-0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 554 012 311 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 554 012 311(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
-0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 554 012 311(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:

|-0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 554 012 311| = 0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 554 012 311


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: 0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 554 012 311.

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) 0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 554 012 311 × 2 = 0 + 0.001 612 529 247 170 725 028 127 308 313 712 210 068 685 108 024 622;
  • 2) 0.001 612 529 247 170 725 028 127 308 313 712 210 068 685 108 024 622 × 2 = 0 + 0.003 225 058 494 341 450 056 254 616 627 424 420 137 370 216 049 244;
  • 3) 0.003 225 058 494 341 450 056 254 616 627 424 420 137 370 216 049 244 × 2 = 0 + 0.006 450 116 988 682 900 112 509 233 254 848 840 274 740 432 098 488;
  • 4) 0.006 450 116 988 682 900 112 509 233 254 848 840 274 740 432 098 488 × 2 = 0 + 0.012 900 233 977 365 800 225 018 466 509 697 680 549 480 864 196 976;
  • 5) 0.012 900 233 977 365 800 225 018 466 509 697 680 549 480 864 196 976 × 2 = 0 + 0.025 800 467 954 731 600 450 036 933 019 395 361 098 961 728 393 952;
  • 6) 0.025 800 467 954 731 600 450 036 933 019 395 361 098 961 728 393 952 × 2 = 0 + 0.051 600 935 909 463 200 900 073 866 038 790 722 197 923 456 787 904;
  • 7) 0.051 600 935 909 463 200 900 073 866 038 790 722 197 923 456 787 904 × 2 = 0 + 0.103 201 871 818 926 401 800 147 732 077 581 444 395 846 913 575 808;
  • 8) 0.103 201 871 818 926 401 800 147 732 077 581 444 395 846 913 575 808 × 2 = 0 + 0.206 403 743 637 852 803 600 295 464 155 162 888 791 693 827 151 616;
  • 9) 0.206 403 743 637 852 803 600 295 464 155 162 888 791 693 827 151 616 × 2 = 0 + 0.412 807 487 275 705 607 200 590 928 310 325 777 583 387 654 303 232;
  • 10) 0.412 807 487 275 705 607 200 590 928 310 325 777 583 387 654 303 232 × 2 = 0 + 0.825 614 974 551 411 214 401 181 856 620 651 555 166 775 308 606 464;
  • 11) 0.825 614 974 551 411 214 401 181 856 620 651 555 166 775 308 606 464 × 2 = 1 + 0.651 229 949 102 822 428 802 363 713 241 303 110 333 550 617 212 928;
  • 12) 0.651 229 949 102 822 428 802 363 713 241 303 110 333 550 617 212 928 × 2 = 1 + 0.302 459 898 205 644 857 604 727 426 482 606 220 667 101 234 425 856;
  • 13) 0.302 459 898 205 644 857 604 727 426 482 606 220 667 101 234 425 856 × 2 = 0 + 0.604 919 796 411 289 715 209 454 852 965 212 441 334 202 468 851 712;
  • 14) 0.604 919 796 411 289 715 209 454 852 965 212 441 334 202 468 851 712 × 2 = 1 + 0.209 839 592 822 579 430 418 909 705 930 424 882 668 404 937 703 424;
  • 15) 0.209 839 592 822 579 430 418 909 705 930 424 882 668 404 937 703 424 × 2 = 0 + 0.419 679 185 645 158 860 837 819 411 860 849 765 336 809 875 406 848;
  • 16) 0.419 679 185 645 158 860 837 819 411 860 849 765 336 809 875 406 848 × 2 = 0 + 0.839 358 371 290 317 721 675 638 823 721 699 530 673 619 750 813 696;
  • 17) 0.839 358 371 290 317 721 675 638 823 721 699 530 673 619 750 813 696 × 2 = 1 + 0.678 716 742 580 635 443 351 277 647 443 399 061 347 239 501 627 392;
  • 18) 0.678 716 742 580 635 443 351 277 647 443 399 061 347 239 501 627 392 × 2 = 1 + 0.357 433 485 161 270 886 702 555 294 886 798 122 694 479 003 254 784;
  • 19) 0.357 433 485 161 270 886 702 555 294 886 798 122 694 479 003 254 784 × 2 = 0 + 0.714 866 970 322 541 773 405 110 589 773 596 245 388 958 006 509 568;
  • 20) 0.714 866 970 322 541 773 405 110 589 773 596 245 388 958 006 509 568 × 2 = 1 + 0.429 733 940 645 083 546 810 221 179 547 192 490 777 916 013 019 136;
  • 21) 0.429 733 940 645 083 546 810 221 179 547 192 490 777 916 013 019 136 × 2 = 0 + 0.859 467 881 290 167 093 620 442 359 094 384 981 555 832 026 038 272;
  • 22) 0.859 467 881 290 167 093 620 442 359 094 384 981 555 832 026 038 272 × 2 = 1 + 0.718 935 762 580 334 187 240 884 718 188 769 963 111 664 052 076 544;
  • 23) 0.718 935 762 580 334 187 240 884 718 188 769 963 111 664 052 076 544 × 2 = 1 + 0.437 871 525 160 668 374 481 769 436 377 539 926 223 328 104 153 088;
  • 24) 0.437 871 525 160 668 374 481 769 436 377 539 926 223 328 104 153 088 × 2 = 0 + 0.875 743 050 321 336 748 963 538 872 755 079 852 446 656 208 306 176;
  • 25) 0.875 743 050 321 336 748 963 538 872 755 079 852 446 656 208 306 176 × 2 = 1 + 0.751 486 100 642 673 497 927 077 745 510 159 704 893 312 416 612 352;
  • 26) 0.751 486 100 642 673 497 927 077 745 510 159 704 893 312 416 612 352 × 2 = 1 + 0.502 972 201 285 346 995 854 155 491 020 319 409 786 624 833 224 704;
  • 27) 0.502 972 201 285 346 995 854 155 491 020 319 409 786 624 833 224 704 × 2 = 1 + 0.005 944 402 570 693 991 708 310 982 040 638 819 573 249 666 449 408;
  • 28) 0.005 944 402 570 693 991 708 310 982 040 638 819 573 249 666 449 408 × 2 = 0 + 0.011 888 805 141 387 983 416 621 964 081 277 639 146 499 332 898 816;
  • 29) 0.011 888 805 141 387 983 416 621 964 081 277 639 146 499 332 898 816 × 2 = 0 + 0.023 777 610 282 775 966 833 243 928 162 555 278 292 998 665 797 632;
  • 30) 0.023 777 610 282 775 966 833 243 928 162 555 278 292 998 665 797 632 × 2 = 0 + 0.047 555 220 565 551 933 666 487 856 325 110 556 585 997 331 595 264;
  • 31) 0.047 555 220 565 551 933 666 487 856 325 110 556 585 997 331 595 264 × 2 = 0 + 0.095 110 441 131 103 867 332 975 712 650 221 113 171 994 663 190 528;
  • 32) 0.095 110 441 131 103 867 332 975 712 650 221 113 171 994 663 190 528 × 2 = 0 + 0.190 220 882 262 207 734 665 951 425 300 442 226 343 989 326 381 056;
  • 33) 0.190 220 882 262 207 734 665 951 425 300 442 226 343 989 326 381 056 × 2 = 0 + 0.380 441 764 524 415 469 331 902 850 600 884 452 687 978 652 762 112;
  • 34) 0.380 441 764 524 415 469 331 902 850 600 884 452 687 978 652 762 112 × 2 = 0 + 0.760 883 529 048 830 938 663 805 701 201 768 905 375 957 305 524 224;
  • 35) 0.760 883 529 048 830 938 663 805 701 201 768 905 375 957 305 524 224 × 2 = 1 + 0.521 767 058 097 661 877 327 611 402 403 537 810 751 914 611 048 448;
  • 36) 0.521 767 058 097 661 877 327 611 402 403 537 810 751 914 611 048 448 × 2 = 1 + 0.043 534 116 195 323 754 655 222 804 807 075 621 503 829 222 096 896;
  • 37) 0.043 534 116 195 323 754 655 222 804 807 075 621 503 829 222 096 896 × 2 = 0 + 0.087 068 232 390 647 509 310 445 609 614 151 243 007 658 444 193 792;
  • 38) 0.087 068 232 390 647 509 310 445 609 614 151 243 007 658 444 193 792 × 2 = 0 + 0.174 136 464 781 295 018 620 891 219 228 302 486 015 316 888 387 584;
  • 39) 0.174 136 464 781 295 018 620 891 219 228 302 486 015 316 888 387 584 × 2 = 0 + 0.348 272 929 562 590 037 241 782 438 456 604 972 030 633 776 775 168;
  • 40) 0.348 272 929 562 590 037 241 782 438 456 604 972 030 633 776 775 168 × 2 = 0 + 0.696 545 859 125 180 074 483 564 876 913 209 944 061 267 553 550 336;
  • 41) 0.696 545 859 125 180 074 483 564 876 913 209 944 061 267 553 550 336 × 2 = 1 + 0.393 091 718 250 360 148 967 129 753 826 419 888 122 535 107 100 672;
  • 42) 0.393 091 718 250 360 148 967 129 753 826 419 888 122 535 107 100 672 × 2 = 0 + 0.786 183 436 500 720 297 934 259 507 652 839 776 245 070 214 201 344;
  • 43) 0.786 183 436 500 720 297 934 259 507 652 839 776 245 070 214 201 344 × 2 = 1 + 0.572 366 873 001 440 595 868 519 015 305 679 552 490 140 428 402 688;
  • 44) 0.572 366 873 001 440 595 868 519 015 305 679 552 490 140 428 402 688 × 2 = 1 + 0.144 733 746 002 881 191 737 038 030 611 359 104 980 280 856 805 376;
  • 45) 0.144 733 746 002 881 191 737 038 030 611 359 104 980 280 856 805 376 × 2 = 0 + 0.289 467 492 005 762 383 474 076 061 222 718 209 960 561 713 610 752;
  • 46) 0.289 467 492 005 762 383 474 076 061 222 718 209 960 561 713 610 752 × 2 = 0 + 0.578 934 984 011 524 766 948 152 122 445 436 419 921 123 427 221 504;
  • 47) 0.578 934 984 011 524 766 948 152 122 445 436 419 921 123 427 221 504 × 2 = 1 + 0.157 869 968 023 049 533 896 304 244 890 872 839 842 246 854 443 008;
  • 48) 0.157 869 968 023 049 533 896 304 244 890 872 839 842 246 854 443 008 × 2 = 0 + 0.315 739 936 046 099 067 792 608 489 781 745 679 684 493 708 886 016;
  • 49) 0.315 739 936 046 099 067 792 608 489 781 745 679 684 493 708 886 016 × 2 = 0 + 0.631 479 872 092 198 135 585 216 979 563 491 359 368 987 417 772 032;
  • 50) 0.631 479 872 092 198 135 585 216 979 563 491 359 368 987 417 772 032 × 2 = 1 + 0.262 959 744 184 396 271 170 433 959 126 982 718 737 974 835 544 064;
  • 51) 0.262 959 744 184 396 271 170 433 959 126 982 718 737 974 835 544 064 × 2 = 0 + 0.525 919 488 368 792 542 340 867 918 253 965 437 475 949 671 088 128;
  • 52) 0.525 919 488 368 792 542 340 867 918 253 965 437 475 949 671 088 128 × 2 = 1 + 0.051 838 976 737 585 084 681 735 836 507 930 874 951 899 342 176 256;
  • 53) 0.051 838 976 737 585 084 681 735 836 507 930 874 951 899 342 176 256 × 2 = 0 + 0.103 677 953 475 170 169 363 471 673 015 861 749 903 798 684 352 512;
  • 54) 0.103 677 953 475 170 169 363 471 673 015 861 749 903 798 684 352 512 × 2 = 0 + 0.207 355 906 950 340 338 726 943 346 031 723 499 807 597 368 705 024;
  • 55) 0.207 355 906 950 340 338 726 943 346 031 723 499 807 597 368 705 024 × 2 = 0 + 0.414 711 813 900 680 677 453 886 692 063 446 999 615 194 737 410 048;
  • 56) 0.414 711 813 900 680 677 453 886 692 063 446 999 615 194 737 410 048 × 2 = 0 + 0.829 423 627 801 361 354 907 773 384 126 893 999 230 389 474 820 096;
  • 57) 0.829 423 627 801 361 354 907 773 384 126 893 999 230 389 474 820 096 × 2 = 1 + 0.658 847 255 602 722 709 815 546 768 253 787 998 460 778 949 640 192;
  • 58) 0.658 847 255 602 722 709 815 546 768 253 787 998 460 778 949 640 192 × 2 = 1 + 0.317 694 511 205 445 419 631 093 536 507 575 996 921 557 899 280 384;
  • 59) 0.317 694 511 205 445 419 631 093 536 507 575 996 921 557 899 280 384 × 2 = 0 + 0.635 389 022 410 890 839 262 187 073 015 151 993 843 115 798 560 768;
  • 60) 0.635 389 022 410 890 839 262 187 073 015 151 993 843 115 798 560 768 × 2 = 1 + 0.270 778 044 821 781 678 524 374 146 030 303 987 686 231 597 121 536;
  • 61) 0.270 778 044 821 781 678 524 374 146 030 303 987 686 231 597 121 536 × 2 = 0 + 0.541 556 089 643 563 357 048 748 292 060 607 975 372 463 194 243 072;
  • 62) 0.541 556 089 643 563 357 048 748 292 060 607 975 372 463 194 243 072 × 2 = 1 + 0.083 112 179 287 126 714 097 496 584 121 215 950 744 926 388 486 144;
  • 63) 0.083 112 179 287 126 714 097 496 584 121 215 950 744 926 388 486 144 × 2 = 0 + 0.166 224 358 574 253 428 194 993 168 242 431 901 489 852 776 972 288;

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:


0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 554 012 311(10) =

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

6. Positive number before normalization:

0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 554 012 311(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:


0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 554 012 311(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 -0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 554 012 311 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