0.009 999 944 444 611 110 827 664 705 285 178 597 291 065 211 927 159 743 720 665 5 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.009 999 944 444 611 110 827 664 705 285 178 597 291 065 211 927 159 743 720 665 5(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
0.009 999 944 444 611 110 827 664 705 285 178 597 291 065 211 927 159 743 720 665 5(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: 0.009 999 944 444 611 110 827 664 705 285 178 597 291 065 211 927 159 743 720 665 5.

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) 0.009 999 944 444 611 110 827 664 705 285 178 597 291 065 211 927 159 743 720 665 5 × 2 = 0 + 0.019 999 888 889 222 221 655 329 410 570 357 194 582 130 423 854 319 487 441 331;
  • 2) 0.019 999 888 889 222 221 655 329 410 570 357 194 582 130 423 854 319 487 441 331 × 2 = 0 + 0.039 999 777 778 444 443 310 658 821 140 714 389 164 260 847 708 638 974 882 662;
  • 3) 0.039 999 777 778 444 443 310 658 821 140 714 389 164 260 847 708 638 974 882 662 × 2 = 0 + 0.079 999 555 556 888 886 621 317 642 281 428 778 328 521 695 417 277 949 765 324;
  • 4) 0.079 999 555 556 888 886 621 317 642 281 428 778 328 521 695 417 277 949 765 324 × 2 = 0 + 0.159 999 111 113 777 773 242 635 284 562 857 556 657 043 390 834 555 899 530 648;
  • 5) 0.159 999 111 113 777 773 242 635 284 562 857 556 657 043 390 834 555 899 530 648 × 2 = 0 + 0.319 998 222 227 555 546 485 270 569 125 715 113 314 086 781 669 111 799 061 296;
  • 6) 0.319 998 222 227 555 546 485 270 569 125 715 113 314 086 781 669 111 799 061 296 × 2 = 0 + 0.639 996 444 455 111 092 970 541 138 251 430 226 628 173 563 338 223 598 122 592;
  • 7) 0.639 996 444 455 111 092 970 541 138 251 430 226 628 173 563 338 223 598 122 592 × 2 = 1 + 0.279 992 888 910 222 185 941 082 276 502 860 453 256 347 126 676 447 196 245 184;
  • 8) 0.279 992 888 910 222 185 941 082 276 502 860 453 256 347 126 676 447 196 245 184 × 2 = 0 + 0.559 985 777 820 444 371 882 164 553 005 720 906 512 694 253 352 894 392 490 368;
  • 9) 0.559 985 777 820 444 371 882 164 553 005 720 906 512 694 253 352 894 392 490 368 × 2 = 1 + 0.119 971 555 640 888 743 764 329 106 011 441 813 025 388 506 705 788 784 980 736;
  • 10) 0.119 971 555 640 888 743 764 329 106 011 441 813 025 388 506 705 788 784 980 736 × 2 = 0 + 0.239 943 111 281 777 487 528 658 212 022 883 626 050 777 013 411 577 569 961 472;
  • 11) 0.239 943 111 281 777 487 528 658 212 022 883 626 050 777 013 411 577 569 961 472 × 2 = 0 + 0.479 886 222 563 554 975 057 316 424 045 767 252 101 554 026 823 155 139 922 944;
  • 12) 0.479 886 222 563 554 975 057 316 424 045 767 252 101 554 026 823 155 139 922 944 × 2 = 0 + 0.959 772 445 127 109 950 114 632 848 091 534 504 203 108 053 646 310 279 845 888;
  • 13) 0.959 772 445 127 109 950 114 632 848 091 534 504 203 108 053 646 310 279 845 888 × 2 = 1 + 0.919 544 890 254 219 900 229 265 696 183 069 008 406 216 107 292 620 559 691 776;
  • 14) 0.919 544 890 254 219 900 229 265 696 183 069 008 406 216 107 292 620 559 691 776 × 2 = 1 + 0.839 089 780 508 439 800 458 531 392 366 138 016 812 432 214 585 241 119 383 552;
  • 15) 0.839 089 780 508 439 800 458 531 392 366 138 016 812 432 214 585 241 119 383 552 × 2 = 1 + 0.678 179 561 016 879 600 917 062 784 732 276 033 624 864 429 170 482 238 767 104;
  • 16) 0.678 179 561 016 879 600 917 062 784 732 276 033 624 864 429 170 482 238 767 104 × 2 = 1 + 0.356 359 122 033 759 201 834 125 569 464 552 067 249 728 858 340 964 477 534 208;
  • 17) 0.356 359 122 033 759 201 834 125 569 464 552 067 249 728 858 340 964 477 534 208 × 2 = 0 + 0.712 718 244 067 518 403 668 251 138 929 104 134 499 457 716 681 928 955 068 416;
  • 18) 0.712 718 244 067 518 403 668 251 138 929 104 134 499 457 716 681 928 955 068 416 × 2 = 1 + 0.425 436 488 135 036 807 336 502 277 858 208 268 998 915 433 363 857 910 136 832;
  • 19) 0.425 436 488 135 036 807 336 502 277 858 208 268 998 915 433 363 857 910 136 832 × 2 = 0 + 0.850 872 976 270 073 614 673 004 555 716 416 537 997 830 866 727 715 820 273 664;
  • 20) 0.850 872 976 270 073 614 673 004 555 716 416 537 997 830 866 727 715 820 273 664 × 2 = 1 + 0.701 745 952 540 147 229 346 009 111 432 833 075 995 661 733 455 431 640 547 328;
  • 21) 0.701 745 952 540 147 229 346 009 111 432 833 075 995 661 733 455 431 640 547 328 × 2 = 1 + 0.403 491 905 080 294 458 692 018 222 865 666 151 991 323 466 910 863 281 094 656;
  • 22) 0.403 491 905 080 294 458 692 018 222 865 666 151 991 323 466 910 863 281 094 656 × 2 = 0 + 0.806 983 810 160 588 917 384 036 445 731 332 303 982 646 933 821 726 562 189 312;
  • 23) 0.806 983 810 160 588 917 384 036 445 731 332 303 982 646 933 821 726 562 189 312 × 2 = 1 + 0.613 967 620 321 177 834 768 072 891 462 664 607 965 293 867 643 453 124 378 624;
  • 24) 0.613 967 620 321 177 834 768 072 891 462 664 607 965 293 867 643 453 124 378 624 × 2 = 1 + 0.227 935 240 642 355 669 536 145 782 925 329 215 930 587 735 286 906 248 757 248;
  • 25) 0.227 935 240 642 355 669 536 145 782 925 329 215 930 587 735 286 906 248 757 248 × 2 = 0 + 0.455 870 481 284 711 339 072 291 565 850 658 431 861 175 470 573 812 497 514 496;
  • 26) 0.455 870 481 284 711 339 072 291 565 850 658 431 861 175 470 573 812 497 514 496 × 2 = 0 + 0.911 740 962 569 422 678 144 583 131 701 316 863 722 350 941 147 624 995 028 992;
  • 27) 0.911 740 962 569 422 678 144 583 131 701 316 863 722 350 941 147 624 995 028 992 × 2 = 1 + 0.823 481 925 138 845 356 289 166 263 402 633 727 444 701 882 295 249 990 057 984;
  • 28) 0.823 481 925 138 845 356 289 166 263 402 633 727 444 701 882 295 249 990 057 984 × 2 = 1 + 0.646 963 850 277 690 712 578 332 526 805 267 454 889 403 764 590 499 980 115 968;
  • 29) 0.646 963 850 277 690 712 578 332 526 805 267 454 889 403 764 590 499 980 115 968 × 2 = 1 + 0.293 927 700 555 381 425 156 665 053 610 534 909 778 807 529 180 999 960 231 936;
  • 30) 0.293 927 700 555 381 425 156 665 053 610 534 909 778 807 529 180 999 960 231 936 × 2 = 0 + 0.587 855 401 110 762 850 313 330 107 221 069 819 557 615 058 361 999 920 463 872;
  • 31) 0.587 855 401 110 762 850 313 330 107 221 069 819 557 615 058 361 999 920 463 872 × 2 = 1 + 0.175 710 802 221 525 700 626 660 214 442 139 639 115 230 116 723 999 840 927 744;
  • 32) 0.175 710 802 221 525 700 626 660 214 442 139 639 115 230 116 723 999 840 927 744 × 2 = 0 + 0.351 421 604 443 051 401 253 320 428 884 279 278 230 460 233 447 999 681 855 488;
  • 33) 0.351 421 604 443 051 401 253 320 428 884 279 278 230 460 233 447 999 681 855 488 × 2 = 0 + 0.702 843 208 886 102 802 506 640 857 768 558 556 460 920 466 895 999 363 710 976;
  • 34) 0.702 843 208 886 102 802 506 640 857 768 558 556 460 920 466 895 999 363 710 976 × 2 = 1 + 0.405 686 417 772 205 605 013 281 715 537 117 112 921 840 933 791 998 727 421 952;
  • 35) 0.405 686 417 772 205 605 013 281 715 537 117 112 921 840 933 791 998 727 421 952 × 2 = 0 + 0.811 372 835 544 411 210 026 563 431 074 234 225 843 681 867 583 997 454 843 904;
  • 36) 0.811 372 835 544 411 210 026 563 431 074 234 225 843 681 867 583 997 454 843 904 × 2 = 1 + 0.622 745 671 088 822 420 053 126 862 148 468 451 687 363 735 167 994 909 687 808;
  • 37) 0.622 745 671 088 822 420 053 126 862 148 468 451 687 363 735 167 994 909 687 808 × 2 = 1 + 0.245 491 342 177 644 840 106 253 724 296 936 903 374 727 470 335 989 819 375 616;
  • 38) 0.245 491 342 177 644 840 106 253 724 296 936 903 374 727 470 335 989 819 375 616 × 2 = 0 + 0.490 982 684 355 289 680 212 507 448 593 873 806 749 454 940 671 979 638 751 232;
  • 39) 0.490 982 684 355 289 680 212 507 448 593 873 806 749 454 940 671 979 638 751 232 × 2 = 0 + 0.981 965 368 710 579 360 425 014 897 187 747 613 498 909 881 343 959 277 502 464;
  • 40) 0.981 965 368 710 579 360 425 014 897 187 747 613 498 909 881 343 959 277 502 464 × 2 = 1 + 0.963 930 737 421 158 720 850 029 794 375 495 226 997 819 762 687 918 555 004 928;
  • 41) 0.963 930 737 421 158 720 850 029 794 375 495 226 997 819 762 687 918 555 004 928 × 2 = 1 + 0.927 861 474 842 317 441 700 059 588 750 990 453 995 639 525 375 837 110 009 856;
  • 42) 0.927 861 474 842 317 441 700 059 588 750 990 453 995 639 525 375 837 110 009 856 × 2 = 1 + 0.855 722 949 684 634 883 400 119 177 501 980 907 991 279 050 751 674 220 019 712;
  • 43) 0.855 722 949 684 634 883 400 119 177 501 980 907 991 279 050 751 674 220 019 712 × 2 = 1 + 0.711 445 899 369 269 766 800 238 355 003 961 815 982 558 101 503 348 440 039 424;
  • 44) 0.711 445 899 369 269 766 800 238 355 003 961 815 982 558 101 503 348 440 039 424 × 2 = 1 + 0.422 891 798 738 539 533 600 476 710 007 923 631 965 116 203 006 696 880 078 848;
  • 45) 0.422 891 798 738 539 533 600 476 710 007 923 631 965 116 203 006 696 880 078 848 × 2 = 0 + 0.845 783 597 477 079 067 200 953 420 015 847 263 930 232 406 013 393 760 157 696;
  • 46) 0.845 783 597 477 079 067 200 953 420 015 847 263 930 232 406 013 393 760 157 696 × 2 = 1 + 0.691 567 194 954 158 134 401 906 840 031 694 527 860 464 812 026 787 520 315 392;
  • 47) 0.691 567 194 954 158 134 401 906 840 031 694 527 860 464 812 026 787 520 315 392 × 2 = 1 + 0.383 134 389 908 316 268 803 813 680 063 389 055 720 929 624 053 575 040 630 784;
  • 48) 0.383 134 389 908 316 268 803 813 680 063 389 055 720 929 624 053 575 040 630 784 × 2 = 0 + 0.766 268 779 816 632 537 607 627 360 126 778 111 441 859 248 107 150 081 261 568;
  • 49) 0.766 268 779 816 632 537 607 627 360 126 778 111 441 859 248 107 150 081 261 568 × 2 = 1 + 0.532 537 559 633 265 075 215 254 720 253 556 222 883 718 496 214 300 162 523 136;
  • 50) 0.532 537 559 633 265 075 215 254 720 253 556 222 883 718 496 214 300 162 523 136 × 2 = 1 + 0.065 075 119 266 530 150 430 509 440 507 112 445 767 436 992 428 600 325 046 272;
  • 51) 0.065 075 119 266 530 150 430 509 440 507 112 445 767 436 992 428 600 325 046 272 × 2 = 0 + 0.130 150 238 533 060 300 861 018 881 014 224 891 534 873 984 857 200 650 092 544;
  • 52) 0.130 150 238 533 060 300 861 018 881 014 224 891 534 873 984 857 200 650 092 544 × 2 = 0 + 0.260 300 477 066 120 601 722 037 762 028 449 783 069 747 969 714 401 300 185 088;
  • 53) 0.260 300 477 066 120 601 722 037 762 028 449 783 069 747 969 714 401 300 185 088 × 2 = 0 + 0.520 600 954 132 241 203 444 075 524 056 899 566 139 495 939 428 802 600 370 176;
  • 54) 0.520 600 954 132 241 203 444 075 524 056 899 566 139 495 939 428 802 600 370 176 × 2 = 1 + 0.041 201 908 264 482 406 888 151 048 113 799 132 278 991 878 857 605 200 740 352;
  • 55) 0.041 201 908 264 482 406 888 151 048 113 799 132 278 991 878 857 605 200 740 352 × 2 = 0 + 0.082 403 816 528 964 813 776 302 096 227 598 264 557 983 757 715 210 401 480 704;
  • 56) 0.082 403 816 528 964 813 776 302 096 227 598 264 557 983 757 715 210 401 480 704 × 2 = 0 + 0.164 807 633 057 929 627 552 604 192 455 196 529 115 967 515 430 420 802 961 408;
  • 57) 0.164 807 633 057 929 627 552 604 192 455 196 529 115 967 515 430 420 802 961 408 × 2 = 0 + 0.329 615 266 115 859 255 105 208 384 910 393 058 231 935 030 860 841 605 922 816;
  • 58) 0.329 615 266 115 859 255 105 208 384 910 393 058 231 935 030 860 841 605 922 816 × 2 = 0 + 0.659 230 532 231 718 510 210 416 769 820 786 116 463 870 061 721 683 211 845 632;
  • 59) 0.659 230 532 231 718 510 210 416 769 820 786 116 463 870 061 721 683 211 845 632 × 2 = 1 + 0.318 461 064 463 437 020 420 833 539 641 572 232 927 740 123 443 366 423 691 264;

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:


0.009 999 944 444 611 110 827 664 705 285 178 597 291 065 211 927 159 743 720 665 5(10) =

0.0000 0010 1000 1111 0101 1011 0011 1010 0101 1001 1111 0110 1100 0100 001(2)

5. Positive number before normalization:

0.009 999 944 444 611 110 827 664 705 285 178 597 291 065 211 927 159 743 720 665 5(10) =

0.0000 0010 1000 1111 0101 1011 0011 1010 0101 1001 1111 0110 1100 0100 001(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 7 positions to the right, so that only one non zero digit remains to the left of it:


0.009 999 944 444 611 110 827 664 705 285 178 597 291 065 211 927 159 743 720 665 5(10) =

0.0000 0010 1000 1111 0101 1011 0011 1010 0101 1001 1111 0110 1100 0100 001(2) =

0.0000 0010 1000 1111 0101 1011 0011 1010 0101 1001 1111 0110 1100 0100 001(2) × 20 =

1.0100 0111 1010 1101 1001 1101 0010 1100 1111 1011 0110 0010 0001(2) × 2-7


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): -7

Mantissa (not normalized):
1.0100 0111 1010 1101 1001 1101 0010 1100 1111 1011 0110 0010 0001


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

-7 + 2(11-1) - 1 =

(-7 + 1 023)(10) =

1 016(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;
  • 1 016 ÷ 2 = 508 + 0;
  • 508 ÷ 2 = 254 + 0;
  • 254 ÷ 2 = 127 + 0;
  • 127 ÷ 2 = 63 + 1;
  • 63 ÷ 2 = 31 + 1;
  • 31 ÷ 2 = 15 + 1;
  • 15 ÷ 2 = 7 + 1;
  • 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) =

1016(10) =

011 1111 1000(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 0111 1010 1101 1001 1101 0010 1100 1111 1011 0110 0010 0001 =

0100 0111 1010 1101 1001 1101 0010 1100 1111 1011 0110 0010 0001


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 1111 1000

Mantissa (52 bits) =
0100 0111 1010 1101 1001 1101 0010 1100 1111 1011 0110 0010 0001


Decimal number 0.009 999 944 444 611 110 827 664 705 285 178 597 291 065 211 927 159 743 720 665 5 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1111 1000 - 0100 0111 1010 1101 1001 1101 0010 1100 1111 1011 0110 0010 0001

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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