0.000 000 000 000 000 000 008 537 088 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 000 000 008 537 088(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
0.000 000 000 000 000 000 008 537 088(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: 0.000 000 000 000 000 000 008 537 088.

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) 0.000 000 000 000 000 000 008 537 088 × 2 = 0 + 0.000 000 000 000 000 000 017 074 176;
  • 2) 0.000 000 000 000 000 000 017 074 176 × 2 = 0 + 0.000 000 000 000 000 000 034 148 352;
  • 3) 0.000 000 000 000 000 000 034 148 352 × 2 = 0 + 0.000 000 000 000 000 000 068 296 704;
  • 4) 0.000 000 000 000 000 000 068 296 704 × 2 = 0 + 0.000 000 000 000 000 000 136 593 408;
  • 5) 0.000 000 000 000 000 000 136 593 408 × 2 = 0 + 0.000 000 000 000 000 000 273 186 816;
  • 6) 0.000 000 000 000 000 000 273 186 816 × 2 = 0 + 0.000 000 000 000 000 000 546 373 632;
  • 7) 0.000 000 000 000 000 000 546 373 632 × 2 = 0 + 0.000 000 000 000 000 001 092 747 264;
  • 8) 0.000 000 000 000 000 001 092 747 264 × 2 = 0 + 0.000 000 000 000 000 002 185 494 528;
  • 9) 0.000 000 000 000 000 002 185 494 528 × 2 = 0 + 0.000 000 000 000 000 004 370 989 056;
  • 10) 0.000 000 000 000 000 004 370 989 056 × 2 = 0 + 0.000 000 000 000 000 008 741 978 112;
  • 11) 0.000 000 000 000 000 008 741 978 112 × 2 = 0 + 0.000 000 000 000 000 017 483 956 224;
  • 12) 0.000 000 000 000 000 017 483 956 224 × 2 = 0 + 0.000 000 000 000 000 034 967 912 448;
  • 13) 0.000 000 000 000 000 034 967 912 448 × 2 = 0 + 0.000 000 000 000 000 069 935 824 896;
  • 14) 0.000 000 000 000 000 069 935 824 896 × 2 = 0 + 0.000 000 000 000 000 139 871 649 792;
  • 15) 0.000 000 000 000 000 139 871 649 792 × 2 = 0 + 0.000 000 000 000 000 279 743 299 584;
  • 16) 0.000 000 000 000 000 279 743 299 584 × 2 = 0 + 0.000 000 000 000 000 559 486 599 168;
  • 17) 0.000 000 000 000 000 559 486 599 168 × 2 = 0 + 0.000 000 000 000 001 118 973 198 336;
  • 18) 0.000 000 000 000 001 118 973 198 336 × 2 = 0 + 0.000 000 000 000 002 237 946 396 672;
  • 19) 0.000 000 000 000 002 237 946 396 672 × 2 = 0 + 0.000 000 000 000 004 475 892 793 344;
  • 20) 0.000 000 000 000 004 475 892 793 344 × 2 = 0 + 0.000 000 000 000 008 951 785 586 688;
  • 21) 0.000 000 000 000 008 951 785 586 688 × 2 = 0 + 0.000 000 000 000 017 903 571 173 376;
  • 22) 0.000 000 000 000 017 903 571 173 376 × 2 = 0 + 0.000 000 000 000 035 807 142 346 752;
  • 23) 0.000 000 000 000 035 807 142 346 752 × 2 = 0 + 0.000 000 000 000 071 614 284 693 504;
  • 24) 0.000 000 000 000 071 614 284 693 504 × 2 = 0 + 0.000 000 000 000 143 228 569 387 008;
  • 25) 0.000 000 000 000 143 228 569 387 008 × 2 = 0 + 0.000 000 000 000 286 457 138 774 016;
  • 26) 0.000 000 000 000 286 457 138 774 016 × 2 = 0 + 0.000 000 000 000 572 914 277 548 032;
  • 27) 0.000 000 000 000 572 914 277 548 032 × 2 = 0 + 0.000 000 000 001 145 828 555 096 064;
  • 28) 0.000 000 000 001 145 828 555 096 064 × 2 = 0 + 0.000 000 000 002 291 657 110 192 128;
  • 29) 0.000 000 000 002 291 657 110 192 128 × 2 = 0 + 0.000 000 000 004 583 314 220 384 256;
  • 30) 0.000 000 000 004 583 314 220 384 256 × 2 = 0 + 0.000 000 000 009 166 628 440 768 512;
  • 31) 0.000 000 000 009 166 628 440 768 512 × 2 = 0 + 0.000 000 000 018 333 256 881 537 024;
  • 32) 0.000 000 000 018 333 256 881 537 024 × 2 = 0 + 0.000 000 000 036 666 513 763 074 048;
  • 33) 0.000 000 000 036 666 513 763 074 048 × 2 = 0 + 0.000 000 000 073 333 027 526 148 096;
  • 34) 0.000 000 000 073 333 027 526 148 096 × 2 = 0 + 0.000 000 000 146 666 055 052 296 192;
  • 35) 0.000 000 000 146 666 055 052 296 192 × 2 = 0 + 0.000 000 000 293 332 110 104 592 384;
  • 36) 0.000 000 000 293 332 110 104 592 384 × 2 = 0 + 0.000 000 000 586 664 220 209 184 768;
  • 37) 0.000 000 000 586 664 220 209 184 768 × 2 = 0 + 0.000 000 001 173 328 440 418 369 536;
  • 38) 0.000 000 001 173 328 440 418 369 536 × 2 = 0 + 0.000 000 002 346 656 880 836 739 072;
  • 39) 0.000 000 002 346 656 880 836 739 072 × 2 = 0 + 0.000 000 004 693 313 761 673 478 144;
  • 40) 0.000 000 004 693 313 761 673 478 144 × 2 = 0 + 0.000 000 009 386 627 523 346 956 288;
  • 41) 0.000 000 009 386 627 523 346 956 288 × 2 = 0 + 0.000 000 018 773 255 046 693 912 576;
  • 42) 0.000 000 018 773 255 046 693 912 576 × 2 = 0 + 0.000 000 037 546 510 093 387 825 152;
  • 43) 0.000 000 037 546 510 093 387 825 152 × 2 = 0 + 0.000 000 075 093 020 186 775 650 304;
  • 44) 0.000 000 075 093 020 186 775 650 304 × 2 = 0 + 0.000 000 150 186 040 373 551 300 608;
  • 45) 0.000 000 150 186 040 373 551 300 608 × 2 = 0 + 0.000 000 300 372 080 747 102 601 216;
  • 46) 0.000 000 300 372 080 747 102 601 216 × 2 = 0 + 0.000 000 600 744 161 494 205 202 432;
  • 47) 0.000 000 600 744 161 494 205 202 432 × 2 = 0 + 0.000 001 201 488 322 988 410 404 864;
  • 48) 0.000 001 201 488 322 988 410 404 864 × 2 = 0 + 0.000 002 402 976 645 976 820 809 728;
  • 49) 0.000 002 402 976 645 976 820 809 728 × 2 = 0 + 0.000 004 805 953 291 953 641 619 456;
  • 50) 0.000 004 805 953 291 953 641 619 456 × 2 = 0 + 0.000 009 611 906 583 907 283 238 912;
  • 51) 0.000 009 611 906 583 907 283 238 912 × 2 = 0 + 0.000 019 223 813 167 814 566 477 824;
  • 52) 0.000 019 223 813 167 814 566 477 824 × 2 = 0 + 0.000 038 447 626 335 629 132 955 648;
  • 53) 0.000 038 447 626 335 629 132 955 648 × 2 = 0 + 0.000 076 895 252 671 258 265 911 296;
  • 54) 0.000 076 895 252 671 258 265 911 296 × 2 = 0 + 0.000 153 790 505 342 516 531 822 592;
  • 55) 0.000 153 790 505 342 516 531 822 592 × 2 = 0 + 0.000 307 581 010 685 033 063 645 184;
  • 56) 0.000 307 581 010 685 033 063 645 184 × 2 = 0 + 0.000 615 162 021 370 066 127 290 368;
  • 57) 0.000 615 162 021 370 066 127 290 368 × 2 = 0 + 0.001 230 324 042 740 132 254 580 736;
  • 58) 0.001 230 324 042 740 132 254 580 736 × 2 = 0 + 0.002 460 648 085 480 264 509 161 472;
  • 59) 0.002 460 648 085 480 264 509 161 472 × 2 = 0 + 0.004 921 296 170 960 529 018 322 944;
  • 60) 0.004 921 296 170 960 529 018 322 944 × 2 = 0 + 0.009 842 592 341 921 058 036 645 888;
  • 61) 0.009 842 592 341 921 058 036 645 888 × 2 = 0 + 0.019 685 184 683 842 116 073 291 776;
  • 62) 0.019 685 184 683 842 116 073 291 776 × 2 = 0 + 0.039 370 369 367 684 232 146 583 552;
  • 63) 0.039 370 369 367 684 232 146 583 552 × 2 = 0 + 0.078 740 738 735 368 464 293 167 104;
  • 64) 0.078 740 738 735 368 464 293 167 104 × 2 = 0 + 0.157 481 477 470 736 928 586 334 208;
  • 65) 0.157 481 477 470 736 928 586 334 208 × 2 = 0 + 0.314 962 954 941 473 857 172 668 416;
  • 66) 0.314 962 954 941 473 857 172 668 416 × 2 = 0 + 0.629 925 909 882 947 714 345 336 832;
  • 67) 0.629 925 909 882 947 714 345 336 832 × 2 = 1 + 0.259 851 819 765 895 428 690 673 664;
  • 68) 0.259 851 819 765 895 428 690 673 664 × 2 = 0 + 0.519 703 639 531 790 857 381 347 328;
  • 69) 0.519 703 639 531 790 857 381 347 328 × 2 = 1 + 0.039 407 279 063 581 714 762 694 656;
  • 70) 0.039 407 279 063 581 714 762 694 656 × 2 = 0 + 0.078 814 558 127 163 429 525 389 312;
  • 71) 0.078 814 558 127 163 429 525 389 312 × 2 = 0 + 0.157 629 116 254 326 859 050 778 624;
  • 72) 0.157 629 116 254 326 859 050 778 624 × 2 = 0 + 0.315 258 232 508 653 718 101 557 248;
  • 73) 0.315 258 232 508 653 718 101 557 248 × 2 = 0 + 0.630 516 465 017 307 436 203 114 496;
  • 74) 0.630 516 465 017 307 436 203 114 496 × 2 = 1 + 0.261 032 930 034 614 872 406 228 992;
  • 75) 0.261 032 930 034 614 872 406 228 992 × 2 = 0 + 0.522 065 860 069 229 744 812 457 984;
  • 76) 0.522 065 860 069 229 744 812 457 984 × 2 = 1 + 0.044 131 720 138 459 489 624 915 968;
  • 77) 0.044 131 720 138 459 489 624 915 968 × 2 = 0 + 0.088 263 440 276 918 979 249 831 936;
  • 78) 0.088 263 440 276 918 979 249 831 936 × 2 = 0 + 0.176 526 880 553 837 958 499 663 872;
  • 79) 0.176 526 880 553 837 958 499 663 872 × 2 = 0 + 0.353 053 761 107 675 916 999 327 744;
  • 80) 0.353 053 761 107 675 916 999 327 744 × 2 = 0 + 0.706 107 522 215 351 833 998 655 488;
  • 81) 0.706 107 522 215 351 833 998 655 488 × 2 = 1 + 0.412 215 044 430 703 667 997 310 976;
  • 82) 0.412 215 044 430 703 667 997 310 976 × 2 = 0 + 0.824 430 088 861 407 335 994 621 952;
  • 83) 0.824 430 088 861 407 335 994 621 952 × 2 = 1 + 0.648 860 177 722 814 671 989 243 904;
  • 84) 0.648 860 177 722 814 671 989 243 904 × 2 = 1 + 0.297 720 355 445 629 343 978 487 808;
  • 85) 0.297 720 355 445 629 343 978 487 808 × 2 = 0 + 0.595 440 710 891 258 687 956 975 616;
  • 86) 0.595 440 710 891 258 687 956 975 616 × 2 = 1 + 0.190 881 421 782 517 375 913 951 232;
  • 87) 0.190 881 421 782 517 375 913 951 232 × 2 = 0 + 0.381 762 843 565 034 751 827 902 464;
  • 88) 0.381 762 843 565 034 751 827 902 464 × 2 = 0 + 0.763 525 687 130 069 503 655 804 928;
  • 89) 0.763 525 687 130 069 503 655 804 928 × 2 = 1 + 0.527 051 374 260 139 007 311 609 856;
  • 90) 0.527 051 374 260 139 007 311 609 856 × 2 = 1 + 0.054 102 748 520 278 014 623 219 712;
  • 91) 0.054 102 748 520 278 014 623 219 712 × 2 = 0 + 0.108 205 497 040 556 029 246 439 424;
  • 92) 0.108 205 497 040 556 029 246 439 424 × 2 = 0 + 0.216 410 994 081 112 058 492 878 848;
  • 93) 0.216 410 994 081 112 058 492 878 848 × 2 = 0 + 0.432 821 988 162 224 116 985 757 696;
  • 94) 0.432 821 988 162 224 116 985 757 696 × 2 = 0 + 0.865 643 976 324 448 233 971 515 392;
  • 95) 0.865 643 976 324 448 233 971 515 392 × 2 = 1 + 0.731 287 952 648 896 467 943 030 784;
  • 96) 0.731 287 952 648 896 467 943 030 784 × 2 = 1 + 0.462 575 905 297 792 935 886 061 568;
  • 97) 0.462 575 905 297 792 935 886 061 568 × 2 = 0 + 0.925 151 810 595 585 871 772 123 136;
  • 98) 0.925 151 810 595 585 871 772 123 136 × 2 = 1 + 0.850 303 621 191 171 743 544 246 272;
  • 99) 0.850 303 621 191 171 743 544 246 272 × 2 = 1 + 0.700 607 242 382 343 487 088 492 544;
  • 100) 0.700 607 242 382 343 487 088 492 544 × 2 = 1 + 0.401 214 484 764 686 974 176 985 088;
  • 101) 0.401 214 484 764 686 974 176 985 088 × 2 = 0 + 0.802 428 969 529 373 948 353 970 176;
  • 102) 0.802 428 969 529 373 948 353 970 176 × 2 = 1 + 0.604 857 939 058 747 896 707 940 352;
  • 103) 0.604 857 939 058 747 896 707 940 352 × 2 = 1 + 0.209 715 878 117 495 793 415 880 704;
  • 104) 0.209 715 878 117 495 793 415 880 704 × 2 = 0 + 0.419 431 756 234 991 586 831 761 408;
  • 105) 0.419 431 756 234 991 586 831 761 408 × 2 = 0 + 0.838 863 512 469 983 173 663 522 816;
  • 106) 0.838 863 512 469 983 173 663 522 816 × 2 = 1 + 0.677 727 024 939 966 347 327 045 632;
  • 107) 0.677 727 024 939 966 347 327 045 632 × 2 = 1 + 0.355 454 049 879 932 694 654 091 264;
  • 108) 0.355 454 049 879 932 694 654 091 264 × 2 = 0 + 0.710 908 099 759 865 389 308 182 528;
  • 109) 0.710 908 099 759 865 389 308 182 528 × 2 = 1 + 0.421 816 199 519 730 778 616 365 056;
  • 110) 0.421 816 199 519 730 778 616 365 056 × 2 = 0 + 0.843 632 399 039 461 557 232 730 112;
  • 111) 0.843 632 399 039 461 557 232 730 112 × 2 = 1 + 0.687 264 798 078 923 114 465 460 224;
  • 112) 0.687 264 798 078 923 114 465 460 224 × 2 = 1 + 0.374 529 596 157 846 228 930 920 448;
  • 113) 0.374 529 596 157 846 228 930 920 448 × 2 = 0 + 0.749 059 192 315 692 457 861 840 896;
  • 114) 0.749 059 192 315 692 457 861 840 896 × 2 = 1 + 0.498 118 384 631 384 915 723 681 792;
  • 115) 0.498 118 384 631 384 915 723 681 792 × 2 = 0 + 0.996 236 769 262 769 831 447 363 584;
  • 116) 0.996 236 769 262 769 831 447 363 584 × 2 = 1 + 0.992 473 538 525 539 662 894 727 168;
  • 117) 0.992 473 538 525 539 662 894 727 168 × 2 = 1 + 0.984 947 077 051 079 325 789 454 336;
  • 118) 0.984 947 077 051 079 325 789 454 336 × 2 = 1 + 0.969 894 154 102 158 651 578 908 672;
  • 119) 0.969 894 154 102 158 651 578 908 672 × 2 = 1 + 0.939 788 308 204 317 303 157 817 344;

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:


0.000 000 000 000 000 000 008 537 088(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0101 0000 1011 0100 1100 0011 0111 0110 0110 1011 0101 111(2)

5. Positive number before normalization:

0.000 000 000 000 000 000 008 537 088(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0101 0000 1011 0100 1100 0011 0111 0110 0110 1011 0101 111(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:


0.000 000 000 000 000 000 008 537 088(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0101 0000 1011 0100 1100 0011 0111 0110 0110 1011 0101 111(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0101 0000 1011 0100 1100 0011 0111 0110 0110 1011 0101 111(2) × 20 =

1.0100 0010 1000 0101 1010 0110 0001 1011 1011 0011 0101 1010 1111(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 1000 0101 1010 0110 0001 1011 1011 0011 0101 1010 1111


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 1000 0101 1010 0110 0001 1011 1011 0011 0101 1010 1111 =

0100 0010 1000 0101 1010 0110 0001 1011 1011 0011 0101 1010 1111


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 1000 0101 1010 0110 0001 1011 1011 0011 0101 1010 1111


Decimal number 0.000 000 000 000 000 000 008 537 088 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1011 1100 - 0100 0010 1000 0101 1010 0110 0001 1011 1011 0011 0101 1010 1111

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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