1.161 834 032 785 773 095 167 300 531 840 609 646 036 411 516 475 047 407 229 817 563 319 072 995 387 15 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 1.161 834 032 785 773 095 167 300 531 840 609 646 036 411 516 475 047 407 229 817 563 319 072 995 387 15(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
1.161 834 032 785 773 095 167 300 531 840 609 646 036 411 516 475 047 407 229 817 563 319 072 995 387 15(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: 1.
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;
  • 1 ÷ 2 = 0 + 1;

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.

1(10) =

1(2)


3. Convert to binary (base 2) the fractional part: 0.161 834 032 785 773 095 167 300 531 840 609 646 036 411 516 475 047 407 229 817 563 319 072 995 387 15.

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) 0.161 834 032 785 773 095 167 300 531 840 609 646 036 411 516 475 047 407 229 817 563 319 072 995 387 15 × 2 = 0 + 0.323 668 065 571 546 190 334 601 063 681 219 292 072 823 032 950 094 814 459 635 126 638 145 990 774 3;
  • 2) 0.323 668 065 571 546 190 334 601 063 681 219 292 072 823 032 950 094 814 459 635 126 638 145 990 774 3 × 2 = 0 + 0.647 336 131 143 092 380 669 202 127 362 438 584 145 646 065 900 189 628 919 270 253 276 291 981 548 6;
  • 3) 0.647 336 131 143 092 380 669 202 127 362 438 584 145 646 065 900 189 628 919 270 253 276 291 981 548 6 × 2 = 1 + 0.294 672 262 286 184 761 338 404 254 724 877 168 291 292 131 800 379 257 838 540 506 552 583 963 097 2;
  • 4) 0.294 672 262 286 184 761 338 404 254 724 877 168 291 292 131 800 379 257 838 540 506 552 583 963 097 2 × 2 = 0 + 0.589 344 524 572 369 522 676 808 509 449 754 336 582 584 263 600 758 515 677 081 013 105 167 926 194 4;
  • 5) 0.589 344 524 572 369 522 676 808 509 449 754 336 582 584 263 600 758 515 677 081 013 105 167 926 194 4 × 2 = 1 + 0.178 689 049 144 739 045 353 617 018 899 508 673 165 168 527 201 517 031 354 162 026 210 335 852 388 8;
  • 6) 0.178 689 049 144 739 045 353 617 018 899 508 673 165 168 527 201 517 031 354 162 026 210 335 852 388 8 × 2 = 0 + 0.357 378 098 289 478 090 707 234 037 799 017 346 330 337 054 403 034 062 708 324 052 420 671 704 777 6;
  • 7) 0.357 378 098 289 478 090 707 234 037 799 017 346 330 337 054 403 034 062 708 324 052 420 671 704 777 6 × 2 = 0 + 0.714 756 196 578 956 181 414 468 075 598 034 692 660 674 108 806 068 125 416 648 104 841 343 409 555 2;
  • 8) 0.714 756 196 578 956 181 414 468 075 598 034 692 660 674 108 806 068 125 416 648 104 841 343 409 555 2 × 2 = 1 + 0.429 512 393 157 912 362 828 936 151 196 069 385 321 348 217 612 136 250 833 296 209 682 686 819 110 4;
  • 9) 0.429 512 393 157 912 362 828 936 151 196 069 385 321 348 217 612 136 250 833 296 209 682 686 819 110 4 × 2 = 0 + 0.859 024 786 315 824 725 657 872 302 392 138 770 642 696 435 224 272 501 666 592 419 365 373 638 220 8;
  • 10) 0.859 024 786 315 824 725 657 872 302 392 138 770 642 696 435 224 272 501 666 592 419 365 373 638 220 8 × 2 = 1 + 0.718 049 572 631 649 451 315 744 604 784 277 541 285 392 870 448 545 003 333 184 838 730 747 276 441 6;
  • 11) 0.718 049 572 631 649 451 315 744 604 784 277 541 285 392 870 448 545 003 333 184 838 730 747 276 441 6 × 2 = 1 + 0.436 099 145 263 298 902 631 489 209 568 555 082 570 785 740 897 090 006 666 369 677 461 494 552 883 2;
  • 12) 0.436 099 145 263 298 902 631 489 209 568 555 082 570 785 740 897 090 006 666 369 677 461 494 552 883 2 × 2 = 0 + 0.872 198 290 526 597 805 262 978 419 137 110 165 141 571 481 794 180 013 332 739 354 922 989 105 766 4;
  • 13) 0.872 198 290 526 597 805 262 978 419 137 110 165 141 571 481 794 180 013 332 739 354 922 989 105 766 4 × 2 = 1 + 0.744 396 581 053 195 610 525 956 838 274 220 330 283 142 963 588 360 026 665 478 709 845 978 211 532 8;
  • 14) 0.744 396 581 053 195 610 525 956 838 274 220 330 283 142 963 588 360 026 665 478 709 845 978 211 532 8 × 2 = 1 + 0.488 793 162 106 391 221 051 913 676 548 440 660 566 285 927 176 720 053 330 957 419 691 956 423 065 6;
  • 15) 0.488 793 162 106 391 221 051 913 676 548 440 660 566 285 927 176 720 053 330 957 419 691 956 423 065 6 × 2 = 0 + 0.977 586 324 212 782 442 103 827 353 096 881 321 132 571 854 353 440 106 661 914 839 383 912 846 131 2;
  • 16) 0.977 586 324 212 782 442 103 827 353 096 881 321 132 571 854 353 440 106 661 914 839 383 912 846 131 2 × 2 = 1 + 0.955 172 648 425 564 884 207 654 706 193 762 642 265 143 708 706 880 213 323 829 678 767 825 692 262 4;
  • 17) 0.955 172 648 425 564 884 207 654 706 193 762 642 265 143 708 706 880 213 323 829 678 767 825 692 262 4 × 2 = 1 + 0.910 345 296 851 129 768 415 309 412 387 525 284 530 287 417 413 760 426 647 659 357 535 651 384 524 8;
  • 18) 0.910 345 296 851 129 768 415 309 412 387 525 284 530 287 417 413 760 426 647 659 357 535 651 384 524 8 × 2 = 1 + 0.820 690 593 702 259 536 830 618 824 775 050 569 060 574 834 827 520 853 295 318 715 071 302 769 049 6;
  • 19) 0.820 690 593 702 259 536 830 618 824 775 050 569 060 574 834 827 520 853 295 318 715 071 302 769 049 6 × 2 = 1 + 0.641 381 187 404 519 073 661 237 649 550 101 138 121 149 669 655 041 706 590 637 430 142 605 538 099 2;
  • 20) 0.641 381 187 404 519 073 661 237 649 550 101 138 121 149 669 655 041 706 590 637 430 142 605 538 099 2 × 2 = 1 + 0.282 762 374 809 038 147 322 475 299 100 202 276 242 299 339 310 083 413 181 274 860 285 211 076 198 4;
  • 21) 0.282 762 374 809 038 147 322 475 299 100 202 276 242 299 339 310 083 413 181 274 860 285 211 076 198 4 × 2 = 0 + 0.565 524 749 618 076 294 644 950 598 200 404 552 484 598 678 620 166 826 362 549 720 570 422 152 396 8;
  • 22) 0.565 524 749 618 076 294 644 950 598 200 404 552 484 598 678 620 166 826 362 549 720 570 422 152 396 8 × 2 = 1 + 0.131 049 499 236 152 589 289 901 196 400 809 104 969 197 357 240 333 652 725 099 441 140 844 304 793 6;
  • 23) 0.131 049 499 236 152 589 289 901 196 400 809 104 969 197 357 240 333 652 725 099 441 140 844 304 793 6 × 2 = 0 + 0.262 098 998 472 305 178 579 802 392 801 618 209 938 394 714 480 667 305 450 198 882 281 688 609 587 2;
  • 24) 0.262 098 998 472 305 178 579 802 392 801 618 209 938 394 714 480 667 305 450 198 882 281 688 609 587 2 × 2 = 0 + 0.524 197 996 944 610 357 159 604 785 603 236 419 876 789 428 961 334 610 900 397 764 563 377 219 174 4;
  • 25) 0.524 197 996 944 610 357 159 604 785 603 236 419 876 789 428 961 334 610 900 397 764 563 377 219 174 4 × 2 = 1 + 0.048 395 993 889 220 714 319 209 571 206 472 839 753 578 857 922 669 221 800 795 529 126 754 438 348 8;
  • 26) 0.048 395 993 889 220 714 319 209 571 206 472 839 753 578 857 922 669 221 800 795 529 126 754 438 348 8 × 2 = 0 + 0.096 791 987 778 441 428 638 419 142 412 945 679 507 157 715 845 338 443 601 591 058 253 508 876 697 6;
  • 27) 0.096 791 987 778 441 428 638 419 142 412 945 679 507 157 715 845 338 443 601 591 058 253 508 876 697 6 × 2 = 0 + 0.193 583 975 556 882 857 276 838 284 825 891 359 014 315 431 690 676 887 203 182 116 507 017 753 395 2;
  • 28) 0.193 583 975 556 882 857 276 838 284 825 891 359 014 315 431 690 676 887 203 182 116 507 017 753 395 2 × 2 = 0 + 0.387 167 951 113 765 714 553 676 569 651 782 718 028 630 863 381 353 774 406 364 233 014 035 506 790 4;
  • 29) 0.387 167 951 113 765 714 553 676 569 651 782 718 028 630 863 381 353 774 406 364 233 014 035 506 790 4 × 2 = 0 + 0.774 335 902 227 531 429 107 353 139 303 565 436 057 261 726 762 707 548 812 728 466 028 071 013 580 8;
  • 30) 0.774 335 902 227 531 429 107 353 139 303 565 436 057 261 726 762 707 548 812 728 466 028 071 013 580 8 × 2 = 1 + 0.548 671 804 455 062 858 214 706 278 607 130 872 114 523 453 525 415 097 625 456 932 056 142 027 161 6;
  • 31) 0.548 671 804 455 062 858 214 706 278 607 130 872 114 523 453 525 415 097 625 456 932 056 142 027 161 6 × 2 = 1 + 0.097 343 608 910 125 716 429 412 557 214 261 744 229 046 907 050 830 195 250 913 864 112 284 054 323 2;
  • 32) 0.097 343 608 910 125 716 429 412 557 214 261 744 229 046 907 050 830 195 250 913 864 112 284 054 323 2 × 2 = 0 + 0.194 687 217 820 251 432 858 825 114 428 523 488 458 093 814 101 660 390 501 827 728 224 568 108 646 4;
  • 33) 0.194 687 217 820 251 432 858 825 114 428 523 488 458 093 814 101 660 390 501 827 728 224 568 108 646 4 × 2 = 0 + 0.389 374 435 640 502 865 717 650 228 857 046 976 916 187 628 203 320 781 003 655 456 449 136 217 292 8;
  • 34) 0.389 374 435 640 502 865 717 650 228 857 046 976 916 187 628 203 320 781 003 655 456 449 136 217 292 8 × 2 = 0 + 0.778 748 871 281 005 731 435 300 457 714 093 953 832 375 256 406 641 562 007 310 912 898 272 434 585 6;
  • 35) 0.778 748 871 281 005 731 435 300 457 714 093 953 832 375 256 406 641 562 007 310 912 898 272 434 585 6 × 2 = 1 + 0.557 497 742 562 011 462 870 600 915 428 187 907 664 750 512 813 283 124 014 621 825 796 544 869 171 2;
  • 36) 0.557 497 742 562 011 462 870 600 915 428 187 907 664 750 512 813 283 124 014 621 825 796 544 869 171 2 × 2 = 1 + 0.114 995 485 124 022 925 741 201 830 856 375 815 329 501 025 626 566 248 029 243 651 593 089 738 342 4;
  • 37) 0.114 995 485 124 022 925 741 201 830 856 375 815 329 501 025 626 566 248 029 243 651 593 089 738 342 4 × 2 = 0 + 0.229 990 970 248 045 851 482 403 661 712 751 630 659 002 051 253 132 496 058 487 303 186 179 476 684 8;
  • 38) 0.229 990 970 248 045 851 482 403 661 712 751 630 659 002 051 253 132 496 058 487 303 186 179 476 684 8 × 2 = 0 + 0.459 981 940 496 091 702 964 807 323 425 503 261 318 004 102 506 264 992 116 974 606 372 358 953 369 6;
  • 39) 0.459 981 940 496 091 702 964 807 323 425 503 261 318 004 102 506 264 992 116 974 606 372 358 953 369 6 × 2 = 0 + 0.919 963 880 992 183 405 929 614 646 851 006 522 636 008 205 012 529 984 233 949 212 744 717 906 739 2;
  • 40) 0.919 963 880 992 183 405 929 614 646 851 006 522 636 008 205 012 529 984 233 949 212 744 717 906 739 2 × 2 = 1 + 0.839 927 761 984 366 811 859 229 293 702 013 045 272 016 410 025 059 968 467 898 425 489 435 813 478 4;
  • 41) 0.839 927 761 984 366 811 859 229 293 702 013 045 272 016 410 025 059 968 467 898 425 489 435 813 478 4 × 2 = 1 + 0.679 855 523 968 733 623 718 458 587 404 026 090 544 032 820 050 119 936 935 796 850 978 871 626 956 8;
  • 42) 0.679 855 523 968 733 623 718 458 587 404 026 090 544 032 820 050 119 936 935 796 850 978 871 626 956 8 × 2 = 1 + 0.359 711 047 937 467 247 436 917 174 808 052 181 088 065 640 100 239 873 871 593 701 957 743 253 913 6;
  • 43) 0.359 711 047 937 467 247 436 917 174 808 052 181 088 065 640 100 239 873 871 593 701 957 743 253 913 6 × 2 = 0 + 0.719 422 095 874 934 494 873 834 349 616 104 362 176 131 280 200 479 747 743 187 403 915 486 507 827 2;
  • 44) 0.719 422 095 874 934 494 873 834 349 616 104 362 176 131 280 200 479 747 743 187 403 915 486 507 827 2 × 2 = 1 + 0.438 844 191 749 868 989 747 668 699 232 208 724 352 262 560 400 959 495 486 374 807 830 973 015 654 4;
  • 45) 0.438 844 191 749 868 989 747 668 699 232 208 724 352 262 560 400 959 495 486 374 807 830 973 015 654 4 × 2 = 0 + 0.877 688 383 499 737 979 495 337 398 464 417 448 704 525 120 801 918 990 972 749 615 661 946 031 308 8;
  • 46) 0.877 688 383 499 737 979 495 337 398 464 417 448 704 525 120 801 918 990 972 749 615 661 946 031 308 8 × 2 = 1 + 0.755 376 766 999 475 958 990 674 796 928 834 897 409 050 241 603 837 981 945 499 231 323 892 062 617 6;
  • 47) 0.755 376 766 999 475 958 990 674 796 928 834 897 409 050 241 603 837 981 945 499 231 323 892 062 617 6 × 2 = 1 + 0.510 753 533 998 951 917 981 349 593 857 669 794 818 100 483 207 675 963 890 998 462 647 784 125 235 2;
  • 48) 0.510 753 533 998 951 917 981 349 593 857 669 794 818 100 483 207 675 963 890 998 462 647 784 125 235 2 × 2 = 1 + 0.021 507 067 997 903 835 962 699 187 715 339 589 636 200 966 415 351 927 781 996 925 295 568 250 470 4;
  • 49) 0.021 507 067 997 903 835 962 699 187 715 339 589 636 200 966 415 351 927 781 996 925 295 568 250 470 4 × 2 = 0 + 0.043 014 135 995 807 671 925 398 375 430 679 179 272 401 932 830 703 855 563 993 850 591 136 500 940 8;
  • 50) 0.043 014 135 995 807 671 925 398 375 430 679 179 272 401 932 830 703 855 563 993 850 591 136 500 940 8 × 2 = 0 + 0.086 028 271 991 615 343 850 796 750 861 358 358 544 803 865 661 407 711 127 987 701 182 273 001 881 6;
  • 51) 0.086 028 271 991 615 343 850 796 750 861 358 358 544 803 865 661 407 711 127 987 701 182 273 001 881 6 × 2 = 0 + 0.172 056 543 983 230 687 701 593 501 722 716 717 089 607 731 322 815 422 255 975 402 364 546 003 763 2;
  • 52) 0.172 056 543 983 230 687 701 593 501 722 716 717 089 607 731 322 815 422 255 975 402 364 546 003 763 2 × 2 = 0 + 0.344 113 087 966 461 375 403 187 003 445 433 434 179 215 462 645 630 844 511 950 804 729 092 007 526 4;
  • 53) 0.344 113 087 966 461 375 403 187 003 445 433 434 179 215 462 645 630 844 511 950 804 729 092 007 526 4 × 2 = 0 + 0.688 226 175 932 922 750 806 374 006 890 866 868 358 430 925 291 261 689 023 901 609 458 184 015 052 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:


0.161 834 032 785 773 095 167 300 531 840 609 646 036 411 516 475 047 407 229 817 563 319 072 995 387 15(10) =

0.0010 1001 0110 1101 1111 0100 1000 0110 0011 0001 1101 0111 0000 0(2)

5. Positive number before normalization:

1.161 834 032 785 773 095 167 300 531 840 609 646 036 411 516 475 047 407 229 817 563 319 072 995 387 15(10) =

1.0010 1001 0110 1101 1111 0100 1000 0110 0011 0001 1101 0111 0000 0(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 0 positions to the left, so that only one non zero digit remains to the left of it:


1.161 834 032 785 773 095 167 300 531 840 609 646 036 411 516 475 047 407 229 817 563 319 072 995 387 15(10) =

1.0010 1001 0110 1101 1111 0100 1000 0110 0011 0001 1101 0111 0000 0(2) =

1.0010 1001 0110 1101 1111 0100 1000 0110 0011 0001 1101 0111 0000 0(2) × 20


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): 0

Mantissa (not normalized):
1.0010 1001 0110 1101 1111 0100 1000 0110 0011 0001 1101 0111 0000 0


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

0 + 2(11-1) - 1 =

(0 + 1 023)(10) =

1 023(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;
  • 1 023 ÷ 2 = 511 + 1;
  • 511 ÷ 2 = 255 + 1;
  • 255 ÷ 2 = 127 + 1;
  • 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) =

1023(10) =

011 1111 1111(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, by removing the excess bits, from the right (if any of the excess bits is set on 1, we are losing precision...).


Mantissa (normalized) =

1. 0010 1001 0110 1101 1111 0100 1000 0110 0011 0001 1101 0111 0000 0 =

0010 1001 0110 1101 1111 0100 1000 0110 0011 0001 1101 0111 0000


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 1111

Mantissa (52 bits) =
0010 1001 0110 1101 1111 0100 1000 0110 0011 0001 1101 0111 0000


Decimal number 1.161 834 032 785 773 095 167 300 531 840 609 646 036 411 516 475 047 407 229 817 563 319 072 995 387 15 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1111 1111 - 0010 1001 0110 1101 1111 0100 1000 0110 0011 0001 1101 0111 0000

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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