 0 + 0.000 000 000 000 000 000 069 84;
  • 4) 0.000 000 000 000 000 000 069 84 × 2 = 0 + 0.000 000 000 000 000 000 139 68;
  • 5) 0.000 000 000 000 000 000 139 68 × 2 = 0 + 0.000 000 000 000 000 000 279 36;
  • 6) 0.000 000 000 000 000 000 279 36 × 2 = 0 + 0.000 000 000 000 000 000 558 72;
  • 7) 0.000 000 000 000 000 000 558 72 × 2 = 0 + 0.000 000 000 000 000 001 117 44;
  • 8) 0.000 000 000 000 000 001 117 44 × 2 = 0 + 0.000 000 000 000 000 002 234 88;
  • 9) 0.000 000 000 000 000 002 234 88 × 2 = 0 + 0.000 000 000 000 000 004 469 76;
  • 10) 0.000 000 000 000 000 004 469 76 × 2 = 0 + 0.000 000 000 000 000 008 939 52;
  • 11) 0.000 000 000 000 000 008 939 52 × 2 = 0 + 0.000 000 000 000 000 017 879 04;
  • 12) 0.000 000 000 000 000 017 879 04 × 2 = 0 + 0.000 000 000 000 000 035 758 08;
  • 13) 0.000 000 000 000 000 035 758 08 × 2 = 0 + 0.000 000 000 000 000 071 516 16;
  • 14) 0.000 000 000 000 000 071 516 16 × 2 = 0 + 0.000 000 000 000 000 143 032 32;
  • 15) 0.000 000 000 000 000 143 032 32 × 2 = 0 + 0.000 000 000 000 000 286 064 64;
  • 16) 0.000 000 000 000 000 286 064 64 × 2 = 0 + 0.000 000 000 000 000 572 129 28;
  • 17) 0.000 000 000 000 000 572 129 28 × 2 = 0 + 0.000 000 000 000 001 144 258 56;
  • 18) 0.000 000 000 000 001 144 258 56 × 2 = 0 + 0.000 000 000 000 002 288 517 12;
  • 19) 0.000 000 000 000 002 288 517 12 × 2 = 0 + 0.000 000 000 000 004 577 034 24;
  • 20) 0.000 000 000 000 004 577 034 24 × 2 = 0 + 0.000 000 000 000 009 154 068 48;
  • 21) 0.000 000 000 000 009 154 068 48 × 2 = 0 + 0.000 000 000 000 018 308 136 96;
  • 22) 0.000 000 000 000 018 308 136 96 × 2 = 0 + 0.000 000 000 000 036 616 273 92;
  • 23) 0.000 000 000 000 036 616 273 92 × 2 = 0 + 0.000 000 000 000 073 232 547 84;
  • 24) 0.000 000 000 000 073 232 547 84 × 2 = 0 + 0.000 000 000 000 146 465 095 68;
  • 25) 0.000 000 000 000 146 465 095 68 × 2 = 0 + 0.000 000 000 000 292 930 191 36;
  • 26) 0.000 000 000 000 292 930 191 36 × 2 = 0 + 0.000 000 000 000 585 860 382 72;
  • 27) 0.000 000 000 000 585 860 382 72 × 2 = 0 + 0.000 000 000 001 171 720 765 44;
  • 28) 0.000 000 000 001 171 720 765 44 × 2 = 0 + 0.000 000 000 002 343 441 530 88;
  • 29) 0.000 000 000 002 343 441 530 88 × 2 = 0 + 0.000 000 000 004 686 883 061 76;
  • 30) 0.000 000 000 004 686 883 061 76 × 2 = 0 + 0.000 000 000 009 373 766 123 52;
  • 31) 0.000 000 000 009 373 766 123 52 × 2 = 0 + 0.000 000 000 018 747 532 247 04;
  • 32) 0.000 000 000 018 747 532 247 04 × 2 = 0 + 0.000 000 000 037 495 064 494 08;
  • 33) 0.000 000 000 037 495 064 494 08 × 2 = 0 + 0.000 000 000 074 990 128 988 16;
  • 34) 0.000 000 000 074 990 128 988 16 × 2 = 0 + 0.000 000 000 149 980 257 976 32;
  • 35) 0.000 000 000 149 980 257 976 32 × 2 = 0 + 0.000 000 000 299 960 515 952 64;
  • 36) 0.000 000 000 299 960 515 952 64 × 2 = 0 + 0.000 000 000 599 921 031 905 28;
  • 37) 0.000 000 000 599 921 031 905 28 × 2 = 0 + 0.000 000 001 199 842 063 810 56;
  • 38) 0.000 000 001 199 842 063 810 56 × 2 = 0 + 0.000 000 002 399 684 127 621 12;
  • 39) 0.000 000 002 399 684 127 621 12 × 2 = 0 + 0.000 000 004 799 368 255 242 24;
  • 40) 0.000 000 004 799 368 255 242 24 × 2 = 0 + 0.000 000 009 598 736 510 484 48;
  • 41) 0.000 000 009 598 736 510 484 48 × 2 = 0 + 0.000 000 019 197 473 020 968 96;
  • 42) 0.000 000 019 197 473 020 968 96 × 2 = 0 + 0.000 000 038 394 946 041 937 92;
  • 43) 0.000 000 038 394 946 041 937 92 × 2 = 0 + 0.000 000 076 789 892 083 875 84;
  • 44) 0.000 000 076 789 892 083 875 84 × 2 = 0 + 0.000 000 153 579 784 167 751 68;
  • 45) 0.000 000 153 579 784 167 751 68 × 2 = 0 + 0.000 000 307 159 568 335 503 36;
  • 46) 0.000 000 307 159 568 335 503 36 × 2 = 0 + 0.000 000 614 319 136 671 006 72;
  • 47) 0.000 000 614 319 136 671 006 72 × 2 = 0 + 0.000 001 228 638 273 342 013 44;
  • 48) 0.000 001 228 638 273 342 013 44 × 2 = 0 + 0.000 002 457 276 546 684 026 88;
  • 49) 0.000 002 457 276 546 684 026 88 × 2 = 0 + 0.000 004 914 553 093 368 053 76;
  • 50) 0.000 004 914 553 093 368 053 76 × 2 = 0 + 0.000 009 829 106 186 736 107 52;
  • 51) 0.000 009 829 106 186 736 107 52 × 2 = 0 + 0.000 019 658 212 373 472 215 04;
  • 52) 0.000 019 658 212 373 472 215 04 × 2 = 0 + 0.000 039 316 424 746 944 430 08;
  • 53) 0.000 039 316 424 746 944 430 08 × 2 = 0 + 0.000 078 632 849 493 888 860 16;
  • 54) 0.000 078 632 849 493 888 860 16 × 2 = 0 + 0.000 157 265 698 987 777 720 32;
  • 55) 0.000 157 265 698 987 777 720 32 × 2 = 0 + 0.000 314 531 397 975 555 440 64;
  • 56) 0.000 314 531 397 975 555 440 64 × 2 = 0 + 0.000 629 062 795 951 110 881 28;
  • 57) 0.000 629 062 795 951 110 881 28 × 2 = 0 + 0.001 258 125 591 902 221 762 56;
  • 58) 0.001 258 125 591 902 221 762 56 × 2 = 0 + 0.002 516 251 183 804 443 525 12;
  • 59) 0.002 516 251 183 804 443 525 12 × 2 = 0 + 0.005 032 502 367 608 887 050 24;
  • 60) 0.005 032 502 367 608 887 050 24 × 2 = 0 + 0.010 065 004 735 217 774 100 48;
  • 61) 0.010 065 004 735 217 774 100 48 × 2 = 0 + 0.020 130 009 470 435 548 200 96;
  • 62) 0.020 130 009 470 435 548 200 96 × 2 = 0 + 0.040 260 018 940 871 096 401 92;
  • 63) 0.040 260 018 940 871 096 401 92 × 2 = 0 + 0.080 520 037 881 742 192 803 84;
  • 64) 0.080 520 037 881 742 192 803 84 × 2 = 0 + 0.161 040 075 763 484 385 607 68;
  • 65) 0.161 040 075 763 484 385 607 68 × 2 = 0 + 0.322 080 151 526 968 771 215 36;
  • 66) 0.322 080 151 526 968 771 215 36 × 2 = 0 + 0.644 160 303 053 937 542 430 72;
  • 67) 0.644 160 303 053 937 542 430 72 × 2 = 1 + 0.288 320 606 107 875 084 861 44;
  • 68) 0.288 320 606 107 875 084 861 44 × 2 = 0 + 0.576 641 212 215 750 169 722 88;
  • 69) 0.576 641 212 215 750 169 722 88 × 2 = 1 + 0.153 282 424 431 500 339 445 76;
  • 70) 0.153 282 424 431 500 339 445 76 × 2 = 0 + 0.306 564 848 863 000 678 891 52;
  • 71) 0.306 564 848 863 000 678 891 52 × 2 = 0 + 0.613 129 697 726 001 357 783 04;
  • 72) 0.613 129 697 726 001 357 783 04 × 2 = 1 + 0.226 259 395 452 002 715 566 08;
  • 73) 0.226 259 395 452 002 715 566 08 × 2 = 0 + 0.452 518 790 904 005 431 132 16;
  • 74) 0.452 518 790 904 005 431 132 16 × 2 = 0 + 0.905 037 581 808 010 862 264 32;
  • 75) 0.905 037 581 808 010 862 264 32 × 2 = 1 + 0.810 075 163 616 021 724 528 64;
  • 76) 0.810 075 163 616 021 724 528 64 × 2 = 1 + 0.620 150 327 232 043 449 057 28;
  • 77) 0.620 150 327 232 043 449 057 28 × 2 = 1 + 0.240 300 654 464 086 898 114 56;
  • 78) 0.240 300 654 464 086 898 114 56 × 2 = 0 + 0.480 601 308 928 173 796 229 12;
  • 79) 0.480 601 308 928 173 796 229 12 × 2 = 0 + 0.961 202 617 856 347 592 458 24;
  • 80) 0.961 202 617 856 347 592 458 24 × 2 = 1 + 0.922 405 235 712 695 184 916 48;
  • 81) 0.922 405 235 712 695 184 916 48 × 2 = 1 + 0.844 810 471 425 390 369 832 96;
  • 82) 0.844 810 471 425 390 369 832 96 × 2 = 1 + 0.689 620 942 850 780 739 665 92;
  • 83) 0.689 620 942 850 780 739 665 92 × 2 = 1 + 0.379 241 885 701 561 479 331 84;
  • 84) 0.379 241 885 701 561 479 331 84 × 2 = 0 + 0.758 483 771 403 122 958 663 68;
  • 85) 0.758 483 771 403 122 958 663 68 × 2 = 1 + 0.516 967 542 806 245 917 327 36;
  • 86) 0.516 967 542 806 245 917 327 36 × 2 = 1 + 0.033 935 085 612 491 834 654 72;
  • 87) 0.033 935 085 612 491 834 654 72 × 2 = 0 + 0.067 870 171 224 983 669 309 44;
  • 88) 0.067 870 171 224 983 669 309 44 × 2 = 0 + 0.135 740 342 449 967 338 618 88;
  • 89) 0.135 740 342 449 967 338 618 88 × 2 = 0 + 0.271 480 684 899 934 677 237 76;
  • 90) 0.271 480 684 899 934 677 237 76 × 2 = 0 + 0.542 961 369 799 869 354 475 52;
  • 91) 0.542 961 369 799 869 354 475 52 × 2 = 1 + 0.085 922 739 599 738 708 951 04;
  • 92) 0.085 922 739 599 738 708 951 04 × 2 = 0 + 0.171 845 479 199 477 417 902 08;
  • 93) 0.171 845 479 199 477 417 902 08 × 2 = 0 + 0.343 690 958 398 954 835 804 16;
  • 94) 0.343 690 958 398 954 835 804 16 × 2 = 0 + 0.687 381 916 797 909 671 608 32;
  • 95) 0.687 381 916 797 909 671 608 32 × 2 = 1 + 0.374 763 833 595 819 343 216 64;
  • 96) 0.374 763 833 595 819 343 216 64 × 2 = 0 + 0.749 527 667 191 638 686 433 28;
  • 97) 0.749 527 667 191 638 686 433 28 × 2 = 1 + 0.499 055 334 383 277 372 866 56;
  • 98) 0.499 055 334 383 277 372 866 56 × 2 = 0 + 0.998 110 668 766 554 745 733 12;
  • 99) 0.998 110 668 766 554 745 733 12 × 2 = 1 + 0.996 221 337 533 109 491 466 24;
  • 100) 0.996 221 337 533 109 491 466 24 × 2 = 1 + 0.992 442 675 066 218 982 932 48;
  • 101) 0.992 442 675 066 218 982 932 48 × 2 = 1 + 0.984 885 350 132 437 965 864 96;
  • 102) 0.984 885 350 132 437 965 864 96 × 2 = 1 + 0.969 770 700 264 875 931 729 92;
  • 103) 0.969 770 700 264 875 931 729 92 × 2 = 1 + 0.939 541 400 529 751 863 459 84;
  • 104) 0.939 541 400 529 751 863 459 84 × 2 = 1 + 0.879 082 801 059 503 726 919 68;
  • 105) 0.879 082 801 059 503 726 919 68 × 2 = 1 + 0.758 165 602 119 007 453 839 36;
  • 106) 0.758 165 602 119 007 453 839 36 × 2 = 1 + 0.516 331 204 238 014 907 678 72;
  • 107) 0.516 331 204 238 014 907 678 72 × 2 = 1 + 0.032 662 408 476 029 815 357 44;
  • 108) 0.032 662 408 476 029 815 357 44 × 2 = 0 + 0.065 324 816 952 059 630 714 88;
  • 109) 0.065 324 816 952 059 630 714 88 × 2 = 0 + 0.130 649 633 904 119 261 429 76;
  • 110) 0.130 649 633 904 119 261 429 76 × 2 = 0 + 0.261 299 267 808 238 522 859 52;
  • 111) 0.261 299 267 808 238 522 859 52 × 2 = 0 + 0.522 598 535 616 477 045 719 04;
  • 112) 0.522 598 535 616 477 045 719 04 × 2 = 1 + 0.045 197 071 232 954 091 438 08;
  • 113) 0.045 197 071 232 954 091 438 08 × 2 = 0 + 0.090 394 142 465 908 182 876 16;
  • 114) 0.090 394 142 465 908 182 876 16 × 2 = 0 + 0.180 788 284 931 816 365 752 32;
  • 115) 0.180 788 284 931 816 365 752 32 × 2 = 0 + 0.361 576 569 863 632 731 504 64;
  • 116) 0.361 576 569 863 632 731 504 64 × 2 = 0 + 0.723 153 139 727 265 463 009 28;
  • 117) 0.723 153 139 727 265 463 009 28 × 2 = 1 + 0.446 306 279 454 530 926 018 56;
  • 118) 0.446 306 279 454 530 926 018 56 × 2 = 0 + 0.892 612 558 909 061 852 037 12;
  • 119) 0.892 612 558 909 061 852 037 12 × 2 = 1 + 0.785 225 117 818 123 704 074 24;

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:


0.000 000 000 000 000 000 008 73(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1001 0011 1001 1110 1100 0010 0010 1011 1111 1110 0001 0000 101(2)

5. Positive number before normalization:

0.000 000 000 000 000 000 008 73(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1001 0011 1001 1110 1100 0010 0010 1011 1111 1110 0001 0000 101(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 67 positions to the right, so that only one non zero digit remains to the left of it:


0.000 000 000 000 000 000 008 73(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1001 0011 1001 1110 1100 0010 0010 1011 1111 1110 0001 0000 101(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1001 0011 1001 1110 1100 0010 0010 1011 1111 1110 0001 0000 101(2) × 20 =

1.0100 1001 1100 1111 0110 0001 0001 0101 1111 1111 0000 1000 0101(2) × 2-67


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): -67

Mantissa (not normalized):
1.0100 1001 1100 1111 0110 0001 0001 0101 1111 1111 0000 1000 0101


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

-67 + 2(11-1) - 1 =

(-67 + 1 023)(10) =

956(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;
  • 956 ÷ 2 = 478 + 0;
  • 478 ÷ 2 = 239 + 0;
  • 239 ÷ 2 = 119 + 1;
  • 119 ÷ 2 = 59 + 1;
  • 59 ÷ 2 = 29 + 1;
  • 29 ÷ 2 = 14 + 1;
  • 14 ÷ 2 = 7 + 0;
  • 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) =

956(10) =

011 1011 1100(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. 0100 1001 1100 1111 0110 0001 0001 0101 1111 1111 0000 1000 0101 =

0100 1001 1100 1111 0110 0001 0001 0101 1111 1111 0000 1000 0101


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 1011 1100

Mantissa (52 bits) =
0100 1001 1100 1111 0110 0001 0001 0101 1111 1111 0000 1000 0101


Decimal number 0.000 000 000 000 000 000 008 73 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1011 1100 - 0100 1001 1100 1111 0110 0001 0001 0101 1111 1111 0000 1000 0101

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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