0.000 000 000 000 000 000 008 538 21 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 000 000 008 538 21(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
0.000 000 000 000 000 000 008 538 21(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: 0.000 000 000 000 000 000 008 538 21.

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) 0.000 000 000 000 000 000 008 538 21 × 2 = 0 + 0.000 000 000 000 000 000 017 076 42;
  • 2) 0.000 000 000 000 000 000 017 076 42 × 2 = 0 + 0.000 000 000 000 000 000 034 152 84;
  • 3) 0.000 000 000 000 000 000 034 152 84 × 2 = 0 + 0.000 000 000 000 000 000 068 305 68;
  • 4) 0.000 000 000 000 000 000 068 305 68 × 2 = 0 + 0.000 000 000 000 000 000 136 611 36;
  • 5) 0.000 000 000 000 000 000 136 611 36 × 2 = 0 + 0.000 000 000 000 000 000 273 222 72;
  • 6) 0.000 000 000 000 000 000 273 222 72 × 2 = 0 + 0.000 000 000 000 000 000 546 445 44;
  • 7) 0.000 000 000 000 000 000 546 445 44 × 2 = 0 + 0.000 000 000 000 000 001 092 890 88;
  • 8) 0.000 000 000 000 000 001 092 890 88 × 2 = 0 + 0.000 000 000 000 000 002 185 781 76;
  • 9) 0.000 000 000 000 000 002 185 781 76 × 2 = 0 + 0.000 000 000 000 000 004 371 563 52;
  • 10) 0.000 000 000 000 000 004 371 563 52 × 2 = 0 + 0.000 000 000 000 000 008 743 127 04;
  • 11) 0.000 000 000 000 000 008 743 127 04 × 2 = 0 + 0.000 000 000 000 000 017 486 254 08;
  • 12) 0.000 000 000 000 000 017 486 254 08 × 2 = 0 + 0.000 000 000 000 000 034 972 508 16;
  • 13) 0.000 000 000 000 000 034 972 508 16 × 2 = 0 + 0.000 000 000 000 000 069 945 016 32;
  • 14) 0.000 000 000 000 000 069 945 016 32 × 2 = 0 + 0.000 000 000 000 000 139 890 032 64;
  • 15) 0.000 000 000 000 000 139 890 032 64 × 2 = 0 + 0.000 000 000 000 000 279 780 065 28;
  • 16) 0.000 000 000 000 000 279 780 065 28 × 2 = 0 + 0.000 000 000 000 000 559 560 130 56;
  • 17) 0.000 000 000 000 000 559 560 130 56 × 2 = 0 + 0.000 000 000 000 001 119 120 261 12;
  • 18) 0.000 000 000 000 001 119 120 261 12 × 2 = 0 + 0.000 000 000 000 002 238 240 522 24;
  • 19) 0.000 000 000 000 002 238 240 522 24 × 2 = 0 + 0.000 000 000 000 004 476 481 044 48;
  • 20) 0.000 000 000 000 004 476 481 044 48 × 2 = 0 + 0.000 000 000 000 008 952 962 088 96;
  • 21) 0.000 000 000 000 008 952 962 088 96 × 2 = 0 + 0.000 000 000 000 017 905 924 177 92;
  • 22) 0.000 000 000 000 017 905 924 177 92 × 2 = 0 + 0.000 000 000 000 035 811 848 355 84;
  • 23) 0.000 000 000 000 035 811 848 355 84 × 2 = 0 + 0.000 000 000 000 071 623 696 711 68;
  • 24) 0.000 000 000 000 071 623 696 711 68 × 2 = 0 + 0.000 000 000 000 143 247 393 423 36;
  • 25) 0.000 000 000 000 143 247 393 423 36 × 2 = 0 + 0.000 000 000 000 286 494 786 846 72;
  • 26) 0.000 000 000 000 286 494 786 846 72 × 2 = 0 + 0.000 000 000 000 572 989 573 693 44;
  • 27) 0.000 000 000 000 572 989 573 693 44 × 2 = 0 + 0.000 000 000 001 145 979 147 386 88;
  • 28) 0.000 000 000 001 145 979 147 386 88 × 2 = 0 + 0.000 000 000 002 291 958 294 773 76;
  • 29) 0.000 000 000 002 291 958 294 773 76 × 2 = 0 + 0.000 000 000 004 583 916 589 547 52;
  • 30) 0.000 000 000 004 583 916 589 547 52 × 2 = 0 + 0.000 000 000 009 167 833 179 095 04;
  • 31) 0.000 000 000 009 167 833 179 095 04 × 2 = 0 + 0.000 000 000 018 335 666 358 190 08;
  • 32) 0.000 000 000 018 335 666 358 190 08 × 2 = 0 + 0.000 000 000 036 671 332 716 380 16;
  • 33) 0.000 000 000 036 671 332 716 380 16 × 2 = 0 + 0.000 000 000 073 342 665 432 760 32;
  • 34) 0.000 000 000 073 342 665 432 760 32 × 2 = 0 + 0.000 000 000 146 685 330 865 520 64;
  • 35) 0.000 000 000 146 685 330 865 520 64 × 2 = 0 + 0.000 000 000 293 370 661 731 041 28;
  • 36) 0.000 000 000 293 370 661 731 041 28 × 2 = 0 + 0.000 000 000 586 741 323 462 082 56;
  • 37) 0.000 000 000 586 741 323 462 082 56 × 2 = 0 + 0.000 000 001 173 482 646 924 165 12;
  • 38) 0.000 000 001 173 482 646 924 165 12 × 2 = 0 + 0.000 000 002 346 965 293 848 330 24;
  • 39) 0.000 000 002 346 965 293 848 330 24 × 2 = 0 + 0.000 000 004 693 930 587 696 660 48;
  • 40) 0.000 000 004 693 930 587 696 660 48 × 2 = 0 + 0.000 000 009 387 861 175 393 320 96;
  • 41) 0.000 000 009 387 861 175 393 320 96 × 2 = 0 + 0.000 000 018 775 722 350 786 641 92;
  • 42) 0.000 000 018 775 722 350 786 641 92 × 2 = 0 + 0.000 000 037 551 444 701 573 283 84;
  • 43) 0.000 000 037 551 444 701 573 283 84 × 2 = 0 + 0.000 000 075 102 889 403 146 567 68;
  • 44) 0.000 000 075 102 889 403 146 567 68 × 2 = 0 + 0.000 000 150 205 778 806 293 135 36;
  • 45) 0.000 000 150 205 778 806 293 135 36 × 2 = 0 + 0.000 000 300 411 557 612 586 270 72;
  • 46) 0.000 000 300 411 557 612 586 270 72 × 2 = 0 + 0.000 000 600 823 115 225 172 541 44;
  • 47) 0.000 000 600 823 115 225 172 541 44 × 2 = 0 + 0.000 001 201 646 230 450 345 082 88;
  • 48) 0.000 001 201 646 230 450 345 082 88 × 2 = 0 + 0.000 002 403 292 460 900 690 165 76;
  • 49) 0.000 002 403 292 460 900 690 165 76 × 2 = 0 + 0.000 004 806 584 921 801 380 331 52;
  • 50) 0.000 004 806 584 921 801 380 331 52 × 2 = 0 + 0.000 009 613 169 843 602 760 663 04;
  • 51) 0.000 009 613 169 843 602 760 663 04 × 2 = 0 + 0.000 019 226 339 687 205 521 326 08;
  • 52) 0.000 019 226 339 687 205 521 326 08 × 2 = 0 + 0.000 038 452 679 374 411 042 652 16;
  • 53) 0.000 038 452 679 374 411 042 652 16 × 2 = 0 + 0.000 076 905 358 748 822 085 304 32;
  • 54) 0.000 076 905 358 748 822 085 304 32 × 2 = 0 + 0.000 153 810 717 497 644 170 608 64;
  • 55) 0.000 153 810 717 497 644 170 608 64 × 2 = 0 + 0.000 307 621 434 995 288 341 217 28;
  • 56) 0.000 307 621 434 995 288 341 217 28 × 2 = 0 + 0.000 615 242 869 990 576 682 434 56;
  • 57) 0.000 615 242 869 990 576 682 434 56 × 2 = 0 + 0.001 230 485 739 981 153 364 869 12;
  • 58) 0.001 230 485 739 981 153 364 869 12 × 2 = 0 + 0.002 460 971 479 962 306 729 738 24;
  • 59) 0.002 460 971 479 962 306 729 738 24 × 2 = 0 + 0.004 921 942 959 924 613 459 476 48;
  • 60) 0.004 921 942 959 924 613 459 476 48 × 2 = 0 + 0.009 843 885 919 849 226 918 952 96;
  • 61) 0.009 843 885 919 849 226 918 952 96 × 2 = 0 + 0.019 687 771 839 698 453 837 905 92;
  • 62) 0.019 687 771 839 698 453 837 905 92 × 2 = 0 + 0.039 375 543 679 396 907 675 811 84;
  • 63) 0.039 375 543 679 396 907 675 811 84 × 2 = 0 + 0.078 751 087 358 793 815 351 623 68;
  • 64) 0.078 751 087 358 793 815 351 623 68 × 2 = 0 + 0.157 502 174 717 587 630 703 247 36;
  • 65) 0.157 502 174 717 587 630 703 247 36 × 2 = 0 + 0.315 004 349 435 175 261 406 494 72;
  • 66) 0.315 004 349 435 175 261 406 494 72 × 2 = 0 + 0.630 008 698 870 350 522 812 989 44;
  • 67) 0.630 008 698 870 350 522 812 989 44 × 2 = 1 + 0.260 017 397 740 701 045 625 978 88;
  • 68) 0.260 017 397 740 701 045 625 978 88 × 2 = 0 + 0.520 034 795 481 402 091 251 957 76;
  • 69) 0.520 034 795 481 402 091 251 957 76 × 2 = 1 + 0.040 069 590 962 804 182 503 915 52;
  • 70) 0.040 069 590 962 804 182 503 915 52 × 2 = 0 + 0.080 139 181 925 608 365 007 831 04;
  • 71) 0.080 139 181 925 608 365 007 831 04 × 2 = 0 + 0.160 278 363 851 216 730 015 662 08;
  • 72) 0.160 278 363 851 216 730 015 662 08 × 2 = 0 + 0.320 556 727 702 433 460 031 324 16;
  • 73) 0.320 556 727 702 433 460 031 324 16 × 2 = 0 + 0.641 113 455 404 866 920 062 648 32;
  • 74) 0.641 113 455 404 866 920 062 648 32 × 2 = 1 + 0.282 226 910 809 733 840 125 296 64;
  • 75) 0.282 226 910 809 733 840 125 296 64 × 2 = 0 + 0.564 453 821 619 467 680 250 593 28;
  • 76) 0.564 453 821 619 467 680 250 593 28 × 2 = 1 + 0.128 907 643 238 935 360 501 186 56;
  • 77) 0.128 907 643 238 935 360 501 186 56 × 2 = 0 + 0.257 815 286 477 870 721 002 373 12;
  • 78) 0.257 815 286 477 870 721 002 373 12 × 2 = 0 + 0.515 630 572 955 741 442 004 746 24;
  • 79) 0.515 630 572 955 741 442 004 746 24 × 2 = 1 + 0.031 261 145 911 482 884 009 492 48;
  • 80) 0.031 261 145 911 482 884 009 492 48 × 2 = 0 + 0.062 522 291 822 965 768 018 984 96;
  • 81) 0.062 522 291 822 965 768 018 984 96 × 2 = 0 + 0.125 044 583 645 931 536 037 969 92;
  • 82) 0.125 044 583 645 931 536 037 969 92 × 2 = 0 + 0.250 089 167 291 863 072 075 939 84;
  • 83) 0.250 089 167 291 863 072 075 939 84 × 2 = 0 + 0.500 178 334 583 726 144 151 879 68;
  • 84) 0.500 178 334 583 726 144 151 879 68 × 2 = 1 + 0.000 356 669 167 452 288 303 759 36;
  • 85) 0.000 356 669 167 452 288 303 759 36 × 2 = 0 + 0.000 713 338 334 904 576 607 518 72;
  • 86) 0.000 713 338 334 904 576 607 518 72 × 2 = 0 + 0.001 426 676 669 809 153 215 037 44;
  • 87) 0.001 426 676 669 809 153 215 037 44 × 2 = 0 + 0.002 853 353 339 618 306 430 074 88;
  • 88) 0.002 853 353 339 618 306 430 074 88 × 2 = 0 + 0.005 706 706 679 236 612 860 149 76;
  • 89) 0.005 706 706 679 236 612 860 149 76 × 2 = 0 + 0.011 413 413 358 473 225 720 299 52;
  • 90) 0.011 413 413 358 473 225 720 299 52 × 2 = 0 + 0.022 826 826 716 946 451 440 599 04;
  • 91) 0.022 826 826 716 946 451 440 599 04 × 2 = 0 + 0.045 653 653 433 892 902 881 198 08;
  • 92) 0.045 653 653 433 892 902 881 198 08 × 2 = 0 + 0.091 307 306 867 785 805 762 396 16;
  • 93) 0.091 307 306 867 785 805 762 396 16 × 2 = 0 + 0.182 614 613 735 571 611 524 792 32;
  • 94) 0.182 614 613 735 571 611 524 792 32 × 2 = 0 + 0.365 229 227 471 143 223 049 584 64;
  • 95) 0.365 229 227 471 143 223 049 584 64 × 2 = 0 + 0.730 458 454 942 286 446 099 169 28;
  • 96) 0.730 458 454 942 286 446 099 169 28 × 2 = 1 + 0.460 916 909 884 572 892 198 338 56;
  • 97) 0.460 916 909 884 572 892 198 338 56 × 2 = 0 + 0.921 833 819 769 145 784 396 677 12;
  • 98) 0.921 833 819 769 145 784 396 677 12 × 2 = 1 + 0.843 667 639 538 291 568 793 354 24;
  • 99) 0.843 667 639 538 291 568 793 354 24 × 2 = 1 + 0.687 335 279 076 583 137 586 708 48;
  • 100) 0.687 335 279 076 583 137 586 708 48 × 2 = 1 + 0.374 670 558 153 166 275 173 416 96;
  • 101) 0.374 670 558 153 166 275 173 416 96 × 2 = 0 + 0.749 341 116 306 332 550 346 833 92;
  • 102) 0.749 341 116 306 332 550 346 833 92 × 2 = 1 + 0.498 682 232 612 665 100 693 667 84;
  • 103) 0.498 682 232 612 665 100 693 667 84 × 2 = 0 + 0.997 364 465 225 330 201 387 335 68;
  • 104) 0.997 364 465 225 330 201 387 335 68 × 2 = 1 + 0.994 728 930 450 660 402 774 671 36;
  • 105) 0.994 728 930 450 660 402 774 671 36 × 2 = 1 + 0.989 457 860 901 320 805 549 342 72;
  • 106) 0.989 457 860 901 320 805 549 342 72 × 2 = 1 + 0.978 915 721 802 641 611 098 685 44;
  • 107) 0.978 915 721 802 641 611 098 685 44 × 2 = 1 + 0.957 831 443 605 283 222 197 370 88;
  • 108) 0.957 831 443 605 283 222 197 370 88 × 2 = 1 + 0.915 662 887 210 566 444 394 741 76;
  • 109) 0.915 662 887 210 566 444 394 741 76 × 2 = 1 + 0.831 325 774 421 132 888 789 483 52;
  • 110) 0.831 325 774 421 132 888 789 483 52 × 2 = 1 + 0.662 651 548 842 265 777 578 967 04;
  • 111) 0.662 651 548 842 265 777 578 967 04 × 2 = 1 + 0.325 303 097 684 531 555 157 934 08;
  • 112) 0.325 303 097 684 531 555 157 934 08 × 2 = 0 + 0.650 606 195 369 063 110 315 868 16;
  • 113) 0.650 606 195 369 063 110 315 868 16 × 2 = 1 + 0.301 212 390 738 126 220 631 736 32;
  • 114) 0.301 212 390 738 126 220 631 736 32 × 2 = 0 + 0.602 424 781 476 252 441 263 472 64;
  • 115) 0.602 424 781 476 252 441 263 472 64 × 2 = 1 + 0.204 849 562 952 504 882 526 945 28;
  • 116) 0.204 849 562 952 504 882 526 945 28 × 2 = 0 + 0.409 699 125 905 009 765 053 890 56;
  • 117) 0.409 699 125 905 009 765 053 890 56 × 2 = 0 + 0.819 398 251 810 019 530 107 781 12;
  • 118) 0.819 398 251 810 019 530 107 781 12 × 2 = 1 + 0.638 796 503 620 039 060 215 562 24;
  • 119) 0.638 796 503 620 039 060 215 562 24 × 2 = 1 + 0.277 593 007 240 078 120 431 124 48;

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:


0.000 000 000 000 000 000 008 538 21(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0101 0010 0001 0000 0000 0001 0111 0101 1111 1110 1010 011(2)

5. Positive number before normalization:

0.000 000 000 000 000 000 008 538 21(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0101 0010 0001 0000 0000 0001 0111 0101 1111 1110 1010 011(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:


0.000 000 000 000 000 000 008 538 21(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0101 0010 0001 0000 0000 0001 0111 0101 1111 1110 1010 011(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0101 0010 0001 0000 0000 0001 0111 0101 1111 1110 1010 011(2) × 20 =

1.0100 0010 1001 0000 1000 0000 0000 1011 1010 1111 1111 0101 0011(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 1001 0000 1000 0000 0000 1011 1010 1111 1111 0101 0011


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 1001 0000 1000 0000 0000 1011 1010 1111 1111 0101 0011 =

0100 0010 1001 0000 1000 0000 0000 1011 1010 1111 1111 0101 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 1011 1100

Mantissa (52 bits) =
0100 0010 1001 0000 1000 0000 0000 1011 1010 1111 1111 0101 0011


Decimal number 0.000 000 000 000 000 000 008 538 21 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1011 1100 - 0100 0010 1001 0000 1000 0000 0000 1011 1010 1111 1111 0101 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