-0.000 000 000 000 000 222 044 604 925 013 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.000 000 000 000 000 222 044 604 925 013(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
-0.000 000 000 000 000 222 044 604 925 013(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. Start with the positive version of the number:

|-0.000 000 000 000 000 222 044 604 925 013| = 0.000 000 000 000 000 222 044 604 925 013


2. 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;

3. 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)


4. Convert to binary (base 2) the fractional part: 0.000 000 000 000 000 222 044 604 925 013.

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) 0.000 000 000 000 000 222 044 604 925 013 × 2 = 0 + 0.000 000 000 000 000 444 089 209 850 026;
  • 2) 0.000 000 000 000 000 444 089 209 850 026 × 2 = 0 + 0.000 000 000 000 000 888 178 419 700 052;
  • 3) 0.000 000 000 000 000 888 178 419 700 052 × 2 = 0 + 0.000 000 000 000 001 776 356 839 400 104;
  • 4) 0.000 000 000 000 001 776 356 839 400 104 × 2 = 0 + 0.000 000 000 000 003 552 713 678 800 208;
  • 5) 0.000 000 000 000 003 552 713 678 800 208 × 2 = 0 + 0.000 000 000 000 007 105 427 357 600 416;
  • 6) 0.000 000 000 000 007 105 427 357 600 416 × 2 = 0 + 0.000 000 000 000 014 210 854 715 200 832;
  • 7) 0.000 000 000 000 014 210 854 715 200 832 × 2 = 0 + 0.000 000 000 000 028 421 709 430 401 664;
  • 8) 0.000 000 000 000 028 421 709 430 401 664 × 2 = 0 + 0.000 000 000 000 056 843 418 860 803 328;
  • 9) 0.000 000 000 000 056 843 418 860 803 328 × 2 = 0 + 0.000 000 000 000 113 686 837 721 606 656;
  • 10) 0.000 000 000 000 113 686 837 721 606 656 × 2 = 0 + 0.000 000 000 000 227 373 675 443 213 312;
  • 11) 0.000 000 000 000 227 373 675 443 213 312 × 2 = 0 + 0.000 000 000 000 454 747 350 886 426 624;
  • 12) 0.000 000 000 000 454 747 350 886 426 624 × 2 = 0 + 0.000 000 000 000 909 494 701 772 853 248;
  • 13) 0.000 000 000 000 909 494 701 772 853 248 × 2 = 0 + 0.000 000 000 001 818 989 403 545 706 496;
  • 14) 0.000 000 000 001 818 989 403 545 706 496 × 2 = 0 + 0.000 000 000 003 637 978 807 091 412 992;
  • 15) 0.000 000 000 003 637 978 807 091 412 992 × 2 = 0 + 0.000 000 000 007 275 957 614 182 825 984;
  • 16) 0.000 000 000 007 275 957 614 182 825 984 × 2 = 0 + 0.000 000 000 014 551 915 228 365 651 968;
  • 17) 0.000 000 000 014 551 915 228 365 651 968 × 2 = 0 + 0.000 000 000 029 103 830 456 731 303 936;
  • 18) 0.000 000 000 029 103 830 456 731 303 936 × 2 = 0 + 0.000 000 000 058 207 660 913 462 607 872;
  • 19) 0.000 000 000 058 207 660 913 462 607 872 × 2 = 0 + 0.000 000 000 116 415 321 826 925 215 744;
  • 20) 0.000 000 000 116 415 321 826 925 215 744 × 2 = 0 + 0.000 000 000 232 830 643 653 850 431 488;
  • 21) 0.000 000 000 232 830 643 653 850 431 488 × 2 = 0 + 0.000 000 000 465 661 287 307 700 862 976;
  • 22) 0.000 000 000 465 661 287 307 700 862 976 × 2 = 0 + 0.000 000 000 931 322 574 615 401 725 952;
  • 23) 0.000 000 000 931 322 574 615 401 725 952 × 2 = 0 + 0.000 000 001 862 645 149 230 803 451 904;
  • 24) 0.000 000 001 862 645 149 230 803 451 904 × 2 = 0 + 0.000 000 003 725 290 298 461 606 903 808;
  • 25) 0.000 000 003 725 290 298 461 606 903 808 × 2 = 0 + 0.000 000 007 450 580 596 923 213 807 616;
  • 26) 0.000 000 007 450 580 596 923 213 807 616 × 2 = 0 + 0.000 000 014 901 161 193 846 427 615 232;
  • 27) 0.000 000 014 901 161 193 846 427 615 232 × 2 = 0 + 0.000 000 029 802 322 387 692 855 230 464;
  • 28) 0.000 000 029 802 322 387 692 855 230 464 × 2 = 0 + 0.000 000 059 604 644 775 385 710 460 928;
  • 29) 0.000 000 059 604 644 775 385 710 460 928 × 2 = 0 + 0.000 000 119 209 289 550 771 420 921 856;
  • 30) 0.000 000 119 209 289 550 771 420 921 856 × 2 = 0 + 0.000 000 238 418 579 101 542 841 843 712;
  • 31) 0.000 000 238 418 579 101 542 841 843 712 × 2 = 0 + 0.000 000 476 837 158 203 085 683 687 424;
  • 32) 0.000 000 476 837 158 203 085 683 687 424 × 2 = 0 + 0.000 000 953 674 316 406 171 367 374 848;
  • 33) 0.000 000 953 674 316 406 171 367 374 848 × 2 = 0 + 0.000 001 907 348 632 812 342 734 749 696;
  • 34) 0.000 001 907 348 632 812 342 734 749 696 × 2 = 0 + 0.000 003 814 697 265 624 685 469 499 392;
  • 35) 0.000 003 814 697 265 624 685 469 499 392 × 2 = 0 + 0.000 007 629 394 531 249 370 938 998 784;
  • 36) 0.000 007 629 394 531 249 370 938 998 784 × 2 = 0 + 0.000 015 258 789 062 498 741 877 997 568;
  • 37) 0.000 015 258 789 062 498 741 877 997 568 × 2 = 0 + 0.000 030 517 578 124 997 483 755 995 136;
  • 38) 0.000 030 517 578 124 997 483 755 995 136 × 2 = 0 + 0.000 061 035 156 249 994 967 511 990 272;
  • 39) 0.000 061 035 156 249 994 967 511 990 272 × 2 = 0 + 0.000 122 070 312 499 989 935 023 980 544;
  • 40) 0.000 122 070 312 499 989 935 023 980 544 × 2 = 0 + 0.000 244 140 624 999 979 870 047 961 088;
  • 41) 0.000 244 140 624 999 979 870 047 961 088 × 2 = 0 + 0.000 488 281 249 999 959 740 095 922 176;
  • 42) 0.000 488 281 249 999 959 740 095 922 176 × 2 = 0 + 0.000 976 562 499 999 919 480 191 844 352;
  • 43) 0.000 976 562 499 999 919 480 191 844 352 × 2 = 0 + 0.001 953 124 999 999 838 960 383 688 704;
  • 44) 0.001 953 124 999 999 838 960 383 688 704 × 2 = 0 + 0.003 906 249 999 999 677 920 767 377 408;
  • 45) 0.003 906 249 999 999 677 920 767 377 408 × 2 = 0 + 0.007 812 499 999 999 355 841 534 754 816;
  • 46) 0.007 812 499 999 999 355 841 534 754 816 × 2 = 0 + 0.015 624 999 999 998 711 683 069 509 632;
  • 47) 0.015 624 999 999 998 711 683 069 509 632 × 2 = 0 + 0.031 249 999 999 997 423 366 139 019 264;
  • 48) 0.031 249 999 999 997 423 366 139 019 264 × 2 = 0 + 0.062 499 999 999 994 846 732 278 038 528;
  • 49) 0.062 499 999 999 994 846 732 278 038 528 × 2 = 0 + 0.124 999 999 999 989 693 464 556 077 056;
  • 50) 0.124 999 999 999 989 693 464 556 077 056 × 2 = 0 + 0.249 999 999 999 979 386 929 112 154 112;
  • 51) 0.249 999 999 999 979 386 929 112 154 112 × 2 = 0 + 0.499 999 999 999 958 773 858 224 308 224;
  • 52) 0.499 999 999 999 958 773 858 224 308 224 × 2 = 0 + 0.999 999 999 999 917 547 716 448 616 448;
  • 53) 0.999 999 999 999 917 547 716 448 616 448 × 2 = 1 + 0.999 999 999 999 835 095 432 897 232 896;
  • 54) 0.999 999 999 999 835 095 432 897 232 896 × 2 = 1 + 0.999 999 999 999 670 190 865 794 465 792;
  • 55) 0.999 999 999 999 670 190 865 794 465 792 × 2 = 1 + 0.999 999 999 999 340 381 731 588 931 584;
  • 56) 0.999 999 999 999 340 381 731 588 931 584 × 2 = 1 + 0.999 999 999 998 680 763 463 177 863 168;
  • 57) 0.999 999 999 998 680 763 463 177 863 168 × 2 = 1 + 0.999 999 999 997 361 526 926 355 726 336;
  • 58) 0.999 999 999 997 361 526 926 355 726 336 × 2 = 1 + 0.999 999 999 994 723 053 852 711 452 672;
  • 59) 0.999 999 999 994 723 053 852 711 452 672 × 2 = 1 + 0.999 999 999 989 446 107 705 422 905 344;
  • 60) 0.999 999 999 989 446 107 705 422 905 344 × 2 = 1 + 0.999 999 999 978 892 215 410 845 810 688;
  • 61) 0.999 999 999 978 892 215 410 845 810 688 × 2 = 1 + 0.999 999 999 957 784 430 821 691 621 376;
  • 62) 0.999 999 999 957 784 430 821 691 621 376 × 2 = 1 + 0.999 999 999 915 568 861 643 383 242 752;
  • 63) 0.999 999 999 915 568 861 643 383 242 752 × 2 = 1 + 0.999 999 999 831 137 723 286 766 485 504;
  • 64) 0.999 999 999 831 137 723 286 766 485 504 × 2 = 1 + 0.999 999 999 662 275 446 573 532 971 008;
  • 65) 0.999 999 999 662 275 446 573 532 971 008 × 2 = 1 + 0.999 999 999 324 550 893 147 065 942 016;
  • 66) 0.999 999 999 324 550 893 147 065 942 016 × 2 = 1 + 0.999 999 998 649 101 786 294 131 884 032;
  • 67) 0.999 999 998 649 101 786 294 131 884 032 × 2 = 1 + 0.999 999 997 298 203 572 588 263 768 064;
  • 68) 0.999 999 997 298 203 572 588 263 768 064 × 2 = 1 + 0.999 999 994 596 407 145 176 527 536 128;
  • 69) 0.999 999 994 596 407 145 176 527 536 128 × 2 = 1 + 0.999 999 989 192 814 290 353 055 072 256;
  • 70) 0.999 999 989 192 814 290 353 055 072 256 × 2 = 1 + 0.999 999 978 385 628 580 706 110 144 512;
  • 71) 0.999 999 978 385 628 580 706 110 144 512 × 2 = 1 + 0.999 999 956 771 257 161 412 220 289 024;
  • 72) 0.999 999 956 771 257 161 412 220 289 024 × 2 = 1 + 0.999 999 913 542 514 322 824 440 578 048;
  • 73) 0.999 999 913 542 514 322 824 440 578 048 × 2 = 1 + 0.999 999 827 085 028 645 648 881 156 096;
  • 74) 0.999 999 827 085 028 645 648 881 156 096 × 2 = 1 + 0.999 999 654 170 057 291 297 762 312 192;
  • 75) 0.999 999 654 170 057 291 297 762 312 192 × 2 = 1 + 0.999 999 308 340 114 582 595 524 624 384;
  • 76) 0.999 999 308 340 114 582 595 524 624 384 × 2 = 1 + 0.999 998 616 680 229 165 191 049 248 768;
  • 77) 0.999 998 616 680 229 165 191 049 248 768 × 2 = 1 + 0.999 997 233 360 458 330 382 098 497 536;
  • 78) 0.999 997 233 360 458 330 382 098 497 536 × 2 = 1 + 0.999 994 466 720 916 660 764 196 995 072;
  • 79) 0.999 994 466 720 916 660 764 196 995 072 × 2 = 1 + 0.999 988 933 441 833 321 528 393 990 144;
  • 80) 0.999 988 933 441 833 321 528 393 990 144 × 2 = 1 + 0.999 977 866 883 666 643 056 787 980 288;
  • 81) 0.999 977 866 883 666 643 056 787 980 288 × 2 = 1 + 0.999 955 733 767 333 286 113 575 960 576;
  • 82) 0.999 955 733 767 333 286 113 575 960 576 × 2 = 1 + 0.999 911 467 534 666 572 227 151 921 152;
  • 83) 0.999 911 467 534 666 572 227 151 921 152 × 2 = 1 + 0.999 822 935 069 333 144 454 303 842 304;
  • 84) 0.999 822 935 069 333 144 454 303 842 304 × 2 = 1 + 0.999 645 870 138 666 288 908 607 684 608;
  • 85) 0.999 645 870 138 666 288 908 607 684 608 × 2 = 1 + 0.999 291 740 277 332 577 817 215 369 216;
  • 86) 0.999 291 740 277 332 577 817 215 369 216 × 2 = 1 + 0.998 583 480 554 665 155 634 430 738 432;
  • 87) 0.998 583 480 554 665 155 634 430 738 432 × 2 = 1 + 0.997 166 961 109 330 311 268 861 476 864;
  • 88) 0.997 166 961 109 330 311 268 861 476 864 × 2 = 1 + 0.994 333 922 218 660 622 537 722 953 728;
  • 89) 0.994 333 922 218 660 622 537 722 953 728 × 2 = 1 + 0.988 667 844 437 321 245 075 445 907 456;
  • 90) 0.988 667 844 437 321 245 075 445 907 456 × 2 = 1 + 0.977 335 688 874 642 490 150 891 814 912;
  • 91) 0.977 335 688 874 642 490 150 891 814 912 × 2 = 1 + 0.954 671 377 749 284 980 301 783 629 824;
  • 92) 0.954 671 377 749 284 980 301 783 629 824 × 2 = 1 + 0.909 342 755 498 569 960 603 567 259 648;
  • 93) 0.909 342 755 498 569 960 603 567 259 648 × 2 = 1 + 0.818 685 510 997 139 921 207 134 519 296;
  • 94) 0.818 685 510 997 139 921 207 134 519 296 × 2 = 1 + 0.637 371 021 994 279 842 414 269 038 592;
  • 95) 0.637 371 021 994 279 842 414 269 038 592 × 2 = 1 + 0.274 742 043 988 559 684 828 538 077 184;
  • 96) 0.274 742 043 988 559 684 828 538 077 184 × 2 = 0 + 0.549 484 087 977 119 369 657 076 154 368;
  • 97) 0.549 484 087 977 119 369 657 076 154 368 × 2 = 1 + 0.098 968 175 954 238 739 314 152 308 736;
  • 98) 0.098 968 175 954 238 739 314 152 308 736 × 2 = 0 + 0.197 936 351 908 477 478 628 304 617 472;
  • 99) 0.197 936 351 908 477 478 628 304 617 472 × 2 = 0 + 0.395 872 703 816 954 957 256 609 234 944;
  • 100) 0.395 872 703 816 954 957 256 609 234 944 × 2 = 0 + 0.791 745 407 633 909 914 513 218 469 888;
  • 101) 0.791 745 407 633 909 914 513 218 469 888 × 2 = 1 + 0.583 490 815 267 819 829 026 436 939 776;
  • 102) 0.583 490 815 267 819 829 026 436 939 776 × 2 = 1 + 0.166 981 630 535 639 658 052 873 879 552;
  • 103) 0.166 981 630 535 639 658 052 873 879 552 × 2 = 0 + 0.333 963 261 071 279 316 105 747 759 104;
  • 104) 0.333 963 261 071 279 316 105 747 759 104 × 2 = 0 + 0.667 926 522 142 558 632 211 495 518 208;
  • 105) 0.667 926 522 142 558 632 211 495 518 208 × 2 = 1 + 0.335 853 044 285 117 264 422 991 036 416;

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).


5. 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:


0.000 000 000 000 000 222 044 604 925 013(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1110 1000 1100 1(2)

6. Positive number before normalization:

0.000 000 000 000 000 222 044 604 925 013(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1110 1000 1100 1(2)

7. Normalize the binary representation of the number.

Shift the decimal mark 53 positions to the right, so that only one non zero digit remains to the left of it:


0.000 000 000 000 000 222 044 604 925 013(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1110 1000 1100 1(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1110 1000 1100 1(2) × 20 =

1.1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1101 0001 1001(2) × 2-53


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): -53

Mantissa (not normalized):
1.1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1101 0001 1001


9. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

-53 + 2(11-1) - 1 =

(-53 + 1 023)(10) =

970(10)


10. 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;
  • 970 ÷ 2 = 485 + 0;
  • 485 ÷ 2 = 242 + 1;
  • 242 ÷ 2 = 121 + 0;
  • 121 ÷ 2 = 60 + 1;
  • 60 ÷ 2 = 30 + 0;
  • 30 ÷ 2 = 15 + 0;
  • 15 ÷ 2 = 7 + 1;
  • 7 ÷ 2 = 3 + 1;
  • 3 ÷ 2 = 1 + 1;
  • 1 ÷ 2 = 0 + 1;

11. Construct the base 2 representation of the adjusted exponent.

Take all the remainders starting from the bottom of the list constructed above.


Exponent (adjusted) =

970(10) =

011 1100 1010(2)


12. 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. 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1101 0001 1001 =

1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1101 0001 1001


13. The three elements that make up the number's 64 bit double precision IEEE 754 binary floating point representation:

Sign (1 bit) =
1 (a negative number)

Exponent (11 bits) =
011 1100 1010

Mantissa (52 bits) =
1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1101 0001 1001


Decimal number -0.000 000 000 000 000 222 044 604 925 013 converted to 64 bit double precision IEEE 754 binary floating point representation:

1 - 011 1100 1010 - 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1101 0001 1001

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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