-0.000 000 349 613 904 800 000 238 907 549 083 000 004 389 232 762 896 453 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.000 000 349 613 904 800 000 238 907 549 083 000 004 389 232 762 896 453(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
-0.000 000 349 613 904 800 000 238 907 549 083 000 004 389 232 762 896 453(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:

|-0.000 000 349 613 904 800 000 238 907 549 083 000 004 389 232 762 896 453| = 0.000 000 349 613 904 800 000 238 907 549 083 000 004 389 232 762 896 453


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: 0.000 000 349 613 904 800 000 238 907 549 083 000 004 389 232 762 896 453.

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) 0.000 000 349 613 904 800 000 238 907 549 083 000 004 389 232 762 896 453 × 2 = 0 + 0.000 000 699 227 809 600 000 477 815 098 166 000 008 778 465 525 792 906;
  • 2) 0.000 000 699 227 809 600 000 477 815 098 166 000 008 778 465 525 792 906 × 2 = 0 + 0.000 001 398 455 619 200 000 955 630 196 332 000 017 556 931 051 585 812;
  • 3) 0.000 001 398 455 619 200 000 955 630 196 332 000 017 556 931 051 585 812 × 2 = 0 + 0.000 002 796 911 238 400 001 911 260 392 664 000 035 113 862 103 171 624;
  • 4) 0.000 002 796 911 238 400 001 911 260 392 664 000 035 113 862 103 171 624 × 2 = 0 + 0.000 005 593 822 476 800 003 822 520 785 328 000 070 227 724 206 343 248;
  • 5) 0.000 005 593 822 476 800 003 822 520 785 328 000 070 227 724 206 343 248 × 2 = 0 + 0.000 011 187 644 953 600 007 645 041 570 656 000 140 455 448 412 686 496;
  • 6) 0.000 011 187 644 953 600 007 645 041 570 656 000 140 455 448 412 686 496 × 2 = 0 + 0.000 022 375 289 907 200 015 290 083 141 312 000 280 910 896 825 372 992;
  • 7) 0.000 022 375 289 907 200 015 290 083 141 312 000 280 910 896 825 372 992 × 2 = 0 + 0.000 044 750 579 814 400 030 580 166 282 624 000 561 821 793 650 745 984;
  • 8) 0.000 044 750 579 814 400 030 580 166 282 624 000 561 821 793 650 745 984 × 2 = 0 + 0.000 089 501 159 628 800 061 160 332 565 248 001 123 643 587 301 491 968;
  • 9) 0.000 089 501 159 628 800 061 160 332 565 248 001 123 643 587 301 491 968 × 2 = 0 + 0.000 179 002 319 257 600 122 320 665 130 496 002 247 287 174 602 983 936;
  • 10) 0.000 179 002 319 257 600 122 320 665 130 496 002 247 287 174 602 983 936 × 2 = 0 + 0.000 358 004 638 515 200 244 641 330 260 992 004 494 574 349 205 967 872;
  • 11) 0.000 358 004 638 515 200 244 641 330 260 992 004 494 574 349 205 967 872 × 2 = 0 + 0.000 716 009 277 030 400 489 282 660 521 984 008 989 148 698 411 935 744;
  • 12) 0.000 716 009 277 030 400 489 282 660 521 984 008 989 148 698 411 935 744 × 2 = 0 + 0.001 432 018 554 060 800 978 565 321 043 968 017 978 297 396 823 871 488;
  • 13) 0.001 432 018 554 060 800 978 565 321 043 968 017 978 297 396 823 871 488 × 2 = 0 + 0.002 864 037 108 121 601 957 130 642 087 936 035 956 594 793 647 742 976;
  • 14) 0.002 864 037 108 121 601 957 130 642 087 936 035 956 594 793 647 742 976 × 2 = 0 + 0.005 728 074 216 243 203 914 261 284 175 872 071 913 189 587 295 485 952;
  • 15) 0.005 728 074 216 243 203 914 261 284 175 872 071 913 189 587 295 485 952 × 2 = 0 + 0.011 456 148 432 486 407 828 522 568 351 744 143 826 379 174 590 971 904;
  • 16) 0.011 456 148 432 486 407 828 522 568 351 744 143 826 379 174 590 971 904 × 2 = 0 + 0.022 912 296 864 972 815 657 045 136 703 488 287 652 758 349 181 943 808;
  • 17) 0.022 912 296 864 972 815 657 045 136 703 488 287 652 758 349 181 943 808 × 2 = 0 + 0.045 824 593 729 945 631 314 090 273 406 976 575 305 516 698 363 887 616;
  • 18) 0.045 824 593 729 945 631 314 090 273 406 976 575 305 516 698 363 887 616 × 2 = 0 + 0.091 649 187 459 891 262 628 180 546 813 953 150 611 033 396 727 775 232;
  • 19) 0.091 649 187 459 891 262 628 180 546 813 953 150 611 033 396 727 775 232 × 2 = 0 + 0.183 298 374 919 782 525 256 361 093 627 906 301 222 066 793 455 550 464;
  • 20) 0.183 298 374 919 782 525 256 361 093 627 906 301 222 066 793 455 550 464 × 2 = 0 + 0.366 596 749 839 565 050 512 722 187 255 812 602 444 133 586 911 100 928;
  • 21) 0.366 596 749 839 565 050 512 722 187 255 812 602 444 133 586 911 100 928 × 2 = 0 + 0.733 193 499 679 130 101 025 444 374 511 625 204 888 267 173 822 201 856;
  • 22) 0.733 193 499 679 130 101 025 444 374 511 625 204 888 267 173 822 201 856 × 2 = 1 + 0.466 386 999 358 260 202 050 888 749 023 250 409 776 534 347 644 403 712;
  • 23) 0.466 386 999 358 260 202 050 888 749 023 250 409 776 534 347 644 403 712 × 2 = 0 + 0.932 773 998 716 520 404 101 777 498 046 500 819 553 068 695 288 807 424;
  • 24) 0.932 773 998 716 520 404 101 777 498 046 500 819 553 068 695 288 807 424 × 2 = 1 + 0.865 547 997 433 040 808 203 554 996 093 001 639 106 137 390 577 614 848;
  • 25) 0.865 547 997 433 040 808 203 554 996 093 001 639 106 137 390 577 614 848 × 2 = 1 + 0.731 095 994 866 081 616 407 109 992 186 003 278 212 274 781 155 229 696;
  • 26) 0.731 095 994 866 081 616 407 109 992 186 003 278 212 274 781 155 229 696 × 2 = 1 + 0.462 191 989 732 163 232 814 219 984 372 006 556 424 549 562 310 459 392;
  • 27) 0.462 191 989 732 163 232 814 219 984 372 006 556 424 549 562 310 459 392 × 2 = 0 + 0.924 383 979 464 326 465 628 439 968 744 013 112 849 099 124 620 918 784;
  • 28) 0.924 383 979 464 326 465 628 439 968 744 013 112 849 099 124 620 918 784 × 2 = 1 + 0.848 767 958 928 652 931 256 879 937 488 026 225 698 198 249 241 837 568;
  • 29) 0.848 767 958 928 652 931 256 879 937 488 026 225 698 198 249 241 837 568 × 2 = 1 + 0.697 535 917 857 305 862 513 759 874 976 052 451 396 396 498 483 675 136;
  • 30) 0.697 535 917 857 305 862 513 759 874 976 052 451 396 396 498 483 675 136 × 2 = 1 + 0.395 071 835 714 611 725 027 519 749 952 104 902 792 792 996 967 350 272;
  • 31) 0.395 071 835 714 611 725 027 519 749 952 104 902 792 792 996 967 350 272 × 2 = 0 + 0.790 143 671 429 223 450 055 039 499 904 209 805 585 585 993 934 700 544;
  • 32) 0.790 143 671 429 223 450 055 039 499 904 209 805 585 585 993 934 700 544 × 2 = 1 + 0.580 287 342 858 446 900 110 078 999 808 419 611 171 171 987 869 401 088;
  • 33) 0.580 287 342 858 446 900 110 078 999 808 419 611 171 171 987 869 401 088 × 2 = 1 + 0.160 574 685 716 893 800 220 157 999 616 839 222 342 343 975 738 802 176;
  • 34) 0.160 574 685 716 893 800 220 157 999 616 839 222 342 343 975 738 802 176 × 2 = 0 + 0.321 149 371 433 787 600 440 315 999 233 678 444 684 687 951 477 604 352;
  • 35) 0.321 149 371 433 787 600 440 315 999 233 678 444 684 687 951 477 604 352 × 2 = 0 + 0.642 298 742 867 575 200 880 631 998 467 356 889 369 375 902 955 208 704;
  • 36) 0.642 298 742 867 575 200 880 631 998 467 356 889 369 375 902 955 208 704 × 2 = 1 + 0.284 597 485 735 150 401 761 263 996 934 713 778 738 751 805 910 417 408;
  • 37) 0.284 597 485 735 150 401 761 263 996 934 713 778 738 751 805 910 417 408 × 2 = 0 + 0.569 194 971 470 300 803 522 527 993 869 427 557 477 503 611 820 834 816;
  • 38) 0.569 194 971 470 300 803 522 527 993 869 427 557 477 503 611 820 834 816 × 2 = 1 + 0.138 389 942 940 601 607 045 055 987 738 855 114 955 007 223 641 669 632;
  • 39) 0.138 389 942 940 601 607 045 055 987 738 855 114 955 007 223 641 669 632 × 2 = 0 + 0.276 779 885 881 203 214 090 111 975 477 710 229 910 014 447 283 339 264;
  • 40) 0.276 779 885 881 203 214 090 111 975 477 710 229 910 014 447 283 339 264 × 2 = 0 + 0.553 559 771 762 406 428 180 223 950 955 420 459 820 028 894 566 678 528;
  • 41) 0.553 559 771 762 406 428 180 223 950 955 420 459 820 028 894 566 678 528 × 2 = 1 + 0.107 119 543 524 812 856 360 447 901 910 840 919 640 057 789 133 357 056;
  • 42) 0.107 119 543 524 812 856 360 447 901 910 840 919 640 057 789 133 357 056 × 2 = 0 + 0.214 239 087 049 625 712 720 895 803 821 681 839 280 115 578 266 714 112;
  • 43) 0.214 239 087 049 625 712 720 895 803 821 681 839 280 115 578 266 714 112 × 2 = 0 + 0.428 478 174 099 251 425 441 791 607 643 363 678 560 231 156 533 428 224;
  • 44) 0.428 478 174 099 251 425 441 791 607 643 363 678 560 231 156 533 428 224 × 2 = 0 + 0.856 956 348 198 502 850 883 583 215 286 727 357 120 462 313 066 856 448;
  • 45) 0.856 956 348 198 502 850 883 583 215 286 727 357 120 462 313 066 856 448 × 2 = 1 + 0.713 912 696 397 005 701 767 166 430 573 454 714 240 924 626 133 712 896;
  • 46) 0.713 912 696 397 005 701 767 166 430 573 454 714 240 924 626 133 712 896 × 2 = 1 + 0.427 825 392 794 011 403 534 332 861 146 909 428 481 849 252 267 425 792;
  • 47) 0.427 825 392 794 011 403 534 332 861 146 909 428 481 849 252 267 425 792 × 2 = 0 + 0.855 650 785 588 022 807 068 665 722 293 818 856 963 698 504 534 851 584;
  • 48) 0.855 650 785 588 022 807 068 665 722 293 818 856 963 698 504 534 851 584 × 2 = 1 + 0.711 301 571 176 045 614 137 331 444 587 637 713 927 397 009 069 703 168;
  • 49) 0.711 301 571 176 045 614 137 331 444 587 637 713 927 397 009 069 703 168 × 2 = 1 + 0.422 603 142 352 091 228 274 662 889 175 275 427 854 794 018 139 406 336;
  • 50) 0.422 603 142 352 091 228 274 662 889 175 275 427 854 794 018 139 406 336 × 2 = 0 + 0.845 206 284 704 182 456 549 325 778 350 550 855 709 588 036 278 812 672;
  • 51) 0.845 206 284 704 182 456 549 325 778 350 550 855 709 588 036 278 812 672 × 2 = 1 + 0.690 412 569 408 364 913 098 651 556 701 101 711 419 176 072 557 625 344;
  • 52) 0.690 412 569 408 364 913 098 651 556 701 101 711 419 176 072 557 625 344 × 2 = 1 + 0.380 825 138 816 729 826 197 303 113 402 203 422 838 352 145 115 250 688;
  • 53) 0.380 825 138 816 729 826 197 303 113 402 203 422 838 352 145 115 250 688 × 2 = 0 + 0.761 650 277 633 459 652 394 606 226 804 406 845 676 704 290 230 501 376;
  • 54) 0.761 650 277 633 459 652 394 606 226 804 406 845 676 704 290 230 501 376 × 2 = 1 + 0.523 300 555 266 919 304 789 212 453 608 813 691 353 408 580 461 002 752;
  • 55) 0.523 300 555 266 919 304 789 212 453 608 813 691 353 408 580 461 002 752 × 2 = 1 + 0.046 601 110 533 838 609 578 424 907 217 627 382 706 817 160 922 005 504;
  • 56) 0.046 601 110 533 838 609 578 424 907 217 627 382 706 817 160 922 005 504 × 2 = 0 + 0.093 202 221 067 677 219 156 849 814 435 254 765 413 634 321 844 011 008;
  • 57) 0.093 202 221 067 677 219 156 849 814 435 254 765 413 634 321 844 011 008 × 2 = 0 + 0.186 404 442 135 354 438 313 699 628 870 509 530 827 268 643 688 022 016;
  • 58) 0.186 404 442 135 354 438 313 699 628 870 509 530 827 268 643 688 022 016 × 2 = 0 + 0.372 808 884 270 708 876 627 399 257 741 019 061 654 537 287 376 044 032;
  • 59) 0.372 808 884 270 708 876 627 399 257 741 019 061 654 537 287 376 044 032 × 2 = 0 + 0.745 617 768 541 417 753 254 798 515 482 038 123 309 074 574 752 088 064;
  • 60) 0.745 617 768 541 417 753 254 798 515 482 038 123 309 074 574 752 088 064 × 2 = 1 + 0.491 235 537 082 835 506 509 597 030 964 076 246 618 149 149 504 176 128;
  • 61) 0.491 235 537 082 835 506 509 597 030 964 076 246 618 149 149 504 176 128 × 2 = 0 + 0.982 471 074 165 671 013 019 194 061 928 152 493 236 298 299 008 352 256;
  • 62) 0.982 471 074 165 671 013 019 194 061 928 152 493 236 298 299 008 352 256 × 2 = 1 + 0.964 942 148 331 342 026 038 388 123 856 304 986 472 596 598 016 704 512;
  • 63) 0.964 942 148 331 342 026 038 388 123 856 304 986 472 596 598 016 704 512 × 2 = 1 + 0.929 884 296 662 684 052 076 776 247 712 609 972 945 193 196 033 409 024;
  • 64) 0.929 884 296 662 684 052 076 776 247 712 609 972 945 193 196 033 409 024 × 2 = 1 + 0.859 768 593 325 368 104 153 552 495 425 219 945 890 386 392 066 818 048;
  • 65) 0.859 768 593 325 368 104 153 552 495 425 219 945 890 386 392 066 818 048 × 2 = 1 + 0.719 537 186 650 736 208 307 104 990 850 439 891 780 772 784 133 636 096;
  • 66) 0.719 537 186 650 736 208 307 104 990 850 439 891 780 772 784 133 636 096 × 2 = 1 + 0.439 074 373 301 472 416 614 209 981 700 879 783 561 545 568 267 272 192;
  • 67) 0.439 074 373 301 472 416 614 209 981 700 879 783 561 545 568 267 272 192 × 2 = 0 + 0.878 148 746 602 944 833 228 419 963 401 759 567 123 091 136 534 544 384;
  • 68) 0.878 148 746 602 944 833 228 419 963 401 759 567 123 091 136 534 544 384 × 2 = 1 + 0.756 297 493 205 889 666 456 839 926 803 519 134 246 182 273 069 088 768;
  • 69) 0.756 297 493 205 889 666 456 839 926 803 519 134 246 182 273 069 088 768 × 2 = 1 + 0.512 594 986 411 779 332 913 679 853 607 038 268 492 364 546 138 177 536;
  • 70) 0.512 594 986 411 779 332 913 679 853 607 038 268 492 364 546 138 177 536 × 2 = 1 + 0.025 189 972 823 558 665 827 359 707 214 076 536 984 729 092 276 355 072;
  • 71) 0.025 189 972 823 558 665 827 359 707 214 076 536 984 729 092 276 355 072 × 2 = 0 + 0.050 379 945 647 117 331 654 719 414 428 153 073 969 458 184 552 710 144;
  • 72) 0.050 379 945 647 117 331 654 719 414 428 153 073 969 458 184 552 710 144 × 2 = 0 + 0.100 759 891 294 234 663 309 438 828 856 306 147 938 916 369 105 420 288;
  • 73) 0.100 759 891 294 234 663 309 438 828 856 306 147 938 916 369 105 420 288 × 2 = 0 + 0.201 519 782 588 469 326 618 877 657 712 612 295 877 832 738 210 840 576;
  • 74) 0.201 519 782 588 469 326 618 877 657 712 612 295 877 832 738 210 840 576 × 2 = 0 + 0.403 039 565 176 938 653 237 755 315 425 224 591 755 665 476 421 681 152;

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:


0.000 000 349 613 904 800 000 238 907 549 083 000 004 389 232 762 896 453(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:

0.000 000 349 613 904 800 000 238 907 549 083 000 004 389 232 762 896 453(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:


0.000 000 349 613 904 800 000 238 907 549 083 000 004 389 232 762 896 453(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 -0.000 000 349 613 904 800 000 238 907 549 083 000 004 389 232 762 896 453 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 -0.000 000 349 613 904 800 000 238 907 549 083 000 004 389 232 762 896 544 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