-0.000 000 349 613 904 800 000 238 907 549 083 000 004 389 232 778 8 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.000 000 349 613 904 800 000 238 907 549 083 000 004 389 232 778 8(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
-0.000 000 349 613 904 800 000 238 907 549 083 000 004 389 232 778 8(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:

|-0.000 000 349 613 904 800 000 238 907 549 083 000 004 389 232 778 8| = 0.000 000 349 613 904 800 000 238 907 549 083 000 004 389 232 778 8


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: 0.000 000 349 613 904 800 000 238 907 549 083 000 004 389 232 778 8.

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) 0.000 000 349 613 904 800 000 238 907 549 083 000 004 389 232 778 8 × 2 = 0 + 0.000 000 699 227 809 600 000 477 815 098 166 000 008 778 465 557 6;
  • 2) 0.000 000 699 227 809 600 000 477 815 098 166 000 008 778 465 557 6 × 2 = 0 + 0.000 001 398 455 619 200 000 955 630 196 332 000 017 556 931 115 2;
  • 3) 0.000 001 398 455 619 200 000 955 630 196 332 000 017 556 931 115 2 × 2 = 0 + 0.000 002 796 911 238 400 001 911 260 392 664 000 035 113 862 230 4;
  • 4) 0.000 002 796 911 238 400 001 911 260 392 664 000 035 113 862 230 4 × 2 = 0 + 0.000 005 593 822 476 800 003 822 520 785 328 000 070 227 724 460 8;
  • 5) 0.000 005 593 822 476 800 003 822 520 785 328 000 070 227 724 460 8 × 2 = 0 + 0.000 011 187 644 953 600 007 645 041 570 656 000 140 455 448 921 6;
  • 6) 0.000 011 187 644 953 600 007 645 041 570 656 000 140 455 448 921 6 × 2 = 0 + 0.000 022 375 289 907 200 015 290 083 141 312 000 280 910 897 843 2;
  • 7) 0.000 022 375 289 907 200 015 290 083 141 312 000 280 910 897 843 2 × 2 = 0 + 0.000 044 750 579 814 400 030 580 166 282 624 000 561 821 795 686 4;
  • 8) 0.000 044 750 579 814 400 030 580 166 282 624 000 561 821 795 686 4 × 2 = 0 + 0.000 089 501 159 628 800 061 160 332 565 248 001 123 643 591 372 8;
  • 9) 0.000 089 501 159 628 800 061 160 332 565 248 001 123 643 591 372 8 × 2 = 0 + 0.000 179 002 319 257 600 122 320 665 130 496 002 247 287 182 745 6;
  • 10) 0.000 179 002 319 257 600 122 320 665 130 496 002 247 287 182 745 6 × 2 = 0 + 0.000 358 004 638 515 200 244 641 330 260 992 004 494 574 365 491 2;
  • 11) 0.000 358 004 638 515 200 244 641 330 260 992 004 494 574 365 491 2 × 2 = 0 + 0.000 716 009 277 030 400 489 282 660 521 984 008 989 148 730 982 4;
  • 12) 0.000 716 009 277 030 400 489 282 660 521 984 008 989 148 730 982 4 × 2 = 0 + 0.001 432 018 554 060 800 978 565 321 043 968 017 978 297 461 964 8;
  • 13) 0.001 432 018 554 060 800 978 565 321 043 968 017 978 297 461 964 8 × 2 = 0 + 0.002 864 037 108 121 601 957 130 642 087 936 035 956 594 923 929 6;
  • 14) 0.002 864 037 108 121 601 957 130 642 087 936 035 956 594 923 929 6 × 2 = 0 + 0.005 728 074 216 243 203 914 261 284 175 872 071 913 189 847 859 2;
  • 15) 0.005 728 074 216 243 203 914 261 284 175 872 071 913 189 847 859 2 × 2 = 0 + 0.011 456 148 432 486 407 828 522 568 351 744 143 826 379 695 718 4;
  • 16) 0.011 456 148 432 486 407 828 522 568 351 744 143 826 379 695 718 4 × 2 = 0 + 0.022 912 296 864 972 815 657 045 136 703 488 287 652 759 391 436 8;
  • 17) 0.022 912 296 864 972 815 657 045 136 703 488 287 652 759 391 436 8 × 2 = 0 + 0.045 824 593 729 945 631 314 090 273 406 976 575 305 518 782 873 6;
  • 18) 0.045 824 593 729 945 631 314 090 273 406 976 575 305 518 782 873 6 × 2 = 0 + 0.091 649 187 459 891 262 628 180 546 813 953 150 611 037 565 747 2;
  • 19) 0.091 649 187 459 891 262 628 180 546 813 953 150 611 037 565 747 2 × 2 = 0 + 0.183 298 374 919 782 525 256 361 093 627 906 301 222 075 131 494 4;
  • 20) 0.183 298 374 919 782 525 256 361 093 627 906 301 222 075 131 494 4 × 2 = 0 + 0.366 596 749 839 565 050 512 722 187 255 812 602 444 150 262 988 8;
  • 21) 0.366 596 749 839 565 050 512 722 187 255 812 602 444 150 262 988 8 × 2 = 0 + 0.733 193 499 679 130 101 025 444 374 511 625 204 888 300 525 977 6;
  • 22) 0.733 193 499 679 130 101 025 444 374 511 625 204 888 300 525 977 6 × 2 = 1 + 0.466 386 999 358 260 202 050 888 749 023 250 409 776 601 051 955 2;
  • 23) 0.466 386 999 358 260 202 050 888 749 023 250 409 776 601 051 955 2 × 2 = 0 + 0.932 773 998 716 520 404 101 777 498 046 500 819 553 202 103 910 4;
  • 24) 0.932 773 998 716 520 404 101 777 498 046 500 819 553 202 103 910 4 × 2 = 1 + 0.865 547 997 433 040 808 203 554 996 093 001 639 106 404 207 820 8;
  • 25) 0.865 547 997 433 040 808 203 554 996 093 001 639 106 404 207 820 8 × 2 = 1 + 0.731 095 994 866 081 616 407 109 992 186 003 278 212 808 415 641 6;
  • 26) 0.731 095 994 866 081 616 407 109 992 186 003 278 212 808 415 641 6 × 2 = 1 + 0.462 191 989 732 163 232 814 219 984 372 006 556 425 616 831 283 2;
  • 27) 0.462 191 989 732 163 232 814 219 984 372 006 556 425 616 831 283 2 × 2 = 0 + 0.924 383 979 464 326 465 628 439 968 744 013 112 851 233 662 566 4;
  • 28) 0.924 383 979 464 326 465 628 439 968 744 013 112 851 233 662 566 4 × 2 = 1 + 0.848 767 958 928 652 931 256 879 937 488 026 225 702 467 325 132 8;
  • 29) 0.848 767 958 928 652 931 256 879 937 488 026 225 702 467 325 132 8 × 2 = 1 + 0.697 535 917 857 305 862 513 759 874 976 052 451 404 934 650 265 6;
  • 30) 0.697 535 917 857 305 862 513 759 874 976 052 451 404 934 650 265 6 × 2 = 1 + 0.395 071 835 714 611 725 027 519 749 952 104 902 809 869 300 531 2;
  • 31) 0.395 071 835 714 611 725 027 519 749 952 104 902 809 869 300 531 2 × 2 = 0 + 0.790 143 671 429 223 450 055 039 499 904 209 805 619 738 601 062 4;
  • 32) 0.790 143 671 429 223 450 055 039 499 904 209 805 619 738 601 062 4 × 2 = 1 + 0.580 287 342 858 446 900 110 078 999 808 419 611 239 477 202 124 8;
  • 33) 0.580 287 342 858 446 900 110 078 999 808 419 611 239 477 202 124 8 × 2 = 1 + 0.160 574 685 716 893 800 220 157 999 616 839 222 478 954 404 249 6;
  • 34) 0.160 574 685 716 893 800 220 157 999 616 839 222 478 954 404 249 6 × 2 = 0 + 0.321 149 371 433 787 600 440 315 999 233 678 444 957 908 808 499 2;
  • 35) 0.321 149 371 433 787 600 440 315 999 233 678 444 957 908 808 499 2 × 2 = 0 + 0.642 298 742 867 575 200 880 631 998 467 356 889 915 817 616 998 4;
  • 36) 0.642 298 742 867 575 200 880 631 998 467 356 889 915 817 616 998 4 × 2 = 1 + 0.284 597 485 735 150 401 761 263 996 934 713 779 831 635 233 996 8;
  • 37) 0.284 597 485 735 150 401 761 263 996 934 713 779 831 635 233 996 8 × 2 = 0 + 0.569 194 971 470 300 803 522 527 993 869 427 559 663 270 467 993 6;
  • 38) 0.569 194 971 470 300 803 522 527 993 869 427 559 663 270 467 993 6 × 2 = 1 + 0.138 389 942 940 601 607 045 055 987 738 855 119 326 540 935 987 2;
  • 39) 0.138 389 942 940 601 607 045 055 987 738 855 119 326 540 935 987 2 × 2 = 0 + 0.276 779 885 881 203 214 090 111 975 477 710 238 653 081 871 974 4;
  • 40) 0.276 779 885 881 203 214 090 111 975 477 710 238 653 081 871 974 4 × 2 = 0 + 0.553 559 771 762 406 428 180 223 950 955 420 477 306 163 743 948 8;
  • 41) 0.553 559 771 762 406 428 180 223 950 955 420 477 306 163 743 948 8 × 2 = 1 + 0.107 119 543 524 812 856 360 447 901 910 840 954 612 327 487 897 6;
  • 42) 0.107 119 543 524 812 856 360 447 901 910 840 954 612 327 487 897 6 × 2 = 0 + 0.214 239 087 049 625 712 720 895 803 821 681 909 224 654 975 795 2;
  • 43) 0.214 239 087 049 625 712 720 895 803 821 681 909 224 654 975 795 2 × 2 = 0 + 0.428 478 174 099 251 425 441 791 607 643 363 818 449 309 951 590 4;
  • 44) 0.428 478 174 099 251 425 441 791 607 643 363 818 449 309 951 590 4 × 2 = 0 + 0.856 956 348 198 502 850 883 583 215 286 727 636 898 619 903 180 8;
  • 45) 0.856 956 348 198 502 850 883 583 215 286 727 636 898 619 903 180 8 × 2 = 1 + 0.713 912 696 397 005 701 767 166 430 573 455 273 797 239 806 361 6;
  • 46) 0.713 912 696 397 005 701 767 166 430 573 455 273 797 239 806 361 6 × 2 = 1 + 0.427 825 392 794 011 403 534 332 861 146 910 547 594 479 612 723 2;
  • 47) 0.427 825 392 794 011 403 534 332 861 146 910 547 594 479 612 723 2 × 2 = 0 + 0.855 650 785 588 022 807 068 665 722 293 821 095 188 959 225 446 4;
  • 48) 0.855 650 785 588 022 807 068 665 722 293 821 095 188 959 225 446 4 × 2 = 1 + 0.711 301 571 176 045 614 137 331 444 587 642 190 377 918 450 892 8;
  • 49) 0.711 301 571 176 045 614 137 331 444 587 642 190 377 918 450 892 8 × 2 = 1 + 0.422 603 142 352 091 228 274 662 889 175 284 380 755 836 901 785 6;
  • 50) 0.422 603 142 352 091 228 274 662 889 175 284 380 755 836 901 785 6 × 2 = 0 + 0.845 206 284 704 182 456 549 325 778 350 568 761 511 673 803 571 2;
  • 51) 0.845 206 284 704 182 456 549 325 778 350 568 761 511 673 803 571 2 × 2 = 1 + 0.690 412 569 408 364 913 098 651 556 701 137 523 023 347 607 142 4;
  • 52) 0.690 412 569 408 364 913 098 651 556 701 137 523 023 347 607 142 4 × 2 = 1 + 0.380 825 138 816 729 826 197 303 113 402 275 046 046 695 214 284 8;
  • 53) 0.380 825 138 816 729 826 197 303 113 402 275 046 046 695 214 284 8 × 2 = 0 + 0.761 650 277 633 459 652 394 606 226 804 550 092 093 390 428 569 6;
  • 54) 0.761 650 277 633 459 652 394 606 226 804 550 092 093 390 428 569 6 × 2 = 1 + 0.523 300 555 266 919 304 789 212 453 609 100 184 186 780 857 139 2;
  • 55) 0.523 300 555 266 919 304 789 212 453 609 100 184 186 780 857 139 2 × 2 = 1 + 0.046 601 110 533 838 609 578 424 907 218 200 368 373 561 714 278 4;
  • 56) 0.046 601 110 533 838 609 578 424 907 218 200 368 373 561 714 278 4 × 2 = 0 + 0.093 202 221 067 677 219 156 849 814 436 400 736 747 123 428 556 8;
  • 57) 0.093 202 221 067 677 219 156 849 814 436 400 736 747 123 428 556 8 × 2 = 0 + 0.186 404 442 135 354 438 313 699 628 872 801 473 494 246 857 113 6;
  • 58) 0.186 404 442 135 354 438 313 699 628 872 801 473 494 246 857 113 6 × 2 = 0 + 0.372 808 884 270 708 876 627 399 257 745 602 946 988 493 714 227 2;
  • 59) 0.372 808 884 270 708 876 627 399 257 745 602 946 988 493 714 227 2 × 2 = 0 + 0.745 617 768 541 417 753 254 798 515 491 205 893 976 987 428 454 4;
  • 60) 0.745 617 768 541 417 753 254 798 515 491 205 893 976 987 428 454 4 × 2 = 1 + 0.491 235 537 082 835 506 509 597 030 982 411 787 953 974 856 908 8;
  • 61) 0.491 235 537 082 835 506 509 597 030 982 411 787 953 974 856 908 8 × 2 = 0 + 0.982 471 074 165 671 013 019 194 061 964 823 575 907 949 713 817 6;
  • 62) 0.982 471 074 165 671 013 019 194 061 964 823 575 907 949 713 817 6 × 2 = 1 + 0.964 942 148 331 342 026 038 388 123 929 647 151 815 899 427 635 2;
  • 63) 0.964 942 148 331 342 026 038 388 123 929 647 151 815 899 427 635 2 × 2 = 1 + 0.929 884 296 662 684 052 076 776 247 859 294 303 631 798 855 270 4;
  • 64) 0.929 884 296 662 684 052 076 776 247 859 294 303 631 798 855 270 4 × 2 = 1 + 0.859 768 593 325 368 104 153 552 495 718 588 607 263 597 710 540 8;
  • 65) 0.859 768 593 325 368 104 153 552 495 718 588 607 263 597 710 540 8 × 2 = 1 + 0.719 537 186 650 736 208 307 104 991 437 177 214 527 195 421 081 6;
  • 66) 0.719 537 186 650 736 208 307 104 991 437 177 214 527 195 421 081 6 × 2 = 1 + 0.439 074 373 301 472 416 614 209 982 874 354 429 054 390 842 163 2;
  • 67) 0.439 074 373 301 472 416 614 209 982 874 354 429 054 390 842 163 2 × 2 = 0 + 0.878 148 746 602 944 833 228 419 965 748 708 858 108 781 684 326 4;
  • 68) 0.878 148 746 602 944 833 228 419 965 748 708 858 108 781 684 326 4 × 2 = 1 + 0.756 297 493 205 889 666 456 839 931 497 417 716 217 563 368 652 8;
  • 69) 0.756 297 493 205 889 666 456 839 931 497 417 716 217 563 368 652 8 × 2 = 1 + 0.512 594 986 411 779 332 913 679 862 994 835 432 435 126 737 305 6;
  • 70) 0.512 594 986 411 779 332 913 679 862 994 835 432 435 126 737 305 6 × 2 = 1 + 0.025 189 972 823 558 665 827 359 725 989 670 864 870 253 474 611 2;
  • 71) 0.025 189 972 823 558 665 827 359 725 989 670 864 870 253 474 611 2 × 2 = 0 + 0.050 379 945 647 117 331 654 719 451 979 341 729 740 506 949 222 4;
  • 72) 0.050 379 945 647 117 331 654 719 451 979 341 729 740 506 949 222 4 × 2 = 0 + 0.100 759 891 294 234 663 309 438 903 958 683 459 481 013 898 444 8;
  • 73) 0.100 759 891 294 234 663 309 438 903 958 683 459 481 013 898 444 8 × 2 = 0 + 0.201 519 782 588 469 326 618 877 807 917 366 918 962 027 796 889 6;
  • 74) 0.201 519 782 588 469 326 618 877 807 917 366 918 962 027 796 889 6 × 2 = 0 + 0.403 039 565 176 938 653 237 755 615 834 733 837 924 055 593 779 2;

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:


0.000 000 349 613 904 800 000 238 907 549 083 000 004 389 232 778 8(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:

0.000 000 349 613 904 800 000 238 907 549 083 000 004 389 232 778 8(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:


0.000 000 349 613 904 800 000 238 907 549 083 000 004 389 232 778 8(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 -0.000 000 349 613 904 800 000 238 907 549 083 000 004 389 232 778 8 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

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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