0.000 000 000 000 000 013 248 735 9 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 000 013 248 735 9(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
0.000 000 000 000 000 013 248 735 9(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: 0.000 000 000 000 000 013 248 735 9.

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) 0.000 000 000 000 000 013 248 735 9 × 2 = 0 + 0.000 000 000 000 000 026 497 471 8;
  • 2) 0.000 000 000 000 000 026 497 471 8 × 2 = 0 + 0.000 000 000 000 000 052 994 943 6;
  • 3) 0.000 000 000 000 000 052 994 943 6 × 2 = 0 + 0.000 000 000 000 000 105 989 887 2;
  • 4) 0.000 000 000 000 000 105 989 887 2 × 2 = 0 + 0.000 000 000 000 000 211 979 774 4;
  • 5) 0.000 000 000 000 000 211 979 774 4 × 2 = 0 + 0.000 000 000 000 000 423 959 548 8;
  • 6) 0.000 000 000 000 000 423 959 548 8 × 2 = 0 + 0.000 000 000 000 000 847 919 097 6;
  • 7) 0.000 000 000 000 000 847 919 097 6 × 2 = 0 + 0.000 000 000 000 001 695 838 195 2;
  • 8) 0.000 000 000 000 001 695 838 195 2 × 2 = 0 + 0.000 000 000 000 003 391 676 390 4;
  • 9) 0.000 000 000 000 003 391 676 390 4 × 2 = 0 + 0.000 000 000 000 006 783 352 780 8;
  • 10) 0.000 000 000 000 006 783 352 780 8 × 2 = 0 + 0.000 000 000 000 013 566 705 561 6;
  • 11) 0.000 000 000 000 013 566 705 561 6 × 2 = 0 + 0.000 000 000 000 027 133 411 123 2;
  • 12) 0.000 000 000 000 027 133 411 123 2 × 2 = 0 + 0.000 000 000 000 054 266 822 246 4;
  • 13) 0.000 000 000 000 054 266 822 246 4 × 2 = 0 + 0.000 000 000 000 108 533 644 492 8;
  • 14) 0.000 000 000 000 108 533 644 492 8 × 2 = 0 + 0.000 000 000 000 217 067 288 985 6;
  • 15) 0.000 000 000 000 217 067 288 985 6 × 2 = 0 + 0.000 000 000 000 434 134 577 971 2;
  • 16) 0.000 000 000 000 434 134 577 971 2 × 2 = 0 + 0.000 000 000 000 868 269 155 942 4;
  • 17) 0.000 000 000 000 868 269 155 942 4 × 2 = 0 + 0.000 000 000 001 736 538 311 884 8;
  • 18) 0.000 000 000 001 736 538 311 884 8 × 2 = 0 + 0.000 000 000 003 473 076 623 769 6;
  • 19) 0.000 000 000 003 473 076 623 769 6 × 2 = 0 + 0.000 000 000 006 946 153 247 539 2;
  • 20) 0.000 000 000 006 946 153 247 539 2 × 2 = 0 + 0.000 000 000 013 892 306 495 078 4;
  • 21) 0.000 000 000 013 892 306 495 078 4 × 2 = 0 + 0.000 000 000 027 784 612 990 156 8;
  • 22) 0.000 000 000 027 784 612 990 156 8 × 2 = 0 + 0.000 000 000 055 569 225 980 313 6;
  • 23) 0.000 000 000 055 569 225 980 313 6 × 2 = 0 + 0.000 000 000 111 138 451 960 627 2;
  • 24) 0.000 000 000 111 138 451 960 627 2 × 2 = 0 + 0.000 000 000 222 276 903 921 254 4;
  • 25) 0.000 000 000 222 276 903 921 254 4 × 2 = 0 + 0.000 000 000 444 553 807 842 508 8;
  • 26) 0.000 000 000 444 553 807 842 508 8 × 2 = 0 + 0.000 000 000 889 107 615 685 017 6;
  • 27) 0.000 000 000 889 107 615 685 017 6 × 2 = 0 + 0.000 000 001 778 215 231 370 035 2;
  • 28) 0.000 000 001 778 215 231 370 035 2 × 2 = 0 + 0.000 000 003 556 430 462 740 070 4;
  • 29) 0.000 000 003 556 430 462 740 070 4 × 2 = 0 + 0.000 000 007 112 860 925 480 140 8;
  • 30) 0.000 000 007 112 860 925 480 140 8 × 2 = 0 + 0.000 000 014 225 721 850 960 281 6;
  • 31) 0.000 000 014 225 721 850 960 281 6 × 2 = 0 + 0.000 000 028 451 443 701 920 563 2;
  • 32) 0.000 000 028 451 443 701 920 563 2 × 2 = 0 + 0.000 000 056 902 887 403 841 126 4;
  • 33) 0.000 000 056 902 887 403 841 126 4 × 2 = 0 + 0.000 000 113 805 774 807 682 252 8;
  • 34) 0.000 000 113 805 774 807 682 252 8 × 2 = 0 + 0.000 000 227 611 549 615 364 505 6;
  • 35) 0.000 000 227 611 549 615 364 505 6 × 2 = 0 + 0.000 000 455 223 099 230 729 011 2;
  • 36) 0.000 000 455 223 099 230 729 011 2 × 2 = 0 + 0.000 000 910 446 198 461 458 022 4;
  • 37) 0.000 000 910 446 198 461 458 022 4 × 2 = 0 + 0.000 001 820 892 396 922 916 044 8;
  • 38) 0.000 001 820 892 396 922 916 044 8 × 2 = 0 + 0.000 003 641 784 793 845 832 089 6;
  • 39) 0.000 003 641 784 793 845 832 089 6 × 2 = 0 + 0.000 007 283 569 587 691 664 179 2;
  • 40) 0.000 007 283 569 587 691 664 179 2 × 2 = 0 + 0.000 014 567 139 175 383 328 358 4;
  • 41) 0.000 014 567 139 175 383 328 358 4 × 2 = 0 + 0.000 029 134 278 350 766 656 716 8;
  • 42) 0.000 029 134 278 350 766 656 716 8 × 2 = 0 + 0.000 058 268 556 701 533 313 433 6;
  • 43) 0.000 058 268 556 701 533 313 433 6 × 2 = 0 + 0.000 116 537 113 403 066 626 867 2;
  • 44) 0.000 116 537 113 403 066 626 867 2 × 2 = 0 + 0.000 233 074 226 806 133 253 734 4;
  • 45) 0.000 233 074 226 806 133 253 734 4 × 2 = 0 + 0.000 466 148 453 612 266 507 468 8;
  • 46) 0.000 466 148 453 612 266 507 468 8 × 2 = 0 + 0.000 932 296 907 224 533 014 937 6;
  • 47) 0.000 932 296 907 224 533 014 937 6 × 2 = 0 + 0.001 864 593 814 449 066 029 875 2;
  • 48) 0.001 864 593 814 449 066 029 875 2 × 2 = 0 + 0.003 729 187 628 898 132 059 750 4;
  • 49) 0.003 729 187 628 898 132 059 750 4 × 2 = 0 + 0.007 458 375 257 796 264 119 500 8;
  • 50) 0.007 458 375 257 796 264 119 500 8 × 2 = 0 + 0.014 916 750 515 592 528 239 001 6;
  • 51) 0.014 916 750 515 592 528 239 001 6 × 2 = 0 + 0.029 833 501 031 185 056 478 003 2;
  • 52) 0.029 833 501 031 185 056 478 003 2 × 2 = 0 + 0.059 667 002 062 370 112 956 006 4;
  • 53) 0.059 667 002 062 370 112 956 006 4 × 2 = 0 + 0.119 334 004 124 740 225 912 012 8;
  • 54) 0.119 334 004 124 740 225 912 012 8 × 2 = 0 + 0.238 668 008 249 480 451 824 025 6;
  • 55) 0.238 668 008 249 480 451 824 025 6 × 2 = 0 + 0.477 336 016 498 960 903 648 051 2;
  • 56) 0.477 336 016 498 960 903 648 051 2 × 2 = 0 + 0.954 672 032 997 921 807 296 102 4;
  • 57) 0.954 672 032 997 921 807 296 102 4 × 2 = 1 + 0.909 344 065 995 843 614 592 204 8;
  • 58) 0.909 344 065 995 843 614 592 204 8 × 2 = 1 + 0.818 688 131 991 687 229 184 409 6;
  • 59) 0.818 688 131 991 687 229 184 409 6 × 2 = 1 + 0.637 376 263 983 374 458 368 819 2;
  • 60) 0.637 376 263 983 374 458 368 819 2 × 2 = 1 + 0.274 752 527 966 748 916 737 638 4;
  • 61) 0.274 752 527 966 748 916 737 638 4 × 2 = 0 + 0.549 505 055 933 497 833 475 276 8;
  • 62) 0.549 505 055 933 497 833 475 276 8 × 2 = 1 + 0.099 010 111 866 995 666 950 553 6;
  • 63) 0.099 010 111 866 995 666 950 553 6 × 2 = 0 + 0.198 020 223 733 991 333 901 107 2;
  • 64) 0.198 020 223 733 991 333 901 107 2 × 2 = 0 + 0.396 040 447 467 982 667 802 214 4;
  • 65) 0.396 040 447 467 982 667 802 214 4 × 2 = 0 + 0.792 080 894 935 965 335 604 428 8;
  • 66) 0.792 080 894 935 965 335 604 428 8 × 2 = 1 + 0.584 161 789 871 930 671 208 857 6;
  • 67) 0.584 161 789 871 930 671 208 857 6 × 2 = 1 + 0.168 323 579 743 861 342 417 715 2;
  • 68) 0.168 323 579 743 861 342 417 715 2 × 2 = 0 + 0.336 647 159 487 722 684 835 430 4;
  • 69) 0.336 647 159 487 722 684 835 430 4 × 2 = 0 + 0.673 294 318 975 445 369 670 860 8;
  • 70) 0.673 294 318 975 445 369 670 860 8 × 2 = 1 + 0.346 588 637 950 890 739 341 721 6;
  • 71) 0.346 588 637 950 890 739 341 721 6 × 2 = 0 + 0.693 177 275 901 781 478 683 443 2;
  • 72) 0.693 177 275 901 781 478 683 443 2 × 2 = 1 + 0.386 354 551 803 562 957 366 886 4;
  • 73) 0.386 354 551 803 562 957 366 886 4 × 2 = 0 + 0.772 709 103 607 125 914 733 772 8;
  • 74) 0.772 709 103 607 125 914 733 772 8 × 2 = 1 + 0.545 418 207 214 251 829 467 545 6;
  • 75) 0.545 418 207 214 251 829 467 545 6 × 2 = 1 + 0.090 836 414 428 503 658 935 091 2;
  • 76) 0.090 836 414 428 503 658 935 091 2 × 2 = 0 + 0.181 672 828 857 007 317 870 182 4;
  • 77) 0.181 672 828 857 007 317 870 182 4 × 2 = 0 + 0.363 345 657 714 014 635 740 364 8;
  • 78) 0.363 345 657 714 014 635 740 364 8 × 2 = 0 + 0.726 691 315 428 029 271 480 729 6;
  • 79) 0.726 691 315 428 029 271 480 729 6 × 2 = 1 + 0.453 382 630 856 058 542 961 459 2;
  • 80) 0.453 382 630 856 058 542 961 459 2 × 2 = 0 + 0.906 765 261 712 117 085 922 918 4;
  • 81) 0.906 765 261 712 117 085 922 918 4 × 2 = 1 + 0.813 530 523 424 234 171 845 836 8;
  • 82) 0.813 530 523 424 234 171 845 836 8 × 2 = 1 + 0.627 061 046 848 468 343 691 673 6;
  • 83) 0.627 061 046 848 468 343 691 673 6 × 2 = 1 + 0.254 122 093 696 936 687 383 347 2;
  • 84) 0.254 122 093 696 936 687 383 347 2 × 2 = 0 + 0.508 244 187 393 873 374 766 694 4;
  • 85) 0.508 244 187 393 873 374 766 694 4 × 2 = 1 + 0.016 488 374 787 746 749 533 388 8;
  • 86) 0.016 488 374 787 746 749 533 388 8 × 2 = 0 + 0.032 976 749 575 493 499 066 777 6;
  • 87) 0.032 976 749 575 493 499 066 777 6 × 2 = 0 + 0.065 953 499 150 986 998 133 555 2;
  • 88) 0.065 953 499 150 986 998 133 555 2 × 2 = 0 + 0.131 906 998 301 973 996 267 110 4;
  • 89) 0.131 906 998 301 973 996 267 110 4 × 2 = 0 + 0.263 813 996 603 947 992 534 220 8;
  • 90) 0.263 813 996 603 947 992 534 220 8 × 2 = 0 + 0.527 627 993 207 895 985 068 441 6;
  • 91) 0.527 627 993 207 895 985 068 441 6 × 2 = 1 + 0.055 255 986 415 791 970 136 883 2;
  • 92) 0.055 255 986 415 791 970 136 883 2 × 2 = 0 + 0.110 511 972 831 583 940 273 766 4;
  • 93) 0.110 511 972 831 583 940 273 766 4 × 2 = 0 + 0.221 023 945 663 167 880 547 532 8;
  • 94) 0.221 023 945 663 167 880 547 532 8 × 2 = 0 + 0.442 047 891 326 335 761 095 065 6;
  • 95) 0.442 047 891 326 335 761 095 065 6 × 2 = 0 + 0.884 095 782 652 671 522 190 131 2;
  • 96) 0.884 095 782 652 671 522 190 131 2 × 2 = 1 + 0.768 191 565 305 343 044 380 262 4;
  • 97) 0.768 191 565 305 343 044 380 262 4 × 2 = 1 + 0.536 383 130 610 686 088 760 524 8;
  • 98) 0.536 383 130 610 686 088 760 524 8 × 2 = 1 + 0.072 766 261 221 372 177 521 049 6;
  • 99) 0.072 766 261 221 372 177 521 049 6 × 2 = 0 + 0.145 532 522 442 744 355 042 099 2;
  • 100) 0.145 532 522 442 744 355 042 099 2 × 2 = 0 + 0.291 065 044 885 488 710 084 198 4;
  • 101) 0.291 065 044 885 488 710 084 198 4 × 2 = 0 + 0.582 130 089 770 977 420 168 396 8;
  • 102) 0.582 130 089 770 977 420 168 396 8 × 2 = 1 + 0.164 260 179 541 954 840 336 793 6;
  • 103) 0.164 260 179 541 954 840 336 793 6 × 2 = 0 + 0.328 520 359 083 909 680 673 587 2;
  • 104) 0.328 520 359 083 909 680 673 587 2 × 2 = 0 + 0.657 040 718 167 819 361 347 174 4;
  • 105) 0.657 040 718 167 819 361 347 174 4 × 2 = 1 + 0.314 081 436 335 638 722 694 348 8;
  • 106) 0.314 081 436 335 638 722 694 348 8 × 2 = 0 + 0.628 162 872 671 277 445 388 697 6;
  • 107) 0.628 162 872 671 277 445 388 697 6 × 2 = 1 + 0.256 325 745 342 554 890 777 395 2;
  • 108) 0.256 325 745 342 554 890 777 395 2 × 2 = 0 + 0.512 651 490 685 109 781 554 790 4;
  • 109) 0.512 651 490 685 109 781 554 790 4 × 2 = 1 + 0.025 302 981 370 219 563 109 580 8;

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:


0.000 000 000 000 000 013 248 735 9(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1111 0100 0110 0101 0110 0010 1110 1000 0010 0001 1100 0100 1010 1(2)

5. Positive number before normalization:

0.000 000 000 000 000 013 248 735 9(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1111 0100 0110 0101 0110 0010 1110 1000 0010 0001 1100 0100 1010 1(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 57 positions to the right, so that only one non zero digit remains to the left of it:


0.000 000 000 000 000 013 248 735 9(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1111 0100 0110 0101 0110 0010 1110 1000 0010 0001 1100 0100 1010 1(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1111 0100 0110 0101 0110 0010 1110 1000 0010 0001 1100 0100 1010 1(2) × 20 =

1.1110 1000 1100 1010 1100 0101 1101 0000 0100 0011 1000 1001 0101(2) × 2-57


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): -57

Mantissa (not normalized):
1.1110 1000 1100 1010 1100 0101 1101 0000 0100 0011 1000 1001 0101


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

-57 + 2(11-1) - 1 =

(-57 + 1 023)(10) =

966(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;
  • 966 ÷ 2 = 483 + 0;
  • 483 ÷ 2 = 241 + 1;
  • 241 ÷ 2 = 120 + 1;
  • 120 ÷ 2 = 60 + 0;
  • 60 ÷ 2 = 30 + 0;
  • 30 ÷ 2 = 15 + 0;
  • 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) =

966(10) =

011 1100 0110(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. 1110 1000 1100 1010 1100 0101 1101 0000 0100 0011 1000 1001 0101 =

1110 1000 1100 1010 1100 0101 1101 0000 0100 0011 1000 1001 0101


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 1100 0110

Mantissa (52 bits) =
1110 1000 1100 1010 1100 0101 1101 0000 0100 0011 1000 1001 0101


Decimal number 0.000 000 000 000 000 013 248 735 9 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1100 0110 - 1110 1000 1100 1010 1100 0101 1101 0000 0100 0011 1000 1001 0101

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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