-0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 554 012 248 668 406 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 554 012 248 668 406(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
-0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 554 012 248 668 406(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:

|-0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 554 012 248 668 406| = 0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 554 012 248 668 406


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: 0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 554 012 248 668 406.

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) 0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 554 012 248 668 406 × 2 = 0 + 0.001 612 529 247 170 725 028 127 308 313 712 210 068 685 108 024 497 336 812;
  • 2) 0.001 612 529 247 170 725 028 127 308 313 712 210 068 685 108 024 497 336 812 × 2 = 0 + 0.003 225 058 494 341 450 056 254 616 627 424 420 137 370 216 048 994 673 624;
  • 3) 0.003 225 058 494 341 450 056 254 616 627 424 420 137 370 216 048 994 673 624 × 2 = 0 + 0.006 450 116 988 682 900 112 509 233 254 848 840 274 740 432 097 989 347 248;
  • 4) 0.006 450 116 988 682 900 112 509 233 254 848 840 274 740 432 097 989 347 248 × 2 = 0 + 0.012 900 233 977 365 800 225 018 466 509 697 680 549 480 864 195 978 694 496;
  • 5) 0.012 900 233 977 365 800 225 018 466 509 697 680 549 480 864 195 978 694 496 × 2 = 0 + 0.025 800 467 954 731 600 450 036 933 019 395 361 098 961 728 391 957 388 992;
  • 6) 0.025 800 467 954 731 600 450 036 933 019 395 361 098 961 728 391 957 388 992 × 2 = 0 + 0.051 600 935 909 463 200 900 073 866 038 790 722 197 923 456 783 914 777 984;
  • 7) 0.051 600 935 909 463 200 900 073 866 038 790 722 197 923 456 783 914 777 984 × 2 = 0 + 0.103 201 871 818 926 401 800 147 732 077 581 444 395 846 913 567 829 555 968;
  • 8) 0.103 201 871 818 926 401 800 147 732 077 581 444 395 846 913 567 829 555 968 × 2 = 0 + 0.206 403 743 637 852 803 600 295 464 155 162 888 791 693 827 135 659 111 936;
  • 9) 0.206 403 743 637 852 803 600 295 464 155 162 888 791 693 827 135 659 111 936 × 2 = 0 + 0.412 807 487 275 705 607 200 590 928 310 325 777 583 387 654 271 318 223 872;
  • 10) 0.412 807 487 275 705 607 200 590 928 310 325 777 583 387 654 271 318 223 872 × 2 = 0 + 0.825 614 974 551 411 214 401 181 856 620 651 555 166 775 308 542 636 447 744;
  • 11) 0.825 614 974 551 411 214 401 181 856 620 651 555 166 775 308 542 636 447 744 × 2 = 1 + 0.651 229 949 102 822 428 802 363 713 241 303 110 333 550 617 085 272 895 488;
  • 12) 0.651 229 949 102 822 428 802 363 713 241 303 110 333 550 617 085 272 895 488 × 2 = 1 + 0.302 459 898 205 644 857 604 727 426 482 606 220 667 101 234 170 545 790 976;
  • 13) 0.302 459 898 205 644 857 604 727 426 482 606 220 667 101 234 170 545 790 976 × 2 = 0 + 0.604 919 796 411 289 715 209 454 852 965 212 441 334 202 468 341 091 581 952;
  • 14) 0.604 919 796 411 289 715 209 454 852 965 212 441 334 202 468 341 091 581 952 × 2 = 1 + 0.209 839 592 822 579 430 418 909 705 930 424 882 668 404 936 682 183 163 904;
  • 15) 0.209 839 592 822 579 430 418 909 705 930 424 882 668 404 936 682 183 163 904 × 2 = 0 + 0.419 679 185 645 158 860 837 819 411 860 849 765 336 809 873 364 366 327 808;
  • 16) 0.419 679 185 645 158 860 837 819 411 860 849 765 336 809 873 364 366 327 808 × 2 = 0 + 0.839 358 371 290 317 721 675 638 823 721 699 530 673 619 746 728 732 655 616;
  • 17) 0.839 358 371 290 317 721 675 638 823 721 699 530 673 619 746 728 732 655 616 × 2 = 1 + 0.678 716 742 580 635 443 351 277 647 443 399 061 347 239 493 457 465 311 232;
  • 18) 0.678 716 742 580 635 443 351 277 647 443 399 061 347 239 493 457 465 311 232 × 2 = 1 + 0.357 433 485 161 270 886 702 555 294 886 798 122 694 478 986 914 930 622 464;
  • 19) 0.357 433 485 161 270 886 702 555 294 886 798 122 694 478 986 914 930 622 464 × 2 = 0 + 0.714 866 970 322 541 773 405 110 589 773 596 245 388 957 973 829 861 244 928;
  • 20) 0.714 866 970 322 541 773 405 110 589 773 596 245 388 957 973 829 861 244 928 × 2 = 1 + 0.429 733 940 645 083 546 810 221 179 547 192 490 777 915 947 659 722 489 856;
  • 21) 0.429 733 940 645 083 546 810 221 179 547 192 490 777 915 947 659 722 489 856 × 2 = 0 + 0.859 467 881 290 167 093 620 442 359 094 384 981 555 831 895 319 444 979 712;
  • 22) 0.859 467 881 290 167 093 620 442 359 094 384 981 555 831 895 319 444 979 712 × 2 = 1 + 0.718 935 762 580 334 187 240 884 718 188 769 963 111 663 790 638 889 959 424;
  • 23) 0.718 935 762 580 334 187 240 884 718 188 769 963 111 663 790 638 889 959 424 × 2 = 1 + 0.437 871 525 160 668 374 481 769 436 377 539 926 223 327 581 277 779 918 848;
  • 24) 0.437 871 525 160 668 374 481 769 436 377 539 926 223 327 581 277 779 918 848 × 2 = 0 + 0.875 743 050 321 336 748 963 538 872 755 079 852 446 655 162 555 559 837 696;
  • 25) 0.875 743 050 321 336 748 963 538 872 755 079 852 446 655 162 555 559 837 696 × 2 = 1 + 0.751 486 100 642 673 497 927 077 745 510 159 704 893 310 325 111 119 675 392;
  • 26) 0.751 486 100 642 673 497 927 077 745 510 159 704 893 310 325 111 119 675 392 × 2 = 1 + 0.502 972 201 285 346 995 854 155 491 020 319 409 786 620 650 222 239 350 784;
  • 27) 0.502 972 201 285 346 995 854 155 491 020 319 409 786 620 650 222 239 350 784 × 2 = 1 + 0.005 944 402 570 693 991 708 310 982 040 638 819 573 241 300 444 478 701 568;
  • 28) 0.005 944 402 570 693 991 708 310 982 040 638 819 573 241 300 444 478 701 568 × 2 = 0 + 0.011 888 805 141 387 983 416 621 964 081 277 639 146 482 600 888 957 403 136;
  • 29) 0.011 888 805 141 387 983 416 621 964 081 277 639 146 482 600 888 957 403 136 × 2 = 0 + 0.023 777 610 282 775 966 833 243 928 162 555 278 292 965 201 777 914 806 272;
  • 30) 0.023 777 610 282 775 966 833 243 928 162 555 278 292 965 201 777 914 806 272 × 2 = 0 + 0.047 555 220 565 551 933 666 487 856 325 110 556 585 930 403 555 829 612 544;
  • 31) 0.047 555 220 565 551 933 666 487 856 325 110 556 585 930 403 555 829 612 544 × 2 = 0 + 0.095 110 441 131 103 867 332 975 712 650 221 113 171 860 807 111 659 225 088;
  • 32) 0.095 110 441 131 103 867 332 975 712 650 221 113 171 860 807 111 659 225 088 × 2 = 0 + 0.190 220 882 262 207 734 665 951 425 300 442 226 343 721 614 223 318 450 176;
  • 33) 0.190 220 882 262 207 734 665 951 425 300 442 226 343 721 614 223 318 450 176 × 2 = 0 + 0.380 441 764 524 415 469 331 902 850 600 884 452 687 443 228 446 636 900 352;
  • 34) 0.380 441 764 524 415 469 331 902 850 600 884 452 687 443 228 446 636 900 352 × 2 = 0 + 0.760 883 529 048 830 938 663 805 701 201 768 905 374 886 456 893 273 800 704;
  • 35) 0.760 883 529 048 830 938 663 805 701 201 768 905 374 886 456 893 273 800 704 × 2 = 1 + 0.521 767 058 097 661 877 327 611 402 403 537 810 749 772 913 786 547 601 408;
  • 36) 0.521 767 058 097 661 877 327 611 402 403 537 810 749 772 913 786 547 601 408 × 2 = 1 + 0.043 534 116 195 323 754 655 222 804 807 075 621 499 545 827 573 095 202 816;
  • 37) 0.043 534 116 195 323 754 655 222 804 807 075 621 499 545 827 573 095 202 816 × 2 = 0 + 0.087 068 232 390 647 509 310 445 609 614 151 242 999 091 655 146 190 405 632;
  • 38) 0.087 068 232 390 647 509 310 445 609 614 151 242 999 091 655 146 190 405 632 × 2 = 0 + 0.174 136 464 781 295 018 620 891 219 228 302 485 998 183 310 292 380 811 264;
  • 39) 0.174 136 464 781 295 018 620 891 219 228 302 485 998 183 310 292 380 811 264 × 2 = 0 + 0.348 272 929 562 590 037 241 782 438 456 604 971 996 366 620 584 761 622 528;
  • 40) 0.348 272 929 562 590 037 241 782 438 456 604 971 996 366 620 584 761 622 528 × 2 = 0 + 0.696 545 859 125 180 074 483 564 876 913 209 943 992 733 241 169 523 245 056;
  • 41) 0.696 545 859 125 180 074 483 564 876 913 209 943 992 733 241 169 523 245 056 × 2 = 1 + 0.393 091 718 250 360 148 967 129 753 826 419 887 985 466 482 339 046 490 112;
  • 42) 0.393 091 718 250 360 148 967 129 753 826 419 887 985 466 482 339 046 490 112 × 2 = 0 + 0.786 183 436 500 720 297 934 259 507 652 839 775 970 932 964 678 092 980 224;
  • 43) 0.786 183 436 500 720 297 934 259 507 652 839 775 970 932 964 678 092 980 224 × 2 = 1 + 0.572 366 873 001 440 595 868 519 015 305 679 551 941 865 929 356 185 960 448;
  • 44) 0.572 366 873 001 440 595 868 519 015 305 679 551 941 865 929 356 185 960 448 × 2 = 1 + 0.144 733 746 002 881 191 737 038 030 611 359 103 883 731 858 712 371 920 896;
  • 45) 0.144 733 746 002 881 191 737 038 030 611 359 103 883 731 858 712 371 920 896 × 2 = 0 + 0.289 467 492 005 762 383 474 076 061 222 718 207 767 463 717 424 743 841 792;
  • 46) 0.289 467 492 005 762 383 474 076 061 222 718 207 767 463 717 424 743 841 792 × 2 = 0 + 0.578 934 984 011 524 766 948 152 122 445 436 415 534 927 434 849 487 683 584;
  • 47) 0.578 934 984 011 524 766 948 152 122 445 436 415 534 927 434 849 487 683 584 × 2 = 1 + 0.157 869 968 023 049 533 896 304 244 890 872 831 069 854 869 698 975 367 168;
  • 48) 0.157 869 968 023 049 533 896 304 244 890 872 831 069 854 869 698 975 367 168 × 2 = 0 + 0.315 739 936 046 099 067 792 608 489 781 745 662 139 709 739 397 950 734 336;
  • 49) 0.315 739 936 046 099 067 792 608 489 781 745 662 139 709 739 397 950 734 336 × 2 = 0 + 0.631 479 872 092 198 135 585 216 979 563 491 324 279 419 478 795 901 468 672;
  • 50) 0.631 479 872 092 198 135 585 216 979 563 491 324 279 419 478 795 901 468 672 × 2 = 1 + 0.262 959 744 184 396 271 170 433 959 126 982 648 558 838 957 591 802 937 344;
  • 51) 0.262 959 744 184 396 271 170 433 959 126 982 648 558 838 957 591 802 937 344 × 2 = 0 + 0.525 919 488 368 792 542 340 867 918 253 965 297 117 677 915 183 605 874 688;
  • 52) 0.525 919 488 368 792 542 340 867 918 253 965 297 117 677 915 183 605 874 688 × 2 = 1 + 0.051 838 976 737 585 084 681 735 836 507 930 594 235 355 830 367 211 749 376;
  • 53) 0.051 838 976 737 585 084 681 735 836 507 930 594 235 355 830 367 211 749 376 × 2 = 0 + 0.103 677 953 475 170 169 363 471 673 015 861 188 470 711 660 734 423 498 752;
  • 54) 0.103 677 953 475 170 169 363 471 673 015 861 188 470 711 660 734 423 498 752 × 2 = 0 + 0.207 355 906 950 340 338 726 943 346 031 722 376 941 423 321 468 846 997 504;
  • 55) 0.207 355 906 950 340 338 726 943 346 031 722 376 941 423 321 468 846 997 504 × 2 = 0 + 0.414 711 813 900 680 677 453 886 692 063 444 753 882 846 642 937 693 995 008;
  • 56) 0.414 711 813 900 680 677 453 886 692 063 444 753 882 846 642 937 693 995 008 × 2 = 0 + 0.829 423 627 801 361 354 907 773 384 126 889 507 765 693 285 875 387 990 016;
  • 57) 0.829 423 627 801 361 354 907 773 384 126 889 507 765 693 285 875 387 990 016 × 2 = 1 + 0.658 847 255 602 722 709 815 546 768 253 779 015 531 386 571 750 775 980 032;
  • 58) 0.658 847 255 602 722 709 815 546 768 253 779 015 531 386 571 750 775 980 032 × 2 = 1 + 0.317 694 511 205 445 419 631 093 536 507 558 031 062 773 143 501 551 960 064;
  • 59) 0.317 694 511 205 445 419 631 093 536 507 558 031 062 773 143 501 551 960 064 × 2 = 0 + 0.635 389 022 410 890 839 262 187 073 015 116 062 125 546 287 003 103 920 128;
  • 60) 0.635 389 022 410 890 839 262 187 073 015 116 062 125 546 287 003 103 920 128 × 2 = 1 + 0.270 778 044 821 781 678 524 374 146 030 232 124 251 092 574 006 207 840 256;
  • 61) 0.270 778 044 821 781 678 524 374 146 030 232 124 251 092 574 006 207 840 256 × 2 = 0 + 0.541 556 089 643 563 357 048 748 292 060 464 248 502 185 148 012 415 680 512;
  • 62) 0.541 556 089 643 563 357 048 748 292 060 464 248 502 185 148 012 415 680 512 × 2 = 1 + 0.083 112 179 287 126 714 097 496 584 120 928 497 004 370 296 024 831 361 024;
  • 63) 0.083 112 179 287 126 714 097 496 584 120 928 497 004 370 296 024 831 361 024 × 2 = 0 + 0.166 224 358 574 253 428 194 993 168 241 856 994 008 740 592 049 662 722 048;

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:


0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 554 012 248 668 406(10) =

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

6. Positive number before normalization:

0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 554 012 248 668 406(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:


0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 554 012 248 668 406(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 -0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 554 012 248 668 406 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