-0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 554 012 248 667 4 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 554 012 248 667 4(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
-0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 554 012 248 667 4(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:

|-0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 554 012 248 667 4| = 0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 554 012 248 667 4


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: 0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 554 012 248 667 4.

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) 0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 554 012 248 667 4 × 2 = 0 + 0.001 612 529 247 170 725 028 127 308 313 712 210 068 685 108 024 497 334 8;
  • 2) 0.001 612 529 247 170 725 028 127 308 313 712 210 068 685 108 024 497 334 8 × 2 = 0 + 0.003 225 058 494 341 450 056 254 616 627 424 420 137 370 216 048 994 669 6;
  • 3) 0.003 225 058 494 341 450 056 254 616 627 424 420 137 370 216 048 994 669 6 × 2 = 0 + 0.006 450 116 988 682 900 112 509 233 254 848 840 274 740 432 097 989 339 2;
  • 4) 0.006 450 116 988 682 900 112 509 233 254 848 840 274 740 432 097 989 339 2 × 2 = 0 + 0.012 900 233 977 365 800 225 018 466 509 697 680 549 480 864 195 978 678 4;
  • 5) 0.012 900 233 977 365 800 225 018 466 509 697 680 549 480 864 195 978 678 4 × 2 = 0 + 0.025 800 467 954 731 600 450 036 933 019 395 361 098 961 728 391 957 356 8;
  • 6) 0.025 800 467 954 731 600 450 036 933 019 395 361 098 961 728 391 957 356 8 × 2 = 0 + 0.051 600 935 909 463 200 900 073 866 038 790 722 197 923 456 783 914 713 6;
  • 7) 0.051 600 935 909 463 200 900 073 866 038 790 722 197 923 456 783 914 713 6 × 2 = 0 + 0.103 201 871 818 926 401 800 147 732 077 581 444 395 846 913 567 829 427 2;
  • 8) 0.103 201 871 818 926 401 800 147 732 077 581 444 395 846 913 567 829 427 2 × 2 = 0 + 0.206 403 743 637 852 803 600 295 464 155 162 888 791 693 827 135 658 854 4;
  • 9) 0.206 403 743 637 852 803 600 295 464 155 162 888 791 693 827 135 658 854 4 × 2 = 0 + 0.412 807 487 275 705 607 200 590 928 310 325 777 583 387 654 271 317 708 8;
  • 10) 0.412 807 487 275 705 607 200 590 928 310 325 777 583 387 654 271 317 708 8 × 2 = 0 + 0.825 614 974 551 411 214 401 181 856 620 651 555 166 775 308 542 635 417 6;
  • 11) 0.825 614 974 551 411 214 401 181 856 620 651 555 166 775 308 542 635 417 6 × 2 = 1 + 0.651 229 949 102 822 428 802 363 713 241 303 110 333 550 617 085 270 835 2;
  • 12) 0.651 229 949 102 822 428 802 363 713 241 303 110 333 550 617 085 270 835 2 × 2 = 1 + 0.302 459 898 205 644 857 604 727 426 482 606 220 667 101 234 170 541 670 4;
  • 13) 0.302 459 898 205 644 857 604 727 426 482 606 220 667 101 234 170 541 670 4 × 2 = 0 + 0.604 919 796 411 289 715 209 454 852 965 212 441 334 202 468 341 083 340 8;
  • 14) 0.604 919 796 411 289 715 209 454 852 965 212 441 334 202 468 341 083 340 8 × 2 = 1 + 0.209 839 592 822 579 430 418 909 705 930 424 882 668 404 936 682 166 681 6;
  • 15) 0.209 839 592 822 579 430 418 909 705 930 424 882 668 404 936 682 166 681 6 × 2 = 0 + 0.419 679 185 645 158 860 837 819 411 860 849 765 336 809 873 364 333 363 2;
  • 16) 0.419 679 185 645 158 860 837 819 411 860 849 765 336 809 873 364 333 363 2 × 2 = 0 + 0.839 358 371 290 317 721 675 638 823 721 699 530 673 619 746 728 666 726 4;
  • 17) 0.839 358 371 290 317 721 675 638 823 721 699 530 673 619 746 728 666 726 4 × 2 = 1 + 0.678 716 742 580 635 443 351 277 647 443 399 061 347 239 493 457 333 452 8;
  • 18) 0.678 716 742 580 635 443 351 277 647 443 399 061 347 239 493 457 333 452 8 × 2 = 1 + 0.357 433 485 161 270 886 702 555 294 886 798 122 694 478 986 914 666 905 6;
  • 19) 0.357 433 485 161 270 886 702 555 294 886 798 122 694 478 986 914 666 905 6 × 2 = 0 + 0.714 866 970 322 541 773 405 110 589 773 596 245 388 957 973 829 333 811 2;
  • 20) 0.714 866 970 322 541 773 405 110 589 773 596 245 388 957 973 829 333 811 2 × 2 = 1 + 0.429 733 940 645 083 546 810 221 179 547 192 490 777 915 947 658 667 622 4;
  • 21) 0.429 733 940 645 083 546 810 221 179 547 192 490 777 915 947 658 667 622 4 × 2 = 0 + 0.859 467 881 290 167 093 620 442 359 094 384 981 555 831 895 317 335 244 8;
  • 22) 0.859 467 881 290 167 093 620 442 359 094 384 981 555 831 895 317 335 244 8 × 2 = 1 + 0.718 935 762 580 334 187 240 884 718 188 769 963 111 663 790 634 670 489 6;
  • 23) 0.718 935 762 580 334 187 240 884 718 188 769 963 111 663 790 634 670 489 6 × 2 = 1 + 0.437 871 525 160 668 374 481 769 436 377 539 926 223 327 581 269 340 979 2;
  • 24) 0.437 871 525 160 668 374 481 769 436 377 539 926 223 327 581 269 340 979 2 × 2 = 0 + 0.875 743 050 321 336 748 963 538 872 755 079 852 446 655 162 538 681 958 4;
  • 25) 0.875 743 050 321 336 748 963 538 872 755 079 852 446 655 162 538 681 958 4 × 2 = 1 + 0.751 486 100 642 673 497 927 077 745 510 159 704 893 310 325 077 363 916 8;
  • 26) 0.751 486 100 642 673 497 927 077 745 510 159 704 893 310 325 077 363 916 8 × 2 = 1 + 0.502 972 201 285 346 995 854 155 491 020 319 409 786 620 650 154 727 833 6;
  • 27) 0.502 972 201 285 346 995 854 155 491 020 319 409 786 620 650 154 727 833 6 × 2 = 1 + 0.005 944 402 570 693 991 708 310 982 040 638 819 573 241 300 309 455 667 2;
  • 28) 0.005 944 402 570 693 991 708 310 982 040 638 819 573 241 300 309 455 667 2 × 2 = 0 + 0.011 888 805 141 387 983 416 621 964 081 277 639 146 482 600 618 911 334 4;
  • 29) 0.011 888 805 141 387 983 416 621 964 081 277 639 146 482 600 618 911 334 4 × 2 = 0 + 0.023 777 610 282 775 966 833 243 928 162 555 278 292 965 201 237 822 668 8;
  • 30) 0.023 777 610 282 775 966 833 243 928 162 555 278 292 965 201 237 822 668 8 × 2 = 0 + 0.047 555 220 565 551 933 666 487 856 325 110 556 585 930 402 475 645 337 6;
  • 31) 0.047 555 220 565 551 933 666 487 856 325 110 556 585 930 402 475 645 337 6 × 2 = 0 + 0.095 110 441 131 103 867 332 975 712 650 221 113 171 860 804 951 290 675 2;
  • 32) 0.095 110 441 131 103 867 332 975 712 650 221 113 171 860 804 951 290 675 2 × 2 = 0 + 0.190 220 882 262 207 734 665 951 425 300 442 226 343 721 609 902 581 350 4;
  • 33) 0.190 220 882 262 207 734 665 951 425 300 442 226 343 721 609 902 581 350 4 × 2 = 0 + 0.380 441 764 524 415 469 331 902 850 600 884 452 687 443 219 805 162 700 8;
  • 34) 0.380 441 764 524 415 469 331 902 850 600 884 452 687 443 219 805 162 700 8 × 2 = 0 + 0.760 883 529 048 830 938 663 805 701 201 768 905 374 886 439 610 325 401 6;
  • 35) 0.760 883 529 048 830 938 663 805 701 201 768 905 374 886 439 610 325 401 6 × 2 = 1 + 0.521 767 058 097 661 877 327 611 402 403 537 810 749 772 879 220 650 803 2;
  • 36) 0.521 767 058 097 661 877 327 611 402 403 537 810 749 772 879 220 650 803 2 × 2 = 1 + 0.043 534 116 195 323 754 655 222 804 807 075 621 499 545 758 441 301 606 4;
  • 37) 0.043 534 116 195 323 754 655 222 804 807 075 621 499 545 758 441 301 606 4 × 2 = 0 + 0.087 068 232 390 647 509 310 445 609 614 151 242 999 091 516 882 603 212 8;
  • 38) 0.087 068 232 390 647 509 310 445 609 614 151 242 999 091 516 882 603 212 8 × 2 = 0 + 0.174 136 464 781 295 018 620 891 219 228 302 485 998 183 033 765 206 425 6;
  • 39) 0.174 136 464 781 295 018 620 891 219 228 302 485 998 183 033 765 206 425 6 × 2 = 0 + 0.348 272 929 562 590 037 241 782 438 456 604 971 996 366 067 530 412 851 2;
  • 40) 0.348 272 929 562 590 037 241 782 438 456 604 971 996 366 067 530 412 851 2 × 2 = 0 + 0.696 545 859 125 180 074 483 564 876 913 209 943 992 732 135 060 825 702 4;
  • 41) 0.696 545 859 125 180 074 483 564 876 913 209 943 992 732 135 060 825 702 4 × 2 = 1 + 0.393 091 718 250 360 148 967 129 753 826 419 887 985 464 270 121 651 404 8;
  • 42) 0.393 091 718 250 360 148 967 129 753 826 419 887 985 464 270 121 651 404 8 × 2 = 0 + 0.786 183 436 500 720 297 934 259 507 652 839 775 970 928 540 243 302 809 6;
  • 43) 0.786 183 436 500 720 297 934 259 507 652 839 775 970 928 540 243 302 809 6 × 2 = 1 + 0.572 366 873 001 440 595 868 519 015 305 679 551 941 857 080 486 605 619 2;
  • 44) 0.572 366 873 001 440 595 868 519 015 305 679 551 941 857 080 486 605 619 2 × 2 = 1 + 0.144 733 746 002 881 191 737 038 030 611 359 103 883 714 160 973 211 238 4;
  • 45) 0.144 733 746 002 881 191 737 038 030 611 359 103 883 714 160 973 211 238 4 × 2 = 0 + 0.289 467 492 005 762 383 474 076 061 222 718 207 767 428 321 946 422 476 8;
  • 46) 0.289 467 492 005 762 383 474 076 061 222 718 207 767 428 321 946 422 476 8 × 2 = 0 + 0.578 934 984 011 524 766 948 152 122 445 436 415 534 856 643 892 844 953 6;
  • 47) 0.578 934 984 011 524 766 948 152 122 445 436 415 534 856 643 892 844 953 6 × 2 = 1 + 0.157 869 968 023 049 533 896 304 244 890 872 831 069 713 287 785 689 907 2;
  • 48) 0.157 869 968 023 049 533 896 304 244 890 872 831 069 713 287 785 689 907 2 × 2 = 0 + 0.315 739 936 046 099 067 792 608 489 781 745 662 139 426 575 571 379 814 4;
  • 49) 0.315 739 936 046 099 067 792 608 489 781 745 662 139 426 575 571 379 814 4 × 2 = 0 + 0.631 479 872 092 198 135 585 216 979 563 491 324 278 853 151 142 759 628 8;
  • 50) 0.631 479 872 092 198 135 585 216 979 563 491 324 278 853 151 142 759 628 8 × 2 = 1 + 0.262 959 744 184 396 271 170 433 959 126 982 648 557 706 302 285 519 257 6;
  • 51) 0.262 959 744 184 396 271 170 433 959 126 982 648 557 706 302 285 519 257 6 × 2 = 0 + 0.525 919 488 368 792 542 340 867 918 253 965 297 115 412 604 571 038 515 2;
  • 52) 0.525 919 488 368 792 542 340 867 918 253 965 297 115 412 604 571 038 515 2 × 2 = 1 + 0.051 838 976 737 585 084 681 735 836 507 930 594 230 825 209 142 077 030 4;
  • 53) 0.051 838 976 737 585 084 681 735 836 507 930 594 230 825 209 142 077 030 4 × 2 = 0 + 0.103 677 953 475 170 169 363 471 673 015 861 188 461 650 418 284 154 060 8;
  • 54) 0.103 677 953 475 170 169 363 471 673 015 861 188 461 650 418 284 154 060 8 × 2 = 0 + 0.207 355 906 950 340 338 726 943 346 031 722 376 923 300 836 568 308 121 6;
  • 55) 0.207 355 906 950 340 338 726 943 346 031 722 376 923 300 836 568 308 121 6 × 2 = 0 + 0.414 711 813 900 680 677 453 886 692 063 444 753 846 601 673 136 616 243 2;
  • 56) 0.414 711 813 900 680 677 453 886 692 063 444 753 846 601 673 136 616 243 2 × 2 = 0 + 0.829 423 627 801 361 354 907 773 384 126 889 507 693 203 346 273 232 486 4;
  • 57) 0.829 423 627 801 361 354 907 773 384 126 889 507 693 203 346 273 232 486 4 × 2 = 1 + 0.658 847 255 602 722 709 815 546 768 253 779 015 386 406 692 546 464 972 8;
  • 58) 0.658 847 255 602 722 709 815 546 768 253 779 015 386 406 692 546 464 972 8 × 2 = 1 + 0.317 694 511 205 445 419 631 093 536 507 558 030 772 813 385 092 929 945 6;
  • 59) 0.317 694 511 205 445 419 631 093 536 507 558 030 772 813 385 092 929 945 6 × 2 = 0 + 0.635 389 022 410 890 839 262 187 073 015 116 061 545 626 770 185 859 891 2;
  • 60) 0.635 389 022 410 890 839 262 187 073 015 116 061 545 626 770 185 859 891 2 × 2 = 1 + 0.270 778 044 821 781 678 524 374 146 030 232 123 091 253 540 371 719 782 4;
  • 61) 0.270 778 044 821 781 678 524 374 146 030 232 123 091 253 540 371 719 782 4 × 2 = 0 + 0.541 556 089 643 563 357 048 748 292 060 464 246 182 507 080 743 439 564 8;
  • 62) 0.541 556 089 643 563 357 048 748 292 060 464 246 182 507 080 743 439 564 8 × 2 = 1 + 0.083 112 179 287 126 714 097 496 584 120 928 492 365 014 161 486 879 129 6;
  • 63) 0.083 112 179 287 126 714 097 496 584 120 928 492 365 014 161 486 879 129 6 × 2 = 0 + 0.166 224 358 574 253 428 194 993 168 241 856 984 730 028 322 973 758 259 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:


0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 554 012 248 667 4(10) =

0.0000 0000 0011 0100 1101 0110 1110 0000 0011 0000 1011 0010 0101 0000 1101 010(2)

6. Positive number before normalization:

0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 554 012 248 667 4(10) =

0.0000 0000 0011 0100 1101 0110 1110 0000 0011 0000 1011 0010 0101 0000 1101 010(2)

7. Normalize the binary representation of the number.

Shift the decimal mark 11 positions to the right, so that only one non zero digit remains to the left of it:


0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 554 012 248 667 4(10) =

0.0000 0000 0011 0100 1101 0110 1110 0000 0011 0000 1011 0010 0101 0000 1101 010(2) =

0.0000 0000 0011 0100 1101 0110 1110 0000 0011 0000 1011 0010 0101 0000 1101 010(2) × 20 =

1.1010 0110 1011 0111 0000 0001 1000 0101 1001 0010 1000 0110 1010(2) × 2-11


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): -11

Mantissa (not normalized):
1.1010 0110 1011 0111 0000 0001 1000 0101 1001 0010 1000 0110 1010


9. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

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

(-11 + 1 023)(10) =

1 012(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;
  • 1 012 ÷ 2 = 506 + 0;
  • 506 ÷ 2 = 253 + 0;
  • 253 ÷ 2 = 126 + 1;
  • 126 ÷ 2 = 63 + 0;
  • 63 ÷ 2 = 31 + 1;
  • 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) =

1012(10) =

011 1111 0100(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. 1010 0110 1011 0111 0000 0001 1000 0101 1001 0010 1000 0110 1010 =

1010 0110 1011 0111 0000 0001 1000 0101 1001 0010 1000 0110 1010


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 1111 0100

Mantissa (52 bits) =
1010 0110 1011 0111 0000 0001 1000 0101 1001 0010 1000 0110 1010


Decimal number -0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 554 012 248 667 4 converted to 64 bit double precision IEEE 754 binary floating point representation:

1 - 011 1111 0100 - 1010 0110 1011 0111 0000 0001 1000 0101 1001 0010 1000 0110 1010

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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