-0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 554 012 142 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 554 012 142(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
-0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 554 012 142(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:

|-0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 554 012 142| = 0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 554 012 142


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: 0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 554 012 142.

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) 0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 554 012 142 × 2 = 0 + 0.001 612 529 247 170 725 028 127 308 313 712 210 068 685 108 024 284;
  • 2) 0.001 612 529 247 170 725 028 127 308 313 712 210 068 685 108 024 284 × 2 = 0 + 0.003 225 058 494 341 450 056 254 616 627 424 420 137 370 216 048 568;
  • 3) 0.003 225 058 494 341 450 056 254 616 627 424 420 137 370 216 048 568 × 2 = 0 + 0.006 450 116 988 682 900 112 509 233 254 848 840 274 740 432 097 136;
  • 4) 0.006 450 116 988 682 900 112 509 233 254 848 840 274 740 432 097 136 × 2 = 0 + 0.012 900 233 977 365 800 225 018 466 509 697 680 549 480 864 194 272;
  • 5) 0.012 900 233 977 365 800 225 018 466 509 697 680 549 480 864 194 272 × 2 = 0 + 0.025 800 467 954 731 600 450 036 933 019 395 361 098 961 728 388 544;
  • 6) 0.025 800 467 954 731 600 450 036 933 019 395 361 098 961 728 388 544 × 2 = 0 + 0.051 600 935 909 463 200 900 073 866 038 790 722 197 923 456 777 088;
  • 7) 0.051 600 935 909 463 200 900 073 866 038 790 722 197 923 456 777 088 × 2 = 0 + 0.103 201 871 818 926 401 800 147 732 077 581 444 395 846 913 554 176;
  • 8) 0.103 201 871 818 926 401 800 147 732 077 581 444 395 846 913 554 176 × 2 = 0 + 0.206 403 743 637 852 803 600 295 464 155 162 888 791 693 827 108 352;
  • 9) 0.206 403 743 637 852 803 600 295 464 155 162 888 791 693 827 108 352 × 2 = 0 + 0.412 807 487 275 705 607 200 590 928 310 325 777 583 387 654 216 704;
  • 10) 0.412 807 487 275 705 607 200 590 928 310 325 777 583 387 654 216 704 × 2 = 0 + 0.825 614 974 551 411 214 401 181 856 620 651 555 166 775 308 433 408;
  • 11) 0.825 614 974 551 411 214 401 181 856 620 651 555 166 775 308 433 408 × 2 = 1 + 0.651 229 949 102 822 428 802 363 713 241 303 110 333 550 616 866 816;
  • 12) 0.651 229 949 102 822 428 802 363 713 241 303 110 333 550 616 866 816 × 2 = 1 + 0.302 459 898 205 644 857 604 727 426 482 606 220 667 101 233 733 632;
  • 13) 0.302 459 898 205 644 857 604 727 426 482 606 220 667 101 233 733 632 × 2 = 0 + 0.604 919 796 411 289 715 209 454 852 965 212 441 334 202 467 467 264;
  • 14) 0.604 919 796 411 289 715 209 454 852 965 212 441 334 202 467 467 264 × 2 = 1 + 0.209 839 592 822 579 430 418 909 705 930 424 882 668 404 934 934 528;
  • 15) 0.209 839 592 822 579 430 418 909 705 930 424 882 668 404 934 934 528 × 2 = 0 + 0.419 679 185 645 158 860 837 819 411 860 849 765 336 809 869 869 056;
  • 16) 0.419 679 185 645 158 860 837 819 411 860 849 765 336 809 869 869 056 × 2 = 0 + 0.839 358 371 290 317 721 675 638 823 721 699 530 673 619 739 738 112;
  • 17) 0.839 358 371 290 317 721 675 638 823 721 699 530 673 619 739 738 112 × 2 = 1 + 0.678 716 742 580 635 443 351 277 647 443 399 061 347 239 479 476 224;
  • 18) 0.678 716 742 580 635 443 351 277 647 443 399 061 347 239 479 476 224 × 2 = 1 + 0.357 433 485 161 270 886 702 555 294 886 798 122 694 478 958 952 448;
  • 19) 0.357 433 485 161 270 886 702 555 294 886 798 122 694 478 958 952 448 × 2 = 0 + 0.714 866 970 322 541 773 405 110 589 773 596 245 388 957 917 904 896;
  • 20) 0.714 866 970 322 541 773 405 110 589 773 596 245 388 957 917 904 896 × 2 = 1 + 0.429 733 940 645 083 546 810 221 179 547 192 490 777 915 835 809 792;
  • 21) 0.429 733 940 645 083 546 810 221 179 547 192 490 777 915 835 809 792 × 2 = 0 + 0.859 467 881 290 167 093 620 442 359 094 384 981 555 831 671 619 584;
  • 22) 0.859 467 881 290 167 093 620 442 359 094 384 981 555 831 671 619 584 × 2 = 1 + 0.718 935 762 580 334 187 240 884 718 188 769 963 111 663 343 239 168;
  • 23) 0.718 935 762 580 334 187 240 884 718 188 769 963 111 663 343 239 168 × 2 = 1 + 0.437 871 525 160 668 374 481 769 436 377 539 926 223 326 686 478 336;
  • 24) 0.437 871 525 160 668 374 481 769 436 377 539 926 223 326 686 478 336 × 2 = 0 + 0.875 743 050 321 336 748 963 538 872 755 079 852 446 653 372 956 672;
  • 25) 0.875 743 050 321 336 748 963 538 872 755 079 852 446 653 372 956 672 × 2 = 1 + 0.751 486 100 642 673 497 927 077 745 510 159 704 893 306 745 913 344;
  • 26) 0.751 486 100 642 673 497 927 077 745 510 159 704 893 306 745 913 344 × 2 = 1 + 0.502 972 201 285 346 995 854 155 491 020 319 409 786 613 491 826 688;
  • 27) 0.502 972 201 285 346 995 854 155 491 020 319 409 786 613 491 826 688 × 2 = 1 + 0.005 944 402 570 693 991 708 310 982 040 638 819 573 226 983 653 376;
  • 28) 0.005 944 402 570 693 991 708 310 982 040 638 819 573 226 983 653 376 × 2 = 0 + 0.011 888 805 141 387 983 416 621 964 081 277 639 146 453 967 306 752;
  • 29) 0.011 888 805 141 387 983 416 621 964 081 277 639 146 453 967 306 752 × 2 = 0 + 0.023 777 610 282 775 966 833 243 928 162 555 278 292 907 934 613 504;
  • 30) 0.023 777 610 282 775 966 833 243 928 162 555 278 292 907 934 613 504 × 2 = 0 + 0.047 555 220 565 551 933 666 487 856 325 110 556 585 815 869 227 008;
  • 31) 0.047 555 220 565 551 933 666 487 856 325 110 556 585 815 869 227 008 × 2 = 0 + 0.095 110 441 131 103 867 332 975 712 650 221 113 171 631 738 454 016;
  • 32) 0.095 110 441 131 103 867 332 975 712 650 221 113 171 631 738 454 016 × 2 = 0 + 0.190 220 882 262 207 734 665 951 425 300 442 226 343 263 476 908 032;
  • 33) 0.190 220 882 262 207 734 665 951 425 300 442 226 343 263 476 908 032 × 2 = 0 + 0.380 441 764 524 415 469 331 902 850 600 884 452 686 526 953 816 064;
  • 34) 0.380 441 764 524 415 469 331 902 850 600 884 452 686 526 953 816 064 × 2 = 0 + 0.760 883 529 048 830 938 663 805 701 201 768 905 373 053 907 632 128;
  • 35) 0.760 883 529 048 830 938 663 805 701 201 768 905 373 053 907 632 128 × 2 = 1 + 0.521 767 058 097 661 877 327 611 402 403 537 810 746 107 815 264 256;
  • 36) 0.521 767 058 097 661 877 327 611 402 403 537 810 746 107 815 264 256 × 2 = 1 + 0.043 534 116 195 323 754 655 222 804 807 075 621 492 215 630 528 512;
  • 37) 0.043 534 116 195 323 754 655 222 804 807 075 621 492 215 630 528 512 × 2 = 0 + 0.087 068 232 390 647 509 310 445 609 614 151 242 984 431 261 057 024;
  • 38) 0.087 068 232 390 647 509 310 445 609 614 151 242 984 431 261 057 024 × 2 = 0 + 0.174 136 464 781 295 018 620 891 219 228 302 485 968 862 522 114 048;
  • 39) 0.174 136 464 781 295 018 620 891 219 228 302 485 968 862 522 114 048 × 2 = 0 + 0.348 272 929 562 590 037 241 782 438 456 604 971 937 725 044 228 096;
  • 40) 0.348 272 929 562 590 037 241 782 438 456 604 971 937 725 044 228 096 × 2 = 0 + 0.696 545 859 125 180 074 483 564 876 913 209 943 875 450 088 456 192;
  • 41) 0.696 545 859 125 180 074 483 564 876 913 209 943 875 450 088 456 192 × 2 = 1 + 0.393 091 718 250 360 148 967 129 753 826 419 887 750 900 176 912 384;
  • 42) 0.393 091 718 250 360 148 967 129 753 826 419 887 750 900 176 912 384 × 2 = 0 + 0.786 183 436 500 720 297 934 259 507 652 839 775 501 800 353 824 768;
  • 43) 0.786 183 436 500 720 297 934 259 507 652 839 775 501 800 353 824 768 × 2 = 1 + 0.572 366 873 001 440 595 868 519 015 305 679 551 003 600 707 649 536;
  • 44) 0.572 366 873 001 440 595 868 519 015 305 679 551 003 600 707 649 536 × 2 = 1 + 0.144 733 746 002 881 191 737 038 030 611 359 102 007 201 415 299 072;
  • 45) 0.144 733 746 002 881 191 737 038 030 611 359 102 007 201 415 299 072 × 2 = 0 + 0.289 467 492 005 762 383 474 076 061 222 718 204 014 402 830 598 144;
  • 46) 0.289 467 492 005 762 383 474 076 061 222 718 204 014 402 830 598 144 × 2 = 0 + 0.578 934 984 011 524 766 948 152 122 445 436 408 028 805 661 196 288;
  • 47) 0.578 934 984 011 524 766 948 152 122 445 436 408 028 805 661 196 288 × 2 = 1 + 0.157 869 968 023 049 533 896 304 244 890 872 816 057 611 322 392 576;
  • 48) 0.157 869 968 023 049 533 896 304 244 890 872 816 057 611 322 392 576 × 2 = 0 + 0.315 739 936 046 099 067 792 608 489 781 745 632 115 222 644 785 152;
  • 49) 0.315 739 936 046 099 067 792 608 489 781 745 632 115 222 644 785 152 × 2 = 0 + 0.631 479 872 092 198 135 585 216 979 563 491 264 230 445 289 570 304;
  • 50) 0.631 479 872 092 198 135 585 216 979 563 491 264 230 445 289 570 304 × 2 = 1 + 0.262 959 744 184 396 271 170 433 959 126 982 528 460 890 579 140 608;
  • 51) 0.262 959 744 184 396 271 170 433 959 126 982 528 460 890 579 140 608 × 2 = 0 + 0.525 919 488 368 792 542 340 867 918 253 965 056 921 781 158 281 216;
  • 52) 0.525 919 488 368 792 542 340 867 918 253 965 056 921 781 158 281 216 × 2 = 1 + 0.051 838 976 737 585 084 681 735 836 507 930 113 843 562 316 562 432;
  • 53) 0.051 838 976 737 585 084 681 735 836 507 930 113 843 562 316 562 432 × 2 = 0 + 0.103 677 953 475 170 169 363 471 673 015 860 227 687 124 633 124 864;
  • 54) 0.103 677 953 475 170 169 363 471 673 015 860 227 687 124 633 124 864 × 2 = 0 + 0.207 355 906 950 340 338 726 943 346 031 720 455 374 249 266 249 728;
  • 55) 0.207 355 906 950 340 338 726 943 346 031 720 455 374 249 266 249 728 × 2 = 0 + 0.414 711 813 900 680 677 453 886 692 063 440 910 748 498 532 499 456;
  • 56) 0.414 711 813 900 680 677 453 886 692 063 440 910 748 498 532 499 456 × 2 = 0 + 0.829 423 627 801 361 354 907 773 384 126 881 821 496 997 064 998 912;
  • 57) 0.829 423 627 801 361 354 907 773 384 126 881 821 496 997 064 998 912 × 2 = 1 + 0.658 847 255 602 722 709 815 546 768 253 763 642 993 994 129 997 824;
  • 58) 0.658 847 255 602 722 709 815 546 768 253 763 642 993 994 129 997 824 × 2 = 1 + 0.317 694 511 205 445 419 631 093 536 507 527 285 987 988 259 995 648;
  • 59) 0.317 694 511 205 445 419 631 093 536 507 527 285 987 988 259 995 648 × 2 = 0 + 0.635 389 022 410 890 839 262 187 073 015 054 571 975 976 519 991 296;
  • 60) 0.635 389 022 410 890 839 262 187 073 015 054 571 975 976 519 991 296 × 2 = 1 + 0.270 778 044 821 781 678 524 374 146 030 109 143 951 953 039 982 592;
  • 61) 0.270 778 044 821 781 678 524 374 146 030 109 143 951 953 039 982 592 × 2 = 0 + 0.541 556 089 643 563 357 048 748 292 060 218 287 903 906 079 965 184;
  • 62) 0.541 556 089 643 563 357 048 748 292 060 218 287 903 906 079 965 184 × 2 = 1 + 0.083 112 179 287 126 714 097 496 584 120 436 575 807 812 159 930 368;
  • 63) 0.083 112 179 287 126 714 097 496 584 120 436 575 807 812 159 930 368 × 2 = 0 + 0.166 224 358 574 253 428 194 993 168 240 873 151 615 624 319 860 736;

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:


0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 554 012 142(10) =

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

6. Positive number before normalization:

0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 554 012 142(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:


0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 554 012 142(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 -0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 554 012 142 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