0.000 000 000 000 000 000 008 534 92 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 000 000 008 534 92(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
0.000 000 000 000 000 000 008 534 92(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: 0.000 000 000 000 000 000 008 534 92.

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) 0.000 000 000 000 000 000 008 534 92 × 2 = 0 + 0.000 000 000 000 000 000 017 069 84;
  • 2) 0.000 000 000 000 000 000 017 069 84 × 2 = 0 + 0.000 000 000 000 000 000 034 139 68;
  • 3) 0.000 000 000 000 000 000 034 139 68 × 2 = 0 + 0.000 000 000 000 000 000 068 279 36;
  • 4) 0.000 000 000 000 000 000 068 279 36 × 2 = 0 + 0.000 000 000 000 000 000 136 558 72;
  • 5) 0.000 000 000 000 000 000 136 558 72 × 2 = 0 + 0.000 000 000 000 000 000 273 117 44;
  • 6) 0.000 000 000 000 000 000 273 117 44 × 2 = 0 + 0.000 000 000 000 000 000 546 234 88;
  • 7) 0.000 000 000 000 000 000 546 234 88 × 2 = 0 + 0.000 000 000 000 000 001 092 469 76;
  • 8) 0.000 000 000 000 000 001 092 469 76 × 2 = 0 + 0.000 000 000 000 000 002 184 939 52;
  • 9) 0.000 000 000 000 000 002 184 939 52 × 2 = 0 + 0.000 000 000 000 000 004 369 879 04;
  • 10) 0.000 000 000 000 000 004 369 879 04 × 2 = 0 + 0.000 000 000 000 000 008 739 758 08;
  • 11) 0.000 000 000 000 000 008 739 758 08 × 2 = 0 + 0.000 000 000 000 000 017 479 516 16;
  • 12) 0.000 000 000 000 000 017 479 516 16 × 2 = 0 + 0.000 000 000 000 000 034 959 032 32;
  • 13) 0.000 000 000 000 000 034 959 032 32 × 2 = 0 + 0.000 000 000 000 000 069 918 064 64;
  • 14) 0.000 000 000 000 000 069 918 064 64 × 2 = 0 + 0.000 000 000 000 000 139 836 129 28;
  • 15) 0.000 000 000 000 000 139 836 129 28 × 2 = 0 + 0.000 000 000 000 000 279 672 258 56;
  • 16) 0.000 000 000 000 000 279 672 258 56 × 2 = 0 + 0.000 000 000 000 000 559 344 517 12;
  • 17) 0.000 000 000 000 000 559 344 517 12 × 2 = 0 + 0.000 000 000 000 001 118 689 034 24;
  • 18) 0.000 000 000 000 001 118 689 034 24 × 2 = 0 + 0.000 000 000 000 002 237 378 068 48;
  • 19) 0.000 000 000 000 002 237 378 068 48 × 2 = 0 + 0.000 000 000 000 004 474 756 136 96;
  • 20) 0.000 000 000 000 004 474 756 136 96 × 2 = 0 + 0.000 000 000 000 008 949 512 273 92;
  • 21) 0.000 000 000 000 008 949 512 273 92 × 2 = 0 + 0.000 000 000 000 017 899 024 547 84;
  • 22) 0.000 000 000 000 017 899 024 547 84 × 2 = 0 + 0.000 000 000 000 035 798 049 095 68;
  • 23) 0.000 000 000 000 035 798 049 095 68 × 2 = 0 + 0.000 000 000 000 071 596 098 191 36;
  • 24) 0.000 000 000 000 071 596 098 191 36 × 2 = 0 + 0.000 000 000 000 143 192 196 382 72;
  • 25) 0.000 000 000 000 143 192 196 382 72 × 2 = 0 + 0.000 000 000 000 286 384 392 765 44;
  • 26) 0.000 000 000 000 286 384 392 765 44 × 2 = 0 + 0.000 000 000 000 572 768 785 530 88;
  • 27) 0.000 000 000 000 572 768 785 530 88 × 2 = 0 + 0.000 000 000 001 145 537 571 061 76;
  • 28) 0.000 000 000 001 145 537 571 061 76 × 2 = 0 + 0.000 000 000 002 291 075 142 123 52;
  • 29) 0.000 000 000 002 291 075 142 123 52 × 2 = 0 + 0.000 000 000 004 582 150 284 247 04;
  • 30) 0.000 000 000 004 582 150 284 247 04 × 2 = 0 + 0.000 000 000 009 164 300 568 494 08;
  • 31) 0.000 000 000 009 164 300 568 494 08 × 2 = 0 + 0.000 000 000 018 328 601 136 988 16;
  • 32) 0.000 000 000 018 328 601 136 988 16 × 2 = 0 + 0.000 000 000 036 657 202 273 976 32;
  • 33) 0.000 000 000 036 657 202 273 976 32 × 2 = 0 + 0.000 000 000 073 314 404 547 952 64;
  • 34) 0.000 000 000 073 314 404 547 952 64 × 2 = 0 + 0.000 000 000 146 628 809 095 905 28;
  • 35) 0.000 000 000 146 628 809 095 905 28 × 2 = 0 + 0.000 000 000 293 257 618 191 810 56;
  • 36) 0.000 000 000 293 257 618 191 810 56 × 2 = 0 + 0.000 000 000 586 515 236 383 621 12;
  • 37) 0.000 000 000 586 515 236 383 621 12 × 2 = 0 + 0.000 000 001 173 030 472 767 242 24;
  • 38) 0.000 000 001 173 030 472 767 242 24 × 2 = 0 + 0.000 000 002 346 060 945 534 484 48;
  • 39) 0.000 000 002 346 060 945 534 484 48 × 2 = 0 + 0.000 000 004 692 121 891 068 968 96;
  • 40) 0.000 000 004 692 121 891 068 968 96 × 2 = 0 + 0.000 000 009 384 243 782 137 937 92;
  • 41) 0.000 000 009 384 243 782 137 937 92 × 2 = 0 + 0.000 000 018 768 487 564 275 875 84;
  • 42) 0.000 000 018 768 487 564 275 875 84 × 2 = 0 + 0.000 000 037 536 975 128 551 751 68;
  • 43) 0.000 000 037 536 975 128 551 751 68 × 2 = 0 + 0.000 000 075 073 950 257 103 503 36;
  • 44) 0.000 000 075 073 950 257 103 503 36 × 2 = 0 + 0.000 000 150 147 900 514 207 006 72;
  • 45) 0.000 000 150 147 900 514 207 006 72 × 2 = 0 + 0.000 000 300 295 801 028 414 013 44;
  • 46) 0.000 000 300 295 801 028 414 013 44 × 2 = 0 + 0.000 000 600 591 602 056 828 026 88;
  • 47) 0.000 000 600 591 602 056 828 026 88 × 2 = 0 + 0.000 001 201 183 204 113 656 053 76;
  • 48) 0.000 001 201 183 204 113 656 053 76 × 2 = 0 + 0.000 002 402 366 408 227 312 107 52;
  • 49) 0.000 002 402 366 408 227 312 107 52 × 2 = 0 + 0.000 004 804 732 816 454 624 215 04;
  • 50) 0.000 004 804 732 816 454 624 215 04 × 2 = 0 + 0.000 009 609 465 632 909 248 430 08;
  • 51) 0.000 009 609 465 632 909 248 430 08 × 2 = 0 + 0.000 019 218 931 265 818 496 860 16;
  • 52) 0.000 019 218 931 265 818 496 860 16 × 2 = 0 + 0.000 038 437 862 531 636 993 720 32;
  • 53) 0.000 038 437 862 531 636 993 720 32 × 2 = 0 + 0.000 076 875 725 063 273 987 440 64;
  • 54) 0.000 076 875 725 063 273 987 440 64 × 2 = 0 + 0.000 153 751 450 126 547 974 881 28;
  • 55) 0.000 153 751 450 126 547 974 881 28 × 2 = 0 + 0.000 307 502 900 253 095 949 762 56;
  • 56) 0.000 307 502 900 253 095 949 762 56 × 2 = 0 + 0.000 615 005 800 506 191 899 525 12;
  • 57) 0.000 615 005 800 506 191 899 525 12 × 2 = 0 + 0.001 230 011 601 012 383 799 050 24;
  • 58) 0.001 230 011 601 012 383 799 050 24 × 2 = 0 + 0.002 460 023 202 024 767 598 100 48;
  • 59) 0.002 460 023 202 024 767 598 100 48 × 2 = 0 + 0.004 920 046 404 049 535 196 200 96;
  • 60) 0.004 920 046 404 049 535 196 200 96 × 2 = 0 + 0.009 840 092 808 099 070 392 401 92;
  • 61) 0.009 840 092 808 099 070 392 401 92 × 2 = 0 + 0.019 680 185 616 198 140 784 803 84;
  • 62) 0.019 680 185 616 198 140 784 803 84 × 2 = 0 + 0.039 360 371 232 396 281 569 607 68;
  • 63) 0.039 360 371 232 396 281 569 607 68 × 2 = 0 + 0.078 720 742 464 792 563 139 215 36;
  • 64) 0.078 720 742 464 792 563 139 215 36 × 2 = 0 + 0.157 441 484 929 585 126 278 430 72;
  • 65) 0.157 441 484 929 585 126 278 430 72 × 2 = 0 + 0.314 882 969 859 170 252 556 861 44;
  • 66) 0.314 882 969 859 170 252 556 861 44 × 2 = 0 + 0.629 765 939 718 340 505 113 722 88;
  • 67) 0.629 765 939 718 340 505 113 722 88 × 2 = 1 + 0.259 531 879 436 681 010 227 445 76;
  • 68) 0.259 531 879 436 681 010 227 445 76 × 2 = 0 + 0.519 063 758 873 362 020 454 891 52;
  • 69) 0.519 063 758 873 362 020 454 891 52 × 2 = 1 + 0.038 127 517 746 724 040 909 783 04;
  • 70) 0.038 127 517 746 724 040 909 783 04 × 2 = 0 + 0.076 255 035 493 448 081 819 566 08;
  • 71) 0.076 255 035 493 448 081 819 566 08 × 2 = 0 + 0.152 510 070 986 896 163 639 132 16;
  • 72) 0.152 510 070 986 896 163 639 132 16 × 2 = 0 + 0.305 020 141 973 792 327 278 264 32;
  • 73) 0.305 020 141 973 792 327 278 264 32 × 2 = 0 + 0.610 040 283 947 584 654 556 528 64;
  • 74) 0.610 040 283 947 584 654 556 528 64 × 2 = 1 + 0.220 080 567 895 169 309 113 057 28;
  • 75) 0.220 080 567 895 169 309 113 057 28 × 2 = 0 + 0.440 161 135 790 338 618 226 114 56;
  • 76) 0.440 161 135 790 338 618 226 114 56 × 2 = 0 + 0.880 322 271 580 677 236 452 229 12;
  • 77) 0.880 322 271 580 677 236 452 229 12 × 2 = 1 + 0.760 644 543 161 354 472 904 458 24;
  • 78) 0.760 644 543 161 354 472 904 458 24 × 2 = 1 + 0.521 289 086 322 708 945 808 916 48;
  • 79) 0.521 289 086 322 708 945 808 916 48 × 2 = 1 + 0.042 578 172 645 417 891 617 832 96;
  • 80) 0.042 578 172 645 417 891 617 832 96 × 2 = 0 + 0.085 156 345 290 835 783 235 665 92;
  • 81) 0.085 156 345 290 835 783 235 665 92 × 2 = 0 + 0.170 312 690 581 671 566 471 331 84;
  • 82) 0.170 312 690 581 671 566 471 331 84 × 2 = 0 + 0.340 625 381 163 343 132 942 663 68;
  • 83) 0.340 625 381 163 343 132 942 663 68 × 2 = 0 + 0.681 250 762 326 686 265 885 327 36;
  • 84) 0.681 250 762 326 686 265 885 327 36 × 2 = 1 + 0.362 501 524 653 372 531 770 654 72;
  • 85) 0.362 501 524 653 372 531 770 654 72 × 2 = 0 + 0.725 003 049 306 745 063 541 309 44;
  • 86) 0.725 003 049 306 745 063 541 309 44 × 2 = 1 + 0.450 006 098 613 490 127 082 618 88;
  • 87) 0.450 006 098 613 490 127 082 618 88 × 2 = 0 + 0.900 012 197 226 980 254 165 237 76;
  • 88) 0.900 012 197 226 980 254 165 237 76 × 2 = 1 + 0.800 024 394 453 960 508 330 475 52;
  • 89) 0.800 024 394 453 960 508 330 475 52 × 2 = 1 + 0.600 048 788 907 921 016 660 951 04;
  • 90) 0.600 048 788 907 921 016 660 951 04 × 2 = 1 + 0.200 097 577 815 842 033 321 902 08;
  • 91) 0.200 097 577 815 842 033 321 902 08 × 2 = 0 + 0.400 195 155 631 684 066 643 804 16;
  • 92) 0.400 195 155 631 684 066 643 804 16 × 2 = 0 + 0.800 390 311 263 368 133 287 608 32;
  • 93) 0.800 390 311 263 368 133 287 608 32 × 2 = 1 + 0.600 780 622 526 736 266 575 216 64;
  • 94) 0.600 780 622 526 736 266 575 216 64 × 2 = 1 + 0.201 561 245 053 472 533 150 433 28;
  • 95) 0.201 561 245 053 472 533 150 433 28 × 2 = 0 + 0.403 122 490 106 945 066 300 866 56;
  • 96) 0.403 122 490 106 945 066 300 866 56 × 2 = 0 + 0.806 244 980 213 890 132 601 733 12;
  • 97) 0.806 244 980 213 890 132 601 733 12 × 2 = 1 + 0.612 489 960 427 780 265 203 466 24;
  • 98) 0.612 489 960 427 780 265 203 466 24 × 2 = 1 + 0.224 979 920 855 560 530 406 932 48;
  • 99) 0.224 979 920 855 560 530 406 932 48 × 2 = 0 + 0.449 959 841 711 121 060 813 864 96;
  • 100) 0.449 959 841 711 121 060 813 864 96 × 2 = 0 + 0.899 919 683 422 242 121 627 729 92;
  • 101) 0.899 919 683 422 242 121 627 729 92 × 2 = 1 + 0.799 839 366 844 484 243 255 459 84;
  • 102) 0.799 839 366 844 484 243 255 459 84 × 2 = 1 + 0.599 678 733 688 968 486 510 919 68;
  • 103) 0.599 678 733 688 968 486 510 919 68 × 2 = 1 + 0.199 357 467 377 936 973 021 839 36;
  • 104) 0.199 357 467 377 936 973 021 839 36 × 2 = 0 + 0.398 714 934 755 873 946 043 678 72;
  • 105) 0.398 714 934 755 873 946 043 678 72 × 2 = 0 + 0.797 429 869 511 747 892 087 357 44;
  • 106) 0.797 429 869 511 747 892 087 357 44 × 2 = 1 + 0.594 859 739 023 495 784 174 714 88;
  • 107) 0.594 859 739 023 495 784 174 714 88 × 2 = 1 + 0.189 719 478 046 991 568 349 429 76;
  • 108) 0.189 719 478 046 991 568 349 429 76 × 2 = 0 + 0.379 438 956 093 983 136 698 859 52;
  • 109) 0.379 438 956 093 983 136 698 859 52 × 2 = 0 + 0.758 877 912 187 966 273 397 719 04;
  • 110) 0.758 877 912 187 966 273 397 719 04 × 2 = 1 + 0.517 755 824 375 932 546 795 438 08;
  • 111) 0.517 755 824 375 932 546 795 438 08 × 2 = 1 + 0.035 511 648 751 865 093 590 876 16;
  • 112) 0.035 511 648 751 865 093 590 876 16 × 2 = 0 + 0.071 023 297 503 730 187 181 752 32;
  • 113) 0.071 023 297 503 730 187 181 752 32 × 2 = 0 + 0.142 046 595 007 460 374 363 504 64;
  • 114) 0.142 046 595 007 460 374 363 504 64 × 2 = 0 + 0.284 093 190 014 920 748 727 009 28;
  • 115) 0.284 093 190 014 920 748 727 009 28 × 2 = 0 + 0.568 186 380 029 841 497 454 018 56;
  • 116) 0.568 186 380 029 841 497 454 018 56 × 2 = 1 + 0.136 372 760 059 682 994 908 037 12;
  • 117) 0.136 372 760 059 682 994 908 037 12 × 2 = 0 + 0.272 745 520 119 365 989 816 074 24;
  • 118) 0.272 745 520 119 365 989 816 074 24 × 2 = 0 + 0.545 491 040 238 731 979 632 148 48;
  • 119) 0.545 491 040 238 731 979 632 148 48 × 2 = 1 + 0.090 982 080 477 463 959 264 296 96;

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:


0.000 000 000 000 000 000 008 534 92(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1110 0001 0101 1100 1100 1100 1110 0110 0110 0001 001(2)

5. Positive number before normalization:

0.000 000 000 000 000 000 008 534 92(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1110 0001 0101 1100 1100 1100 1110 0110 0110 0001 001(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:


0.000 000 000 000 000 000 008 534 92(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1110 0001 0101 1100 1100 1100 1110 0110 0110 0001 001(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1110 0001 0101 1100 1100 1100 1110 0110 0110 0001 001(2) × 20 =

1.0100 0010 0111 0000 1010 1110 0110 0110 0111 0011 0011 0000 1001(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 0010 0111 0000 1010 1110 0110 0110 0111 0011 0011 0000 1001


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 0010 0111 0000 1010 1110 0110 0110 0111 0011 0011 0000 1001 =

0100 0010 0111 0000 1010 1110 0110 0110 0111 0011 0011 0000 1001


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 0010 0111 0000 1010 1110 0110 0110 0111 0011 0011 0000 1001


Decimal number 0.000 000 000 000 000 000 008 534 92 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1011 1100 - 0100 0010 0111 0000 1010 1110 0110 0110 0111 0011 0011 0000 1001

Share

» Convert decimal 0.000 000 000 000 000 000 008 534 48 to 64 bit double precision IEEE 754 binary floating point representation standard

» Calculations Performed by Our Visitors: Decimal Numbers Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard. Data organized on a Monthly Basis

» Month 06, 2026 [June]: Decimal Numbers Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard. Calculations performed during the month of: June - by our visitors

» Improve your social skills: Are you using communication by design, or by default? NotebookLM YouTube Video of 6.33 minutes

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