3.141 592 653 589 793 238 462 643 383 279 502 884 197 169 399 375 105 820 974 944 592 307 815 84 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 3.141 592 653 589 793 238 462 643 383 279 502 884 197 169 399 375 105 820 974 944 592 307 815 84(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
3.141 592 653 589 793 238 462 643 383 279 502 884 197 169 399 375 105 820 974 944 592 307 815 84(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: 3.
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;
  • 3 ÷ 2 = 1 + 1;
  • 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.

3(10) =

11(2)


3. Convert to binary (base 2) the fractional part: 0.141 592 653 589 793 238 462 643 383 279 502 884 197 169 399 375 105 820 974 944 592 307 815 84.

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) 0.141 592 653 589 793 238 462 643 383 279 502 884 197 169 399 375 105 820 974 944 592 307 815 84 × 2 = 0 + 0.283 185 307 179 586 476 925 286 766 559 005 768 394 338 798 750 211 641 949 889 184 615 631 68;
  • 2) 0.283 185 307 179 586 476 925 286 766 559 005 768 394 338 798 750 211 641 949 889 184 615 631 68 × 2 = 0 + 0.566 370 614 359 172 953 850 573 533 118 011 536 788 677 597 500 423 283 899 778 369 231 263 36;
  • 3) 0.566 370 614 359 172 953 850 573 533 118 011 536 788 677 597 500 423 283 899 778 369 231 263 36 × 2 = 1 + 0.132 741 228 718 345 907 701 147 066 236 023 073 577 355 195 000 846 567 799 556 738 462 526 72;
  • 4) 0.132 741 228 718 345 907 701 147 066 236 023 073 577 355 195 000 846 567 799 556 738 462 526 72 × 2 = 0 + 0.265 482 457 436 691 815 402 294 132 472 046 147 154 710 390 001 693 135 599 113 476 925 053 44;
  • 5) 0.265 482 457 436 691 815 402 294 132 472 046 147 154 710 390 001 693 135 599 113 476 925 053 44 × 2 = 0 + 0.530 964 914 873 383 630 804 588 264 944 092 294 309 420 780 003 386 271 198 226 953 850 106 88;
  • 6) 0.530 964 914 873 383 630 804 588 264 944 092 294 309 420 780 003 386 271 198 226 953 850 106 88 × 2 = 1 + 0.061 929 829 746 767 261 609 176 529 888 184 588 618 841 560 006 772 542 396 453 907 700 213 76;
  • 7) 0.061 929 829 746 767 261 609 176 529 888 184 588 618 841 560 006 772 542 396 453 907 700 213 76 × 2 = 0 + 0.123 859 659 493 534 523 218 353 059 776 369 177 237 683 120 013 545 084 792 907 815 400 427 52;
  • 8) 0.123 859 659 493 534 523 218 353 059 776 369 177 237 683 120 013 545 084 792 907 815 400 427 52 × 2 = 0 + 0.247 719 318 987 069 046 436 706 119 552 738 354 475 366 240 027 090 169 585 815 630 800 855 04;
  • 9) 0.247 719 318 987 069 046 436 706 119 552 738 354 475 366 240 027 090 169 585 815 630 800 855 04 × 2 = 0 + 0.495 438 637 974 138 092 873 412 239 105 476 708 950 732 480 054 180 339 171 631 261 601 710 08;
  • 10) 0.495 438 637 974 138 092 873 412 239 105 476 708 950 732 480 054 180 339 171 631 261 601 710 08 × 2 = 0 + 0.990 877 275 948 276 185 746 824 478 210 953 417 901 464 960 108 360 678 343 262 523 203 420 16;
  • 11) 0.990 877 275 948 276 185 746 824 478 210 953 417 901 464 960 108 360 678 343 262 523 203 420 16 × 2 = 1 + 0.981 754 551 896 552 371 493 648 956 421 906 835 802 929 920 216 721 356 686 525 046 406 840 32;
  • 12) 0.981 754 551 896 552 371 493 648 956 421 906 835 802 929 920 216 721 356 686 525 046 406 840 32 × 2 = 1 + 0.963 509 103 793 104 742 987 297 912 843 813 671 605 859 840 433 442 713 373 050 092 813 680 64;
  • 13) 0.963 509 103 793 104 742 987 297 912 843 813 671 605 859 840 433 442 713 373 050 092 813 680 64 × 2 = 1 + 0.927 018 207 586 209 485 974 595 825 687 627 343 211 719 680 866 885 426 746 100 185 627 361 28;
  • 14) 0.927 018 207 586 209 485 974 595 825 687 627 343 211 719 680 866 885 426 746 100 185 627 361 28 × 2 = 1 + 0.854 036 415 172 418 971 949 191 651 375 254 686 423 439 361 733 770 853 492 200 371 254 722 56;
  • 15) 0.854 036 415 172 418 971 949 191 651 375 254 686 423 439 361 733 770 853 492 200 371 254 722 56 × 2 = 1 + 0.708 072 830 344 837 943 898 383 302 750 509 372 846 878 723 467 541 706 984 400 742 509 445 12;
  • 16) 0.708 072 830 344 837 943 898 383 302 750 509 372 846 878 723 467 541 706 984 400 742 509 445 12 × 2 = 1 + 0.416 145 660 689 675 887 796 766 605 501 018 745 693 757 446 935 083 413 968 801 485 018 890 24;
  • 17) 0.416 145 660 689 675 887 796 766 605 501 018 745 693 757 446 935 083 413 968 801 485 018 890 24 × 2 = 0 + 0.832 291 321 379 351 775 593 533 211 002 037 491 387 514 893 870 166 827 937 602 970 037 780 48;
  • 18) 0.832 291 321 379 351 775 593 533 211 002 037 491 387 514 893 870 166 827 937 602 970 037 780 48 × 2 = 1 + 0.664 582 642 758 703 551 187 066 422 004 074 982 775 029 787 740 333 655 875 205 940 075 560 96;
  • 19) 0.664 582 642 758 703 551 187 066 422 004 074 982 775 029 787 740 333 655 875 205 940 075 560 96 × 2 = 1 + 0.329 165 285 517 407 102 374 132 844 008 149 965 550 059 575 480 667 311 750 411 880 151 121 92;
  • 20) 0.329 165 285 517 407 102 374 132 844 008 149 965 550 059 575 480 667 311 750 411 880 151 121 92 × 2 = 0 + 0.658 330 571 034 814 204 748 265 688 016 299 931 100 119 150 961 334 623 500 823 760 302 243 84;
  • 21) 0.658 330 571 034 814 204 748 265 688 016 299 931 100 119 150 961 334 623 500 823 760 302 243 84 × 2 = 1 + 0.316 661 142 069 628 409 496 531 376 032 599 862 200 238 301 922 669 247 001 647 520 604 487 68;
  • 22) 0.316 661 142 069 628 409 496 531 376 032 599 862 200 238 301 922 669 247 001 647 520 604 487 68 × 2 = 0 + 0.633 322 284 139 256 818 993 062 752 065 199 724 400 476 603 845 338 494 003 295 041 208 975 36;
  • 23) 0.633 322 284 139 256 818 993 062 752 065 199 724 400 476 603 845 338 494 003 295 041 208 975 36 × 2 = 1 + 0.266 644 568 278 513 637 986 125 504 130 399 448 800 953 207 690 676 988 006 590 082 417 950 72;
  • 24) 0.266 644 568 278 513 637 986 125 504 130 399 448 800 953 207 690 676 988 006 590 082 417 950 72 × 2 = 0 + 0.533 289 136 557 027 275 972 251 008 260 798 897 601 906 415 381 353 976 013 180 164 835 901 44;
  • 25) 0.533 289 136 557 027 275 972 251 008 260 798 897 601 906 415 381 353 976 013 180 164 835 901 44 × 2 = 1 + 0.066 578 273 114 054 551 944 502 016 521 597 795 203 812 830 762 707 952 026 360 329 671 802 88;
  • 26) 0.066 578 273 114 054 551 944 502 016 521 597 795 203 812 830 762 707 952 026 360 329 671 802 88 × 2 = 0 + 0.133 156 546 228 109 103 889 004 033 043 195 590 407 625 661 525 415 904 052 720 659 343 605 76;
  • 27) 0.133 156 546 228 109 103 889 004 033 043 195 590 407 625 661 525 415 904 052 720 659 343 605 76 × 2 = 0 + 0.266 313 092 456 218 207 778 008 066 086 391 180 815 251 323 050 831 808 105 441 318 687 211 52;
  • 28) 0.266 313 092 456 218 207 778 008 066 086 391 180 815 251 323 050 831 808 105 441 318 687 211 52 × 2 = 0 + 0.532 626 184 912 436 415 556 016 132 172 782 361 630 502 646 101 663 616 210 882 637 374 423 04;
  • 29) 0.532 626 184 912 436 415 556 016 132 172 782 361 630 502 646 101 663 616 210 882 637 374 423 04 × 2 = 1 + 0.065 252 369 824 872 831 112 032 264 345 564 723 261 005 292 203 327 232 421 765 274 748 846 08;
  • 30) 0.065 252 369 824 872 831 112 032 264 345 564 723 261 005 292 203 327 232 421 765 274 748 846 08 × 2 = 0 + 0.130 504 739 649 745 662 224 064 528 691 129 446 522 010 584 406 654 464 843 530 549 497 692 16;
  • 31) 0.130 504 739 649 745 662 224 064 528 691 129 446 522 010 584 406 654 464 843 530 549 497 692 16 × 2 = 0 + 0.261 009 479 299 491 324 448 129 057 382 258 893 044 021 168 813 308 929 687 061 098 995 384 32;
  • 32) 0.261 009 479 299 491 324 448 129 057 382 258 893 044 021 168 813 308 929 687 061 098 995 384 32 × 2 = 0 + 0.522 018 958 598 982 648 896 258 114 764 517 786 088 042 337 626 617 859 374 122 197 990 768 64;
  • 33) 0.522 018 958 598 982 648 896 258 114 764 517 786 088 042 337 626 617 859 374 122 197 990 768 64 × 2 = 1 + 0.044 037 917 197 965 297 792 516 229 529 035 572 176 084 675 253 235 718 748 244 395 981 537 28;
  • 34) 0.044 037 917 197 965 297 792 516 229 529 035 572 176 084 675 253 235 718 748 244 395 981 537 28 × 2 = 0 + 0.088 075 834 395 930 595 585 032 459 058 071 144 352 169 350 506 471 437 496 488 791 963 074 56;
  • 35) 0.088 075 834 395 930 595 585 032 459 058 071 144 352 169 350 506 471 437 496 488 791 963 074 56 × 2 = 0 + 0.176 151 668 791 861 191 170 064 918 116 142 288 704 338 701 012 942 874 992 977 583 926 149 12;
  • 36) 0.176 151 668 791 861 191 170 064 918 116 142 288 704 338 701 012 942 874 992 977 583 926 149 12 × 2 = 0 + 0.352 303 337 583 722 382 340 129 836 232 284 577 408 677 402 025 885 749 985 955 167 852 298 24;
  • 37) 0.352 303 337 583 722 382 340 129 836 232 284 577 408 677 402 025 885 749 985 955 167 852 298 24 × 2 = 0 + 0.704 606 675 167 444 764 680 259 672 464 569 154 817 354 804 051 771 499 971 910 335 704 596 48;
  • 38) 0.704 606 675 167 444 764 680 259 672 464 569 154 817 354 804 051 771 499 971 910 335 704 596 48 × 2 = 1 + 0.409 213 350 334 889 529 360 519 344 929 138 309 634 709 608 103 542 999 943 820 671 409 192 96;
  • 39) 0.409 213 350 334 889 529 360 519 344 929 138 309 634 709 608 103 542 999 943 820 671 409 192 96 × 2 = 0 + 0.818 426 700 669 779 058 721 038 689 858 276 619 269 419 216 207 085 999 887 641 342 818 385 92;
  • 40) 0.818 426 700 669 779 058 721 038 689 858 276 619 269 419 216 207 085 999 887 641 342 818 385 92 × 2 = 1 + 0.636 853 401 339 558 117 442 077 379 716 553 238 538 838 432 414 171 999 775 282 685 636 771 84;
  • 41) 0.636 853 401 339 558 117 442 077 379 716 553 238 538 838 432 414 171 999 775 282 685 636 771 84 × 2 = 1 + 0.273 706 802 679 116 234 884 154 759 433 106 477 077 676 864 828 343 999 550 565 371 273 543 68;
  • 42) 0.273 706 802 679 116 234 884 154 759 433 106 477 077 676 864 828 343 999 550 565 371 273 543 68 × 2 = 0 + 0.547 413 605 358 232 469 768 309 518 866 212 954 155 353 729 656 687 999 101 130 742 547 087 36;
  • 43) 0.547 413 605 358 232 469 768 309 518 866 212 954 155 353 729 656 687 999 101 130 742 547 087 36 × 2 = 1 + 0.094 827 210 716 464 939 536 619 037 732 425 908 310 707 459 313 375 998 202 261 485 094 174 72;
  • 44) 0.094 827 210 716 464 939 536 619 037 732 425 908 310 707 459 313 375 998 202 261 485 094 174 72 × 2 = 0 + 0.189 654 421 432 929 879 073 238 075 464 851 816 621 414 918 626 751 996 404 522 970 188 349 44;
  • 45) 0.189 654 421 432 929 879 073 238 075 464 851 816 621 414 918 626 751 996 404 522 970 188 349 44 × 2 = 0 + 0.379 308 842 865 859 758 146 476 150 929 703 633 242 829 837 253 503 992 809 045 940 376 698 88;
  • 46) 0.379 308 842 865 859 758 146 476 150 929 703 633 242 829 837 253 503 992 809 045 940 376 698 88 × 2 = 0 + 0.758 617 685 731 719 516 292 952 301 859 407 266 485 659 674 507 007 985 618 091 880 753 397 76;
  • 47) 0.758 617 685 731 719 516 292 952 301 859 407 266 485 659 674 507 007 985 618 091 880 753 397 76 × 2 = 1 + 0.517 235 371 463 439 032 585 904 603 718 814 532 971 319 349 014 015 971 236 183 761 506 795 52;
  • 48) 0.517 235 371 463 439 032 585 904 603 718 814 532 971 319 349 014 015 971 236 183 761 506 795 52 × 2 = 1 + 0.034 470 742 926 878 065 171 809 207 437 629 065 942 638 698 028 031 942 472 367 523 013 591 04;
  • 49) 0.034 470 742 926 878 065 171 809 207 437 629 065 942 638 698 028 031 942 472 367 523 013 591 04 × 2 = 0 + 0.068 941 485 853 756 130 343 618 414 875 258 131 885 277 396 056 063 884 944 735 046 027 182 08;
  • 50) 0.068 941 485 853 756 130 343 618 414 875 258 131 885 277 396 056 063 884 944 735 046 027 182 08 × 2 = 0 + 0.137 882 971 707 512 260 687 236 829 750 516 263 770 554 792 112 127 769 889 470 092 054 364 16;
  • 51) 0.137 882 971 707 512 260 687 236 829 750 516 263 770 554 792 112 127 769 889 470 092 054 364 16 × 2 = 0 + 0.275 765 943 415 024 521 374 473 659 501 032 527 541 109 584 224 255 539 778 940 184 108 728 32;
  • 52) 0.275 765 943 415 024 521 374 473 659 501 032 527 541 109 584 224 255 539 778 940 184 108 728 32 × 2 = 0 + 0.551 531 886 830 049 042 748 947 319 002 065 055 082 219 168 448 511 079 557 880 368 217 456 64;
  • 53) 0.551 531 886 830 049 042 748 947 319 002 065 055 082 219 168 448 511 079 557 880 368 217 456 64 × 2 = 1 + 0.103 063 773 660 098 085 497 894 638 004 130 110 164 438 336 897 022 159 115 760 736 434 913 28;

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:


0.141 592 653 589 793 238 462 643 383 279 502 884 197 169 399 375 105 820 974 944 592 307 815 84(10) =

0.0010 0100 0011 1111 0110 1010 1000 1000 1000 0101 1010 0011 0000 1(2)

5. Positive number before normalization:

3.141 592 653 589 793 238 462 643 383 279 502 884 197 169 399 375 105 820 974 944 592 307 815 84(10) =

11.0010 0100 0011 1111 0110 1010 1000 1000 1000 0101 1010 0011 0000 1(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 1 positions to the left, so that only one non zero digit remains to the left of it:


3.141 592 653 589 793 238 462 643 383 279 502 884 197 169 399 375 105 820 974 944 592 307 815 84(10) =

11.0010 0100 0011 1111 0110 1010 1000 1000 1000 0101 1010 0011 0000 1(2) =

11.0010 0100 0011 1111 0110 1010 1000 1000 1000 0101 1010 0011 0000 1(2) × 20 =

1.1001 0010 0001 1111 1011 0101 0100 0100 0100 0010 1101 0001 1000 01(2) × 21


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): 1

Mantissa (not normalized):
1.1001 0010 0001 1111 1011 0101 0100 0100 0100 0010 1101 0001 1000 01


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

1 + 2(11-1) - 1 =

(1 + 1 023)(10) =

1 024(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;
  • 1 024 ÷ 2 = 512 + 0;
  • 512 ÷ 2 = 256 + 0;
  • 256 ÷ 2 = 128 + 0;
  • 128 ÷ 2 = 64 + 0;
  • 64 ÷ 2 = 32 + 0;
  • 32 ÷ 2 = 16 + 0;
  • 16 ÷ 2 = 8 + 0;
  • 8 ÷ 2 = 4 + 0;
  • 4 ÷ 2 = 2 + 0;
  • 2 ÷ 2 = 1 + 0;
  • 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) =

1024(10) =

100 0000 0000(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. 1001 0010 0001 1111 1011 0101 0100 0100 0100 0010 1101 0001 1000 01 =

1001 0010 0001 1111 1011 0101 0100 0100 0100 0010 1101 0001 1000


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) =
100 0000 0000

Mantissa (52 bits) =
1001 0010 0001 1111 1011 0101 0100 0100 0100 0010 1101 0001 1000


Decimal number 3.141 592 653 589 793 238 462 643 383 279 502 884 197 169 399 375 105 820 974 944 592 307 815 84 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 100 0000 0000 - 1001 0010 0001 1111 1011 0101 0100 0100 0100 0010 1101 0001 1000

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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