-0.000 000 349 613 904 800 000 238 907 549 083 000 004 389 232 771 2 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.000 000 349 613 904 800 000 238 907 549 083 000 004 389 232 771 2(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
-0.000 000 349 613 904 800 000 238 907 549 083 000 004 389 232 771 2(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:

|-0.000 000 349 613 904 800 000 238 907 549 083 000 004 389 232 771 2| = 0.000 000 349 613 904 800 000 238 907 549 083 000 004 389 232 771 2


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: 0.000 000 349 613 904 800 000 238 907 549 083 000 004 389 232 771 2.

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) 0.000 000 349 613 904 800 000 238 907 549 083 000 004 389 232 771 2 × 2 = 0 + 0.000 000 699 227 809 600 000 477 815 098 166 000 008 778 465 542 4;
  • 2) 0.000 000 699 227 809 600 000 477 815 098 166 000 008 778 465 542 4 × 2 = 0 + 0.000 001 398 455 619 200 000 955 630 196 332 000 017 556 931 084 8;
  • 3) 0.000 001 398 455 619 200 000 955 630 196 332 000 017 556 931 084 8 × 2 = 0 + 0.000 002 796 911 238 400 001 911 260 392 664 000 035 113 862 169 6;
  • 4) 0.000 002 796 911 238 400 001 911 260 392 664 000 035 113 862 169 6 × 2 = 0 + 0.000 005 593 822 476 800 003 822 520 785 328 000 070 227 724 339 2;
  • 5) 0.000 005 593 822 476 800 003 822 520 785 328 000 070 227 724 339 2 × 2 = 0 + 0.000 011 187 644 953 600 007 645 041 570 656 000 140 455 448 678 4;
  • 6) 0.000 011 187 644 953 600 007 645 041 570 656 000 140 455 448 678 4 × 2 = 0 + 0.000 022 375 289 907 200 015 290 083 141 312 000 280 910 897 356 8;
  • 7) 0.000 022 375 289 907 200 015 290 083 141 312 000 280 910 897 356 8 × 2 = 0 + 0.000 044 750 579 814 400 030 580 166 282 624 000 561 821 794 713 6;
  • 8) 0.000 044 750 579 814 400 030 580 166 282 624 000 561 821 794 713 6 × 2 = 0 + 0.000 089 501 159 628 800 061 160 332 565 248 001 123 643 589 427 2;
  • 9) 0.000 089 501 159 628 800 061 160 332 565 248 001 123 643 589 427 2 × 2 = 0 + 0.000 179 002 319 257 600 122 320 665 130 496 002 247 287 178 854 4;
  • 10) 0.000 179 002 319 257 600 122 320 665 130 496 002 247 287 178 854 4 × 2 = 0 + 0.000 358 004 638 515 200 244 641 330 260 992 004 494 574 357 708 8;
  • 11) 0.000 358 004 638 515 200 244 641 330 260 992 004 494 574 357 708 8 × 2 = 0 + 0.000 716 009 277 030 400 489 282 660 521 984 008 989 148 715 417 6;
  • 12) 0.000 716 009 277 030 400 489 282 660 521 984 008 989 148 715 417 6 × 2 = 0 + 0.001 432 018 554 060 800 978 565 321 043 968 017 978 297 430 835 2;
  • 13) 0.001 432 018 554 060 800 978 565 321 043 968 017 978 297 430 835 2 × 2 = 0 + 0.002 864 037 108 121 601 957 130 642 087 936 035 956 594 861 670 4;
  • 14) 0.002 864 037 108 121 601 957 130 642 087 936 035 956 594 861 670 4 × 2 = 0 + 0.005 728 074 216 243 203 914 261 284 175 872 071 913 189 723 340 8;
  • 15) 0.005 728 074 216 243 203 914 261 284 175 872 071 913 189 723 340 8 × 2 = 0 + 0.011 456 148 432 486 407 828 522 568 351 744 143 826 379 446 681 6;
  • 16) 0.011 456 148 432 486 407 828 522 568 351 744 143 826 379 446 681 6 × 2 = 0 + 0.022 912 296 864 972 815 657 045 136 703 488 287 652 758 893 363 2;
  • 17) 0.022 912 296 864 972 815 657 045 136 703 488 287 652 758 893 363 2 × 2 = 0 + 0.045 824 593 729 945 631 314 090 273 406 976 575 305 517 786 726 4;
  • 18) 0.045 824 593 729 945 631 314 090 273 406 976 575 305 517 786 726 4 × 2 = 0 + 0.091 649 187 459 891 262 628 180 546 813 953 150 611 035 573 452 8;
  • 19) 0.091 649 187 459 891 262 628 180 546 813 953 150 611 035 573 452 8 × 2 = 0 + 0.183 298 374 919 782 525 256 361 093 627 906 301 222 071 146 905 6;
  • 20) 0.183 298 374 919 782 525 256 361 093 627 906 301 222 071 146 905 6 × 2 = 0 + 0.366 596 749 839 565 050 512 722 187 255 812 602 444 142 293 811 2;
  • 21) 0.366 596 749 839 565 050 512 722 187 255 812 602 444 142 293 811 2 × 2 = 0 + 0.733 193 499 679 130 101 025 444 374 511 625 204 888 284 587 622 4;
  • 22) 0.733 193 499 679 130 101 025 444 374 511 625 204 888 284 587 622 4 × 2 = 1 + 0.466 386 999 358 260 202 050 888 749 023 250 409 776 569 175 244 8;
  • 23) 0.466 386 999 358 260 202 050 888 749 023 250 409 776 569 175 244 8 × 2 = 0 + 0.932 773 998 716 520 404 101 777 498 046 500 819 553 138 350 489 6;
  • 24) 0.932 773 998 716 520 404 101 777 498 046 500 819 553 138 350 489 6 × 2 = 1 + 0.865 547 997 433 040 808 203 554 996 093 001 639 106 276 700 979 2;
  • 25) 0.865 547 997 433 040 808 203 554 996 093 001 639 106 276 700 979 2 × 2 = 1 + 0.731 095 994 866 081 616 407 109 992 186 003 278 212 553 401 958 4;
  • 26) 0.731 095 994 866 081 616 407 109 992 186 003 278 212 553 401 958 4 × 2 = 1 + 0.462 191 989 732 163 232 814 219 984 372 006 556 425 106 803 916 8;
  • 27) 0.462 191 989 732 163 232 814 219 984 372 006 556 425 106 803 916 8 × 2 = 0 + 0.924 383 979 464 326 465 628 439 968 744 013 112 850 213 607 833 6;
  • 28) 0.924 383 979 464 326 465 628 439 968 744 013 112 850 213 607 833 6 × 2 = 1 + 0.848 767 958 928 652 931 256 879 937 488 026 225 700 427 215 667 2;
  • 29) 0.848 767 958 928 652 931 256 879 937 488 026 225 700 427 215 667 2 × 2 = 1 + 0.697 535 917 857 305 862 513 759 874 976 052 451 400 854 431 334 4;
  • 30) 0.697 535 917 857 305 862 513 759 874 976 052 451 400 854 431 334 4 × 2 = 1 + 0.395 071 835 714 611 725 027 519 749 952 104 902 801 708 862 668 8;
  • 31) 0.395 071 835 714 611 725 027 519 749 952 104 902 801 708 862 668 8 × 2 = 0 + 0.790 143 671 429 223 450 055 039 499 904 209 805 603 417 725 337 6;
  • 32) 0.790 143 671 429 223 450 055 039 499 904 209 805 603 417 725 337 6 × 2 = 1 + 0.580 287 342 858 446 900 110 078 999 808 419 611 206 835 450 675 2;
  • 33) 0.580 287 342 858 446 900 110 078 999 808 419 611 206 835 450 675 2 × 2 = 1 + 0.160 574 685 716 893 800 220 157 999 616 839 222 413 670 901 350 4;
  • 34) 0.160 574 685 716 893 800 220 157 999 616 839 222 413 670 901 350 4 × 2 = 0 + 0.321 149 371 433 787 600 440 315 999 233 678 444 827 341 802 700 8;
  • 35) 0.321 149 371 433 787 600 440 315 999 233 678 444 827 341 802 700 8 × 2 = 0 + 0.642 298 742 867 575 200 880 631 998 467 356 889 654 683 605 401 6;
  • 36) 0.642 298 742 867 575 200 880 631 998 467 356 889 654 683 605 401 6 × 2 = 1 + 0.284 597 485 735 150 401 761 263 996 934 713 779 309 367 210 803 2;
  • 37) 0.284 597 485 735 150 401 761 263 996 934 713 779 309 367 210 803 2 × 2 = 0 + 0.569 194 971 470 300 803 522 527 993 869 427 558 618 734 421 606 4;
  • 38) 0.569 194 971 470 300 803 522 527 993 869 427 558 618 734 421 606 4 × 2 = 1 + 0.138 389 942 940 601 607 045 055 987 738 855 117 237 468 843 212 8;
  • 39) 0.138 389 942 940 601 607 045 055 987 738 855 117 237 468 843 212 8 × 2 = 0 + 0.276 779 885 881 203 214 090 111 975 477 710 234 474 937 686 425 6;
  • 40) 0.276 779 885 881 203 214 090 111 975 477 710 234 474 937 686 425 6 × 2 = 0 + 0.553 559 771 762 406 428 180 223 950 955 420 468 949 875 372 851 2;
  • 41) 0.553 559 771 762 406 428 180 223 950 955 420 468 949 875 372 851 2 × 2 = 1 + 0.107 119 543 524 812 856 360 447 901 910 840 937 899 750 745 702 4;
  • 42) 0.107 119 543 524 812 856 360 447 901 910 840 937 899 750 745 702 4 × 2 = 0 + 0.214 239 087 049 625 712 720 895 803 821 681 875 799 501 491 404 8;
  • 43) 0.214 239 087 049 625 712 720 895 803 821 681 875 799 501 491 404 8 × 2 = 0 + 0.428 478 174 099 251 425 441 791 607 643 363 751 599 002 982 809 6;
  • 44) 0.428 478 174 099 251 425 441 791 607 643 363 751 599 002 982 809 6 × 2 = 0 + 0.856 956 348 198 502 850 883 583 215 286 727 503 198 005 965 619 2;
  • 45) 0.856 956 348 198 502 850 883 583 215 286 727 503 198 005 965 619 2 × 2 = 1 + 0.713 912 696 397 005 701 767 166 430 573 455 006 396 011 931 238 4;
  • 46) 0.713 912 696 397 005 701 767 166 430 573 455 006 396 011 931 238 4 × 2 = 1 + 0.427 825 392 794 011 403 534 332 861 146 910 012 792 023 862 476 8;
  • 47) 0.427 825 392 794 011 403 534 332 861 146 910 012 792 023 862 476 8 × 2 = 0 + 0.855 650 785 588 022 807 068 665 722 293 820 025 584 047 724 953 6;
  • 48) 0.855 650 785 588 022 807 068 665 722 293 820 025 584 047 724 953 6 × 2 = 1 + 0.711 301 571 176 045 614 137 331 444 587 640 051 168 095 449 907 2;
  • 49) 0.711 301 571 176 045 614 137 331 444 587 640 051 168 095 449 907 2 × 2 = 1 + 0.422 603 142 352 091 228 274 662 889 175 280 102 336 190 899 814 4;
  • 50) 0.422 603 142 352 091 228 274 662 889 175 280 102 336 190 899 814 4 × 2 = 0 + 0.845 206 284 704 182 456 549 325 778 350 560 204 672 381 799 628 8;
  • 51) 0.845 206 284 704 182 456 549 325 778 350 560 204 672 381 799 628 8 × 2 = 1 + 0.690 412 569 408 364 913 098 651 556 701 120 409 344 763 599 257 6;
  • 52) 0.690 412 569 408 364 913 098 651 556 701 120 409 344 763 599 257 6 × 2 = 1 + 0.380 825 138 816 729 826 197 303 113 402 240 818 689 527 198 515 2;
  • 53) 0.380 825 138 816 729 826 197 303 113 402 240 818 689 527 198 515 2 × 2 = 0 + 0.761 650 277 633 459 652 394 606 226 804 481 637 379 054 397 030 4;
  • 54) 0.761 650 277 633 459 652 394 606 226 804 481 637 379 054 397 030 4 × 2 = 1 + 0.523 300 555 266 919 304 789 212 453 608 963 274 758 108 794 060 8;
  • 55) 0.523 300 555 266 919 304 789 212 453 608 963 274 758 108 794 060 8 × 2 = 1 + 0.046 601 110 533 838 609 578 424 907 217 926 549 516 217 588 121 6;
  • 56) 0.046 601 110 533 838 609 578 424 907 217 926 549 516 217 588 121 6 × 2 = 0 + 0.093 202 221 067 677 219 156 849 814 435 853 099 032 435 176 243 2;
  • 57) 0.093 202 221 067 677 219 156 849 814 435 853 099 032 435 176 243 2 × 2 = 0 + 0.186 404 442 135 354 438 313 699 628 871 706 198 064 870 352 486 4;
  • 58) 0.186 404 442 135 354 438 313 699 628 871 706 198 064 870 352 486 4 × 2 = 0 + 0.372 808 884 270 708 876 627 399 257 743 412 396 129 740 704 972 8;
  • 59) 0.372 808 884 270 708 876 627 399 257 743 412 396 129 740 704 972 8 × 2 = 0 + 0.745 617 768 541 417 753 254 798 515 486 824 792 259 481 409 945 6;
  • 60) 0.745 617 768 541 417 753 254 798 515 486 824 792 259 481 409 945 6 × 2 = 1 + 0.491 235 537 082 835 506 509 597 030 973 649 584 518 962 819 891 2;
  • 61) 0.491 235 537 082 835 506 509 597 030 973 649 584 518 962 819 891 2 × 2 = 0 + 0.982 471 074 165 671 013 019 194 061 947 299 169 037 925 639 782 4;
  • 62) 0.982 471 074 165 671 013 019 194 061 947 299 169 037 925 639 782 4 × 2 = 1 + 0.964 942 148 331 342 026 038 388 123 894 598 338 075 851 279 564 8;
  • 63) 0.964 942 148 331 342 026 038 388 123 894 598 338 075 851 279 564 8 × 2 = 1 + 0.929 884 296 662 684 052 076 776 247 789 196 676 151 702 559 129 6;
  • 64) 0.929 884 296 662 684 052 076 776 247 789 196 676 151 702 559 129 6 × 2 = 1 + 0.859 768 593 325 368 104 153 552 495 578 393 352 303 405 118 259 2;
  • 65) 0.859 768 593 325 368 104 153 552 495 578 393 352 303 405 118 259 2 × 2 = 1 + 0.719 537 186 650 736 208 307 104 991 156 786 704 606 810 236 518 4;
  • 66) 0.719 537 186 650 736 208 307 104 991 156 786 704 606 810 236 518 4 × 2 = 1 + 0.439 074 373 301 472 416 614 209 982 313 573 409 213 620 473 036 8;
  • 67) 0.439 074 373 301 472 416 614 209 982 313 573 409 213 620 473 036 8 × 2 = 0 + 0.878 148 746 602 944 833 228 419 964 627 146 818 427 240 946 073 6;
  • 68) 0.878 148 746 602 944 833 228 419 964 627 146 818 427 240 946 073 6 × 2 = 1 + 0.756 297 493 205 889 666 456 839 929 254 293 636 854 481 892 147 2;
  • 69) 0.756 297 493 205 889 666 456 839 929 254 293 636 854 481 892 147 2 × 2 = 1 + 0.512 594 986 411 779 332 913 679 858 508 587 273 708 963 784 294 4;
  • 70) 0.512 594 986 411 779 332 913 679 858 508 587 273 708 963 784 294 4 × 2 = 1 + 0.025 189 972 823 558 665 827 359 717 017 174 547 417 927 568 588 8;
  • 71) 0.025 189 972 823 558 665 827 359 717 017 174 547 417 927 568 588 8 × 2 = 0 + 0.050 379 945 647 117 331 654 719 434 034 349 094 835 855 137 177 6;
  • 72) 0.050 379 945 647 117 331 654 719 434 034 349 094 835 855 137 177 6 × 2 = 0 + 0.100 759 891 294 234 663 309 438 868 068 698 189 671 710 274 355 2;
  • 73) 0.100 759 891 294 234 663 309 438 868 068 698 189 671 710 274 355 2 × 2 = 0 + 0.201 519 782 588 469 326 618 877 736 137 396 379 343 420 548 710 4;
  • 74) 0.201 519 782 588 469 326 618 877 736 137 396 379 343 420 548 710 4 × 2 = 0 + 0.403 039 565 176 938 653 237 755 472 274 792 758 686 841 097 420 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).


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:


0.000 000 349 613 904 800 000 238 907 549 083 000 004 389 232 771 2(10) =

0.0000 0000 0000 0000 0000 0101 1101 1101 1001 0100 1000 1101 1011 0110 0001 0111 1101 1100 00(2)

6. Positive number before normalization:

0.000 000 349 613 904 800 000 238 907 549 083 000 004 389 232 771 2(10) =

0.0000 0000 0000 0000 0000 0101 1101 1101 1001 0100 1000 1101 1011 0110 0001 0111 1101 1100 00(2)

7. Normalize the binary representation of the number.

Shift the decimal mark 22 positions to the right, so that only one non zero digit remains to the left of it:


0.000 000 349 613 904 800 000 238 907 549 083 000 004 389 232 771 2(10) =

0.0000 0000 0000 0000 0000 0101 1101 1101 1001 0100 1000 1101 1011 0110 0001 0111 1101 1100 00(2) =

0.0000 0000 0000 0000 0000 0101 1101 1101 1001 0100 1000 1101 1011 0110 0001 0111 1101 1100 00(2) × 20 =

1.0111 0111 0110 0101 0010 0011 0110 1101 1000 0101 1111 0111 0000(2) × 2-22


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): -22

Mantissa (not normalized):
1.0111 0111 0110 0101 0010 0011 0110 1101 1000 0101 1111 0111 0000


9. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

-22 + 2(11-1) - 1 =

(-22 + 1 023)(10) =

1 001(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;
  • 1 001 ÷ 2 = 500 + 1;
  • 500 ÷ 2 = 250 + 0;
  • 250 ÷ 2 = 125 + 0;
  • 125 ÷ 2 = 62 + 1;
  • 62 ÷ 2 = 31 + 0;
  • 31 ÷ 2 = 15 + 1;
  • 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) =

1001(10) =

011 1110 1001(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. 0111 0111 0110 0101 0010 0011 0110 1101 1000 0101 1111 0111 0000 =

0111 0111 0110 0101 0010 0011 0110 1101 1000 0101 1111 0111 0000


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 1110 1001

Mantissa (52 bits) =
0111 0111 0110 0101 0010 0011 0110 1101 1000 0101 1111 0111 0000


Decimal number -0.000 000 349 613 904 800 000 238 907 549 083 000 004 389 232 771 2 converted to 64 bit double precision IEEE 754 binary floating point representation:

1 - 011 1110 1001 - 0111 0111 0110 0101 0010 0011 0110 1101 1000 0101 1111 0111 0000

» Convert decimal -0.000 000 349 613 904 800 000 238 907 549 083 000 004 389 232 779 2 to 64 bit double precision IEEE 754 binary floating point representation standard

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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