0.000 000 000 000 000 000 008 534 87 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 000 000 008 534 87(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 87(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 87.

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 87 × 2 = 0 + 0.000 000 000 000 000 000 017 069 74;
  • 2) 0.000 000 000 000 000 000 017 069 74 × 2 = 0 + 0.000 000 000 000 000 000 034 139 48;
  • 3) 0.000 000 000 000 000 000 034 139 48 × 2 = 0 + 0.000 000 000 000 000 000 068 278 96;
  • 4) 0.000 000 000 000 000 000 068 278 96 × 2 = 0 + 0.000 000 000 000 000 000 136 557 92;
  • 5) 0.000 000 000 000 000 000 136 557 92 × 2 = 0 + 0.000 000 000 000 000 000 273 115 84;
  • 6) 0.000 000 000 000 000 000 273 115 84 × 2 = 0 + 0.000 000 000 000 000 000 546 231 68;
  • 7) 0.000 000 000 000 000 000 546 231 68 × 2 = 0 + 0.000 000 000 000 000 001 092 463 36;
  • 8) 0.000 000 000 000 000 001 092 463 36 × 2 = 0 + 0.000 000 000 000 000 002 184 926 72;
  • 9) 0.000 000 000 000 000 002 184 926 72 × 2 = 0 + 0.000 000 000 000 000 004 369 853 44;
  • 10) 0.000 000 000 000 000 004 369 853 44 × 2 = 0 + 0.000 000 000 000 000 008 739 706 88;
  • 11) 0.000 000 000 000 000 008 739 706 88 × 2 = 0 + 0.000 000 000 000 000 017 479 413 76;
  • 12) 0.000 000 000 000 000 017 479 413 76 × 2 = 0 + 0.000 000 000 000 000 034 958 827 52;
  • 13) 0.000 000 000 000 000 034 958 827 52 × 2 = 0 + 0.000 000 000 000 000 069 917 655 04;
  • 14) 0.000 000 000 000 000 069 917 655 04 × 2 = 0 + 0.000 000 000 000 000 139 835 310 08;
  • 15) 0.000 000 000 000 000 139 835 310 08 × 2 = 0 + 0.000 000 000 000 000 279 670 620 16;
  • 16) 0.000 000 000 000 000 279 670 620 16 × 2 = 0 + 0.000 000 000 000 000 559 341 240 32;
  • 17) 0.000 000 000 000 000 559 341 240 32 × 2 = 0 + 0.000 000 000 000 001 118 682 480 64;
  • 18) 0.000 000 000 000 001 118 682 480 64 × 2 = 0 + 0.000 000 000 000 002 237 364 961 28;
  • 19) 0.000 000 000 000 002 237 364 961 28 × 2 = 0 + 0.000 000 000 000 004 474 729 922 56;
  • 20) 0.000 000 000 000 004 474 729 922 56 × 2 = 0 + 0.000 000 000 000 008 949 459 845 12;
  • 21) 0.000 000 000 000 008 949 459 845 12 × 2 = 0 + 0.000 000 000 000 017 898 919 690 24;
  • 22) 0.000 000 000 000 017 898 919 690 24 × 2 = 0 + 0.000 000 000 000 035 797 839 380 48;
  • 23) 0.000 000 000 000 035 797 839 380 48 × 2 = 0 + 0.000 000 000 000 071 595 678 760 96;
  • 24) 0.000 000 000 000 071 595 678 760 96 × 2 = 0 + 0.000 000 000 000 143 191 357 521 92;
  • 25) 0.000 000 000 000 143 191 357 521 92 × 2 = 0 + 0.000 000 000 000 286 382 715 043 84;
  • 26) 0.000 000 000 000 286 382 715 043 84 × 2 = 0 + 0.000 000 000 000 572 765 430 087 68;
  • 27) 0.000 000 000 000 572 765 430 087 68 × 2 = 0 + 0.000 000 000 001 145 530 860 175 36;
  • 28) 0.000 000 000 001 145 530 860 175 36 × 2 = 0 + 0.000 000 000 002 291 061 720 350 72;
  • 29) 0.000 000 000 002 291 061 720 350 72 × 2 = 0 + 0.000 000 000 004 582 123 440 701 44;
  • 30) 0.000 000 000 004 582 123 440 701 44 × 2 = 0 + 0.000 000 000 009 164 246 881 402 88;
  • 31) 0.000 000 000 009 164 246 881 402 88 × 2 = 0 + 0.000 000 000 018 328 493 762 805 76;
  • 32) 0.000 000 000 018 328 493 762 805 76 × 2 = 0 + 0.000 000 000 036 656 987 525 611 52;
  • 33) 0.000 000 000 036 656 987 525 611 52 × 2 = 0 + 0.000 000 000 073 313 975 051 223 04;
  • 34) 0.000 000 000 073 313 975 051 223 04 × 2 = 0 + 0.000 000 000 146 627 950 102 446 08;
  • 35) 0.000 000 000 146 627 950 102 446 08 × 2 = 0 + 0.000 000 000 293 255 900 204 892 16;
  • 36) 0.000 000 000 293 255 900 204 892 16 × 2 = 0 + 0.000 000 000 586 511 800 409 784 32;
  • 37) 0.000 000 000 586 511 800 409 784 32 × 2 = 0 + 0.000 000 001 173 023 600 819 568 64;
  • 38) 0.000 000 001 173 023 600 819 568 64 × 2 = 0 + 0.000 000 002 346 047 201 639 137 28;
  • 39) 0.000 000 002 346 047 201 639 137 28 × 2 = 0 + 0.000 000 004 692 094 403 278 274 56;
  • 40) 0.000 000 004 692 094 403 278 274 56 × 2 = 0 + 0.000 000 009 384 188 806 556 549 12;
  • 41) 0.000 000 009 384 188 806 556 549 12 × 2 = 0 + 0.000 000 018 768 377 613 113 098 24;
  • 42) 0.000 000 018 768 377 613 113 098 24 × 2 = 0 + 0.000 000 037 536 755 226 226 196 48;
  • 43) 0.000 000 037 536 755 226 226 196 48 × 2 = 0 + 0.000 000 075 073 510 452 452 392 96;
  • 44) 0.000 000 075 073 510 452 452 392 96 × 2 = 0 + 0.000 000 150 147 020 904 904 785 92;
  • 45) 0.000 000 150 147 020 904 904 785 92 × 2 = 0 + 0.000 000 300 294 041 809 809 571 84;
  • 46) 0.000 000 300 294 041 809 809 571 84 × 2 = 0 + 0.000 000 600 588 083 619 619 143 68;
  • 47) 0.000 000 600 588 083 619 619 143 68 × 2 = 0 + 0.000 001 201 176 167 239 238 287 36;
  • 48) 0.000 001 201 176 167 239 238 287 36 × 2 = 0 + 0.000 002 402 352 334 478 476 574 72;
  • 49) 0.000 002 402 352 334 478 476 574 72 × 2 = 0 + 0.000 004 804 704 668 956 953 149 44;
  • 50) 0.000 004 804 704 668 956 953 149 44 × 2 = 0 + 0.000 009 609 409 337 913 906 298 88;
  • 51) 0.000 009 609 409 337 913 906 298 88 × 2 = 0 + 0.000 019 218 818 675 827 812 597 76;
  • 52) 0.000 019 218 818 675 827 812 597 76 × 2 = 0 + 0.000 038 437 637 351 655 625 195 52;
  • 53) 0.000 038 437 637 351 655 625 195 52 × 2 = 0 + 0.000 076 875 274 703 311 250 391 04;
  • 54) 0.000 076 875 274 703 311 250 391 04 × 2 = 0 + 0.000 153 750 549 406 622 500 782 08;
  • 55) 0.000 153 750 549 406 622 500 782 08 × 2 = 0 + 0.000 307 501 098 813 245 001 564 16;
  • 56) 0.000 307 501 098 813 245 001 564 16 × 2 = 0 + 0.000 615 002 197 626 490 003 128 32;
  • 57) 0.000 615 002 197 626 490 003 128 32 × 2 = 0 + 0.001 230 004 395 252 980 006 256 64;
  • 58) 0.001 230 004 395 252 980 006 256 64 × 2 = 0 + 0.002 460 008 790 505 960 012 513 28;
  • 59) 0.002 460 008 790 505 960 012 513 28 × 2 = 0 + 0.004 920 017 581 011 920 025 026 56;
  • 60) 0.004 920 017 581 011 920 025 026 56 × 2 = 0 + 0.009 840 035 162 023 840 050 053 12;
  • 61) 0.009 840 035 162 023 840 050 053 12 × 2 = 0 + 0.019 680 070 324 047 680 100 106 24;
  • 62) 0.019 680 070 324 047 680 100 106 24 × 2 = 0 + 0.039 360 140 648 095 360 200 212 48;
  • 63) 0.039 360 140 648 095 360 200 212 48 × 2 = 0 + 0.078 720 281 296 190 720 400 424 96;
  • 64) 0.078 720 281 296 190 720 400 424 96 × 2 = 0 + 0.157 440 562 592 381 440 800 849 92;
  • 65) 0.157 440 562 592 381 440 800 849 92 × 2 = 0 + 0.314 881 125 184 762 881 601 699 84;
  • 66) 0.314 881 125 184 762 881 601 699 84 × 2 = 0 + 0.629 762 250 369 525 763 203 399 68;
  • 67) 0.629 762 250 369 525 763 203 399 68 × 2 = 1 + 0.259 524 500 739 051 526 406 799 36;
  • 68) 0.259 524 500 739 051 526 406 799 36 × 2 = 0 + 0.519 049 001 478 103 052 813 598 72;
  • 69) 0.519 049 001 478 103 052 813 598 72 × 2 = 1 + 0.038 098 002 956 206 105 627 197 44;
  • 70) 0.038 098 002 956 206 105 627 197 44 × 2 = 0 + 0.076 196 005 912 412 211 254 394 88;
  • 71) 0.076 196 005 912 412 211 254 394 88 × 2 = 0 + 0.152 392 011 824 824 422 508 789 76;
  • 72) 0.152 392 011 824 824 422 508 789 76 × 2 = 0 + 0.304 784 023 649 648 845 017 579 52;
  • 73) 0.304 784 023 649 648 845 017 579 52 × 2 = 0 + 0.609 568 047 299 297 690 035 159 04;
  • 74) 0.609 568 047 299 297 690 035 159 04 × 2 = 1 + 0.219 136 094 598 595 380 070 318 08;
  • 75) 0.219 136 094 598 595 380 070 318 08 × 2 = 0 + 0.438 272 189 197 190 760 140 636 16;
  • 76) 0.438 272 189 197 190 760 140 636 16 × 2 = 0 + 0.876 544 378 394 381 520 281 272 32;
  • 77) 0.876 544 378 394 381 520 281 272 32 × 2 = 1 + 0.753 088 756 788 763 040 562 544 64;
  • 78) 0.753 088 756 788 763 040 562 544 64 × 2 = 1 + 0.506 177 513 577 526 081 125 089 28;
  • 79) 0.506 177 513 577 526 081 125 089 28 × 2 = 1 + 0.012 355 027 155 052 162 250 178 56;
  • 80) 0.012 355 027 155 052 162 250 178 56 × 2 = 0 + 0.024 710 054 310 104 324 500 357 12;
  • 81) 0.024 710 054 310 104 324 500 357 12 × 2 = 0 + 0.049 420 108 620 208 649 000 714 24;
  • 82) 0.049 420 108 620 208 649 000 714 24 × 2 = 0 + 0.098 840 217 240 417 298 001 428 48;
  • 83) 0.098 840 217 240 417 298 001 428 48 × 2 = 0 + 0.197 680 434 480 834 596 002 856 96;
  • 84) 0.197 680 434 480 834 596 002 856 96 × 2 = 0 + 0.395 360 868 961 669 192 005 713 92;
  • 85) 0.395 360 868 961 669 192 005 713 92 × 2 = 0 + 0.790 721 737 923 338 384 011 427 84;
  • 86) 0.790 721 737 923 338 384 011 427 84 × 2 = 1 + 0.581 443 475 846 676 768 022 855 68;
  • 87) 0.581 443 475 846 676 768 022 855 68 × 2 = 1 + 0.162 886 951 693 353 536 045 711 36;
  • 88) 0.162 886 951 693 353 536 045 711 36 × 2 = 0 + 0.325 773 903 386 707 072 091 422 72;
  • 89) 0.325 773 903 386 707 072 091 422 72 × 2 = 0 + 0.651 547 806 773 414 144 182 845 44;
  • 90) 0.651 547 806 773 414 144 182 845 44 × 2 = 1 + 0.303 095 613 546 828 288 365 690 88;
  • 91) 0.303 095 613 546 828 288 365 690 88 × 2 = 0 + 0.606 191 227 093 656 576 731 381 76;
  • 92) 0.606 191 227 093 656 576 731 381 76 × 2 = 1 + 0.212 382 454 187 313 153 462 763 52;
  • 93) 0.212 382 454 187 313 153 462 763 52 × 2 = 0 + 0.424 764 908 374 626 306 925 527 04;
  • 94) 0.424 764 908 374 626 306 925 527 04 × 2 = 0 + 0.849 529 816 749 252 613 851 054 08;
  • 95) 0.849 529 816 749 252 613 851 054 08 × 2 = 1 + 0.699 059 633 498 505 227 702 108 16;
  • 96) 0.699 059 633 498 505 227 702 108 16 × 2 = 1 + 0.398 119 266 997 010 455 404 216 32;
  • 97) 0.398 119 266 997 010 455 404 216 32 × 2 = 0 + 0.796 238 533 994 020 910 808 432 64;
  • 98) 0.796 238 533 994 020 910 808 432 64 × 2 = 1 + 0.592 477 067 988 041 821 616 865 28;
  • 99) 0.592 477 067 988 041 821 616 865 28 × 2 = 1 + 0.184 954 135 976 083 643 233 730 56;
  • 100) 0.184 954 135 976 083 643 233 730 56 × 2 = 0 + 0.369 908 271 952 167 286 467 461 12;
  • 101) 0.369 908 271 952 167 286 467 461 12 × 2 = 0 + 0.739 816 543 904 334 572 934 922 24;
  • 102) 0.739 816 543 904 334 572 934 922 24 × 2 = 1 + 0.479 633 087 808 669 145 869 844 48;
  • 103) 0.479 633 087 808 669 145 869 844 48 × 2 = 0 + 0.959 266 175 617 338 291 739 688 96;
  • 104) 0.959 266 175 617 338 291 739 688 96 × 2 = 1 + 0.918 532 351 234 676 583 479 377 92;
  • 105) 0.918 532 351 234 676 583 479 377 92 × 2 = 1 + 0.837 064 702 469 353 166 958 755 84;
  • 106) 0.837 064 702 469 353 166 958 755 84 × 2 = 1 + 0.674 129 404 938 706 333 917 511 68;
  • 107) 0.674 129 404 938 706 333 917 511 68 × 2 = 1 + 0.348 258 809 877 412 667 835 023 36;
  • 108) 0.348 258 809 877 412 667 835 023 36 × 2 = 0 + 0.696 517 619 754 825 335 670 046 72;
  • 109) 0.696 517 619 754 825 335 670 046 72 × 2 = 1 + 0.393 035 239 509 650 671 340 093 44;
  • 110) 0.393 035 239 509 650 671 340 093 44 × 2 = 0 + 0.786 070 479 019 301 342 680 186 88;
  • 111) 0.786 070 479 019 301 342 680 186 88 × 2 = 1 + 0.572 140 958 038 602 685 360 373 76;
  • 112) 0.572 140 958 038 602 685 360 373 76 × 2 = 1 + 0.144 281 916 077 205 370 720 747 52;
  • 113) 0.144 281 916 077 205 370 720 747 52 × 2 = 0 + 0.288 563 832 154 410 741 441 495 04;
  • 114) 0.288 563 832 154 410 741 441 495 04 × 2 = 0 + 0.577 127 664 308 821 482 882 990 08;
  • 115) 0.577 127 664 308 821 482 882 990 08 × 2 = 1 + 0.154 255 328 617 642 965 765 980 16;
  • 116) 0.154 255 328 617 642 965 765 980 16 × 2 = 0 + 0.308 510 657 235 285 931 531 960 32;
  • 117) 0.308 510 657 235 285 931 531 960 32 × 2 = 0 + 0.617 021 314 470 571 863 063 920 64;
  • 118) 0.617 021 314 470 571 863 063 920 64 × 2 = 1 + 0.234 042 628 941 143 726 127 841 28;
  • 119) 0.234 042 628 941 143 726 127 841 28 × 2 = 0 + 0.468 085 257 882 287 452 255 682 56;

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 87(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1110 0000 0110 0101 0011 0110 0101 1110 1011 0010 010(2)

5. Positive number before normalization:

0.000 000 000 000 000 000 008 534 87(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1110 0000 0110 0101 0011 0110 0101 1110 1011 0010 010(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 87(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1110 0000 0110 0101 0011 0110 0101 1110 1011 0010 010(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1110 0000 0110 0101 0011 0110 0101 1110 1011 0010 010(2) × 20 =

1.0100 0010 0111 0000 0011 0010 1001 1011 0010 1111 0101 1001 0010(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 0011 0010 1001 1011 0010 1111 0101 1001 0010


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 0011 0010 1001 1011 0010 1111 0101 1001 0010 =

0100 0010 0111 0000 0011 0010 1001 1011 0010 1111 0101 1001 0010


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 0011 0010 1001 1011 0010 1111 0101 1001 0010


Decimal number 0.000 000 000 000 000 000 008 534 87 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1011 1100 - 0100 0010 0111 0000 0011 0010 1001 1011 0010 1111 0101 1001 0010

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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