-0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 553 990 9 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 553 990 9(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
-0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 553 990 9(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:

|-0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 553 990 9| = 0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 553 990 9


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: 0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 553 990 9.

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) 0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 553 990 9 × 2 = 0 + 0.001 612 529 247 170 725 028 127 308 313 712 210 068 685 107 981 8;
  • 2) 0.001 612 529 247 170 725 028 127 308 313 712 210 068 685 107 981 8 × 2 = 0 + 0.003 225 058 494 341 450 056 254 616 627 424 420 137 370 215 963 6;
  • 3) 0.003 225 058 494 341 450 056 254 616 627 424 420 137 370 215 963 6 × 2 = 0 + 0.006 450 116 988 682 900 112 509 233 254 848 840 274 740 431 927 2;
  • 4) 0.006 450 116 988 682 900 112 509 233 254 848 840 274 740 431 927 2 × 2 = 0 + 0.012 900 233 977 365 800 225 018 466 509 697 680 549 480 863 854 4;
  • 5) 0.012 900 233 977 365 800 225 018 466 509 697 680 549 480 863 854 4 × 2 = 0 + 0.025 800 467 954 731 600 450 036 933 019 395 361 098 961 727 708 8;
  • 6) 0.025 800 467 954 731 600 450 036 933 019 395 361 098 961 727 708 8 × 2 = 0 + 0.051 600 935 909 463 200 900 073 866 038 790 722 197 923 455 417 6;
  • 7) 0.051 600 935 909 463 200 900 073 866 038 790 722 197 923 455 417 6 × 2 = 0 + 0.103 201 871 818 926 401 800 147 732 077 581 444 395 846 910 835 2;
  • 8) 0.103 201 871 818 926 401 800 147 732 077 581 444 395 846 910 835 2 × 2 = 0 + 0.206 403 743 637 852 803 600 295 464 155 162 888 791 693 821 670 4;
  • 9) 0.206 403 743 637 852 803 600 295 464 155 162 888 791 693 821 670 4 × 2 = 0 + 0.412 807 487 275 705 607 200 590 928 310 325 777 583 387 643 340 8;
  • 10) 0.412 807 487 275 705 607 200 590 928 310 325 777 583 387 643 340 8 × 2 = 0 + 0.825 614 974 551 411 214 401 181 856 620 651 555 166 775 286 681 6;
  • 11) 0.825 614 974 551 411 214 401 181 856 620 651 555 166 775 286 681 6 × 2 = 1 + 0.651 229 949 102 822 428 802 363 713 241 303 110 333 550 573 363 2;
  • 12) 0.651 229 949 102 822 428 802 363 713 241 303 110 333 550 573 363 2 × 2 = 1 + 0.302 459 898 205 644 857 604 727 426 482 606 220 667 101 146 726 4;
  • 13) 0.302 459 898 205 644 857 604 727 426 482 606 220 667 101 146 726 4 × 2 = 0 + 0.604 919 796 411 289 715 209 454 852 965 212 441 334 202 293 452 8;
  • 14) 0.604 919 796 411 289 715 209 454 852 965 212 441 334 202 293 452 8 × 2 = 1 + 0.209 839 592 822 579 430 418 909 705 930 424 882 668 404 586 905 6;
  • 15) 0.209 839 592 822 579 430 418 909 705 930 424 882 668 404 586 905 6 × 2 = 0 + 0.419 679 185 645 158 860 837 819 411 860 849 765 336 809 173 811 2;
  • 16) 0.419 679 185 645 158 860 837 819 411 860 849 765 336 809 173 811 2 × 2 = 0 + 0.839 358 371 290 317 721 675 638 823 721 699 530 673 618 347 622 4;
  • 17) 0.839 358 371 290 317 721 675 638 823 721 699 530 673 618 347 622 4 × 2 = 1 + 0.678 716 742 580 635 443 351 277 647 443 399 061 347 236 695 244 8;
  • 18) 0.678 716 742 580 635 443 351 277 647 443 399 061 347 236 695 244 8 × 2 = 1 + 0.357 433 485 161 270 886 702 555 294 886 798 122 694 473 390 489 6;
  • 19) 0.357 433 485 161 270 886 702 555 294 886 798 122 694 473 390 489 6 × 2 = 0 + 0.714 866 970 322 541 773 405 110 589 773 596 245 388 946 780 979 2;
  • 20) 0.714 866 970 322 541 773 405 110 589 773 596 245 388 946 780 979 2 × 2 = 1 + 0.429 733 940 645 083 546 810 221 179 547 192 490 777 893 561 958 4;
  • 21) 0.429 733 940 645 083 546 810 221 179 547 192 490 777 893 561 958 4 × 2 = 0 + 0.859 467 881 290 167 093 620 442 359 094 384 981 555 787 123 916 8;
  • 22) 0.859 467 881 290 167 093 620 442 359 094 384 981 555 787 123 916 8 × 2 = 1 + 0.718 935 762 580 334 187 240 884 718 188 769 963 111 574 247 833 6;
  • 23) 0.718 935 762 580 334 187 240 884 718 188 769 963 111 574 247 833 6 × 2 = 1 + 0.437 871 525 160 668 374 481 769 436 377 539 926 223 148 495 667 2;
  • 24) 0.437 871 525 160 668 374 481 769 436 377 539 926 223 148 495 667 2 × 2 = 0 + 0.875 743 050 321 336 748 963 538 872 755 079 852 446 296 991 334 4;
  • 25) 0.875 743 050 321 336 748 963 538 872 755 079 852 446 296 991 334 4 × 2 = 1 + 0.751 486 100 642 673 497 927 077 745 510 159 704 892 593 982 668 8;
  • 26) 0.751 486 100 642 673 497 927 077 745 510 159 704 892 593 982 668 8 × 2 = 1 + 0.502 972 201 285 346 995 854 155 491 020 319 409 785 187 965 337 6;
  • 27) 0.502 972 201 285 346 995 854 155 491 020 319 409 785 187 965 337 6 × 2 = 1 + 0.005 944 402 570 693 991 708 310 982 040 638 819 570 375 930 675 2;
  • 28) 0.005 944 402 570 693 991 708 310 982 040 638 819 570 375 930 675 2 × 2 = 0 + 0.011 888 805 141 387 983 416 621 964 081 277 639 140 751 861 350 4;
  • 29) 0.011 888 805 141 387 983 416 621 964 081 277 639 140 751 861 350 4 × 2 = 0 + 0.023 777 610 282 775 966 833 243 928 162 555 278 281 503 722 700 8;
  • 30) 0.023 777 610 282 775 966 833 243 928 162 555 278 281 503 722 700 8 × 2 = 0 + 0.047 555 220 565 551 933 666 487 856 325 110 556 563 007 445 401 6;
  • 31) 0.047 555 220 565 551 933 666 487 856 325 110 556 563 007 445 401 6 × 2 = 0 + 0.095 110 441 131 103 867 332 975 712 650 221 113 126 014 890 803 2;
  • 32) 0.095 110 441 131 103 867 332 975 712 650 221 113 126 014 890 803 2 × 2 = 0 + 0.190 220 882 262 207 734 665 951 425 300 442 226 252 029 781 606 4;
  • 33) 0.190 220 882 262 207 734 665 951 425 300 442 226 252 029 781 606 4 × 2 = 0 + 0.380 441 764 524 415 469 331 902 850 600 884 452 504 059 563 212 8;
  • 34) 0.380 441 764 524 415 469 331 902 850 600 884 452 504 059 563 212 8 × 2 = 0 + 0.760 883 529 048 830 938 663 805 701 201 768 905 008 119 126 425 6;
  • 35) 0.760 883 529 048 830 938 663 805 701 201 768 905 008 119 126 425 6 × 2 = 1 + 0.521 767 058 097 661 877 327 611 402 403 537 810 016 238 252 851 2;
  • 36) 0.521 767 058 097 661 877 327 611 402 403 537 810 016 238 252 851 2 × 2 = 1 + 0.043 534 116 195 323 754 655 222 804 807 075 620 032 476 505 702 4;
  • 37) 0.043 534 116 195 323 754 655 222 804 807 075 620 032 476 505 702 4 × 2 = 0 + 0.087 068 232 390 647 509 310 445 609 614 151 240 064 953 011 404 8;
  • 38) 0.087 068 232 390 647 509 310 445 609 614 151 240 064 953 011 404 8 × 2 = 0 + 0.174 136 464 781 295 018 620 891 219 228 302 480 129 906 022 809 6;
  • 39) 0.174 136 464 781 295 018 620 891 219 228 302 480 129 906 022 809 6 × 2 = 0 + 0.348 272 929 562 590 037 241 782 438 456 604 960 259 812 045 619 2;
  • 40) 0.348 272 929 562 590 037 241 782 438 456 604 960 259 812 045 619 2 × 2 = 0 + 0.696 545 859 125 180 074 483 564 876 913 209 920 519 624 091 238 4;
  • 41) 0.696 545 859 125 180 074 483 564 876 913 209 920 519 624 091 238 4 × 2 = 1 + 0.393 091 718 250 360 148 967 129 753 826 419 841 039 248 182 476 8;
  • 42) 0.393 091 718 250 360 148 967 129 753 826 419 841 039 248 182 476 8 × 2 = 0 + 0.786 183 436 500 720 297 934 259 507 652 839 682 078 496 364 953 6;
  • 43) 0.786 183 436 500 720 297 934 259 507 652 839 682 078 496 364 953 6 × 2 = 1 + 0.572 366 873 001 440 595 868 519 015 305 679 364 156 992 729 907 2;
  • 44) 0.572 366 873 001 440 595 868 519 015 305 679 364 156 992 729 907 2 × 2 = 1 + 0.144 733 746 002 881 191 737 038 030 611 358 728 313 985 459 814 4;
  • 45) 0.144 733 746 002 881 191 737 038 030 611 358 728 313 985 459 814 4 × 2 = 0 + 0.289 467 492 005 762 383 474 076 061 222 717 456 627 970 919 628 8;
  • 46) 0.289 467 492 005 762 383 474 076 061 222 717 456 627 970 919 628 8 × 2 = 0 + 0.578 934 984 011 524 766 948 152 122 445 434 913 255 941 839 257 6;
  • 47) 0.578 934 984 011 524 766 948 152 122 445 434 913 255 941 839 257 6 × 2 = 1 + 0.157 869 968 023 049 533 896 304 244 890 869 826 511 883 678 515 2;
  • 48) 0.157 869 968 023 049 533 896 304 244 890 869 826 511 883 678 515 2 × 2 = 0 + 0.315 739 936 046 099 067 792 608 489 781 739 653 023 767 357 030 4;
  • 49) 0.315 739 936 046 099 067 792 608 489 781 739 653 023 767 357 030 4 × 2 = 0 + 0.631 479 872 092 198 135 585 216 979 563 479 306 047 534 714 060 8;
  • 50) 0.631 479 872 092 198 135 585 216 979 563 479 306 047 534 714 060 8 × 2 = 1 + 0.262 959 744 184 396 271 170 433 959 126 958 612 095 069 428 121 6;
  • 51) 0.262 959 744 184 396 271 170 433 959 126 958 612 095 069 428 121 6 × 2 = 0 + 0.525 919 488 368 792 542 340 867 918 253 917 224 190 138 856 243 2;
  • 52) 0.525 919 488 368 792 542 340 867 918 253 917 224 190 138 856 243 2 × 2 = 1 + 0.051 838 976 737 585 084 681 735 836 507 834 448 380 277 712 486 4;
  • 53) 0.051 838 976 737 585 084 681 735 836 507 834 448 380 277 712 486 4 × 2 = 0 + 0.103 677 953 475 170 169 363 471 673 015 668 896 760 555 424 972 8;
  • 54) 0.103 677 953 475 170 169 363 471 673 015 668 896 760 555 424 972 8 × 2 = 0 + 0.207 355 906 950 340 338 726 943 346 031 337 793 521 110 849 945 6;
  • 55) 0.207 355 906 950 340 338 726 943 346 031 337 793 521 110 849 945 6 × 2 = 0 + 0.414 711 813 900 680 677 453 886 692 062 675 587 042 221 699 891 2;
  • 56) 0.414 711 813 900 680 677 453 886 692 062 675 587 042 221 699 891 2 × 2 = 0 + 0.829 423 627 801 361 354 907 773 384 125 351 174 084 443 399 782 4;
  • 57) 0.829 423 627 801 361 354 907 773 384 125 351 174 084 443 399 782 4 × 2 = 1 + 0.658 847 255 602 722 709 815 546 768 250 702 348 168 886 799 564 8;
  • 58) 0.658 847 255 602 722 709 815 546 768 250 702 348 168 886 799 564 8 × 2 = 1 + 0.317 694 511 205 445 419 631 093 536 501 404 696 337 773 599 129 6;
  • 59) 0.317 694 511 205 445 419 631 093 536 501 404 696 337 773 599 129 6 × 2 = 0 + 0.635 389 022 410 890 839 262 187 073 002 809 392 675 547 198 259 2;
  • 60) 0.635 389 022 410 890 839 262 187 073 002 809 392 675 547 198 259 2 × 2 = 1 + 0.270 778 044 821 781 678 524 374 146 005 618 785 351 094 396 518 4;
  • 61) 0.270 778 044 821 781 678 524 374 146 005 618 785 351 094 396 518 4 × 2 = 0 + 0.541 556 089 643 563 357 048 748 292 011 237 570 702 188 793 036 8;
  • 62) 0.541 556 089 643 563 357 048 748 292 011 237 570 702 188 793 036 8 × 2 = 1 + 0.083 112 179 287 126 714 097 496 584 022 475 141 404 377 586 073 6;
  • 63) 0.083 112 179 287 126 714 097 496 584 022 475 141 404 377 586 073 6 × 2 = 0 + 0.166 224 358 574 253 428 194 993 168 044 950 282 808 755 172 147 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:


0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 553 990 9(10) =

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

6. Positive number before normalization:

0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 553 990 9(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:


0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 553 990 9(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 -0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 553 990 9 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