-0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 554 010 31 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 554 010 31(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
-0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 554 010 31(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:

|-0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 554 010 31| = 0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 554 010 31


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: 0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 554 010 31.

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) 0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 554 010 31 × 2 = 0 + 0.001 612 529 247 170 725 028 127 308 313 712 210 068 685 108 020 62;
  • 2) 0.001 612 529 247 170 725 028 127 308 313 712 210 068 685 108 020 62 × 2 = 0 + 0.003 225 058 494 341 450 056 254 616 627 424 420 137 370 216 041 24;
  • 3) 0.003 225 058 494 341 450 056 254 616 627 424 420 137 370 216 041 24 × 2 = 0 + 0.006 450 116 988 682 900 112 509 233 254 848 840 274 740 432 082 48;
  • 4) 0.006 450 116 988 682 900 112 509 233 254 848 840 274 740 432 082 48 × 2 = 0 + 0.012 900 233 977 365 800 225 018 466 509 697 680 549 480 864 164 96;
  • 5) 0.012 900 233 977 365 800 225 018 466 509 697 680 549 480 864 164 96 × 2 = 0 + 0.025 800 467 954 731 600 450 036 933 019 395 361 098 961 728 329 92;
  • 6) 0.025 800 467 954 731 600 450 036 933 019 395 361 098 961 728 329 92 × 2 = 0 + 0.051 600 935 909 463 200 900 073 866 038 790 722 197 923 456 659 84;
  • 7) 0.051 600 935 909 463 200 900 073 866 038 790 722 197 923 456 659 84 × 2 = 0 + 0.103 201 871 818 926 401 800 147 732 077 581 444 395 846 913 319 68;
  • 8) 0.103 201 871 818 926 401 800 147 732 077 581 444 395 846 913 319 68 × 2 = 0 + 0.206 403 743 637 852 803 600 295 464 155 162 888 791 693 826 639 36;
  • 9) 0.206 403 743 637 852 803 600 295 464 155 162 888 791 693 826 639 36 × 2 = 0 + 0.412 807 487 275 705 607 200 590 928 310 325 777 583 387 653 278 72;
  • 10) 0.412 807 487 275 705 607 200 590 928 310 325 777 583 387 653 278 72 × 2 = 0 + 0.825 614 974 551 411 214 401 181 856 620 651 555 166 775 306 557 44;
  • 11) 0.825 614 974 551 411 214 401 181 856 620 651 555 166 775 306 557 44 × 2 = 1 + 0.651 229 949 102 822 428 802 363 713 241 303 110 333 550 613 114 88;
  • 12) 0.651 229 949 102 822 428 802 363 713 241 303 110 333 550 613 114 88 × 2 = 1 + 0.302 459 898 205 644 857 604 727 426 482 606 220 667 101 226 229 76;
  • 13) 0.302 459 898 205 644 857 604 727 426 482 606 220 667 101 226 229 76 × 2 = 0 + 0.604 919 796 411 289 715 209 454 852 965 212 441 334 202 452 459 52;
  • 14) 0.604 919 796 411 289 715 209 454 852 965 212 441 334 202 452 459 52 × 2 = 1 + 0.209 839 592 822 579 430 418 909 705 930 424 882 668 404 904 919 04;
  • 15) 0.209 839 592 822 579 430 418 909 705 930 424 882 668 404 904 919 04 × 2 = 0 + 0.419 679 185 645 158 860 837 819 411 860 849 765 336 809 809 838 08;
  • 16) 0.419 679 185 645 158 860 837 819 411 860 849 765 336 809 809 838 08 × 2 = 0 + 0.839 358 371 290 317 721 675 638 823 721 699 530 673 619 619 676 16;
  • 17) 0.839 358 371 290 317 721 675 638 823 721 699 530 673 619 619 676 16 × 2 = 1 + 0.678 716 742 580 635 443 351 277 647 443 399 061 347 239 239 352 32;
  • 18) 0.678 716 742 580 635 443 351 277 647 443 399 061 347 239 239 352 32 × 2 = 1 + 0.357 433 485 161 270 886 702 555 294 886 798 122 694 478 478 704 64;
  • 19) 0.357 433 485 161 270 886 702 555 294 886 798 122 694 478 478 704 64 × 2 = 0 + 0.714 866 970 322 541 773 405 110 589 773 596 245 388 956 957 409 28;
  • 20) 0.714 866 970 322 541 773 405 110 589 773 596 245 388 956 957 409 28 × 2 = 1 + 0.429 733 940 645 083 546 810 221 179 547 192 490 777 913 914 818 56;
  • 21) 0.429 733 940 645 083 546 810 221 179 547 192 490 777 913 914 818 56 × 2 = 0 + 0.859 467 881 290 167 093 620 442 359 094 384 981 555 827 829 637 12;
  • 22) 0.859 467 881 290 167 093 620 442 359 094 384 981 555 827 829 637 12 × 2 = 1 + 0.718 935 762 580 334 187 240 884 718 188 769 963 111 655 659 274 24;
  • 23) 0.718 935 762 580 334 187 240 884 718 188 769 963 111 655 659 274 24 × 2 = 1 + 0.437 871 525 160 668 374 481 769 436 377 539 926 223 311 318 548 48;
  • 24) 0.437 871 525 160 668 374 481 769 436 377 539 926 223 311 318 548 48 × 2 = 0 + 0.875 743 050 321 336 748 963 538 872 755 079 852 446 622 637 096 96;
  • 25) 0.875 743 050 321 336 748 963 538 872 755 079 852 446 622 637 096 96 × 2 = 1 + 0.751 486 100 642 673 497 927 077 745 510 159 704 893 245 274 193 92;
  • 26) 0.751 486 100 642 673 497 927 077 745 510 159 704 893 245 274 193 92 × 2 = 1 + 0.502 972 201 285 346 995 854 155 491 020 319 409 786 490 548 387 84;
  • 27) 0.502 972 201 285 346 995 854 155 491 020 319 409 786 490 548 387 84 × 2 = 1 + 0.005 944 402 570 693 991 708 310 982 040 638 819 572 981 096 775 68;
  • 28) 0.005 944 402 570 693 991 708 310 982 040 638 819 572 981 096 775 68 × 2 = 0 + 0.011 888 805 141 387 983 416 621 964 081 277 639 145 962 193 551 36;
  • 29) 0.011 888 805 141 387 983 416 621 964 081 277 639 145 962 193 551 36 × 2 = 0 + 0.023 777 610 282 775 966 833 243 928 162 555 278 291 924 387 102 72;
  • 30) 0.023 777 610 282 775 966 833 243 928 162 555 278 291 924 387 102 72 × 2 = 0 + 0.047 555 220 565 551 933 666 487 856 325 110 556 583 848 774 205 44;
  • 31) 0.047 555 220 565 551 933 666 487 856 325 110 556 583 848 774 205 44 × 2 = 0 + 0.095 110 441 131 103 867 332 975 712 650 221 113 167 697 548 410 88;
  • 32) 0.095 110 441 131 103 867 332 975 712 650 221 113 167 697 548 410 88 × 2 = 0 + 0.190 220 882 262 207 734 665 951 425 300 442 226 335 395 096 821 76;
  • 33) 0.190 220 882 262 207 734 665 951 425 300 442 226 335 395 096 821 76 × 2 = 0 + 0.380 441 764 524 415 469 331 902 850 600 884 452 670 790 193 643 52;
  • 34) 0.380 441 764 524 415 469 331 902 850 600 884 452 670 790 193 643 52 × 2 = 0 + 0.760 883 529 048 830 938 663 805 701 201 768 905 341 580 387 287 04;
  • 35) 0.760 883 529 048 830 938 663 805 701 201 768 905 341 580 387 287 04 × 2 = 1 + 0.521 767 058 097 661 877 327 611 402 403 537 810 683 160 774 574 08;
  • 36) 0.521 767 058 097 661 877 327 611 402 403 537 810 683 160 774 574 08 × 2 = 1 + 0.043 534 116 195 323 754 655 222 804 807 075 621 366 321 549 148 16;
  • 37) 0.043 534 116 195 323 754 655 222 804 807 075 621 366 321 549 148 16 × 2 = 0 + 0.087 068 232 390 647 509 310 445 609 614 151 242 732 643 098 296 32;
  • 38) 0.087 068 232 390 647 509 310 445 609 614 151 242 732 643 098 296 32 × 2 = 0 + 0.174 136 464 781 295 018 620 891 219 228 302 485 465 286 196 592 64;
  • 39) 0.174 136 464 781 295 018 620 891 219 228 302 485 465 286 196 592 64 × 2 = 0 + 0.348 272 929 562 590 037 241 782 438 456 604 970 930 572 393 185 28;
  • 40) 0.348 272 929 562 590 037 241 782 438 456 604 970 930 572 393 185 28 × 2 = 0 + 0.696 545 859 125 180 074 483 564 876 913 209 941 861 144 786 370 56;
  • 41) 0.696 545 859 125 180 074 483 564 876 913 209 941 861 144 786 370 56 × 2 = 1 + 0.393 091 718 250 360 148 967 129 753 826 419 883 722 289 572 741 12;
  • 42) 0.393 091 718 250 360 148 967 129 753 826 419 883 722 289 572 741 12 × 2 = 0 + 0.786 183 436 500 720 297 934 259 507 652 839 767 444 579 145 482 24;
  • 43) 0.786 183 436 500 720 297 934 259 507 652 839 767 444 579 145 482 24 × 2 = 1 + 0.572 366 873 001 440 595 868 519 015 305 679 534 889 158 290 964 48;
  • 44) 0.572 366 873 001 440 595 868 519 015 305 679 534 889 158 290 964 48 × 2 = 1 + 0.144 733 746 002 881 191 737 038 030 611 359 069 778 316 581 928 96;
  • 45) 0.144 733 746 002 881 191 737 038 030 611 359 069 778 316 581 928 96 × 2 = 0 + 0.289 467 492 005 762 383 474 076 061 222 718 139 556 633 163 857 92;
  • 46) 0.289 467 492 005 762 383 474 076 061 222 718 139 556 633 163 857 92 × 2 = 0 + 0.578 934 984 011 524 766 948 152 122 445 436 279 113 266 327 715 84;
  • 47) 0.578 934 984 011 524 766 948 152 122 445 436 279 113 266 327 715 84 × 2 = 1 + 0.157 869 968 023 049 533 896 304 244 890 872 558 226 532 655 431 68;
  • 48) 0.157 869 968 023 049 533 896 304 244 890 872 558 226 532 655 431 68 × 2 = 0 + 0.315 739 936 046 099 067 792 608 489 781 745 116 453 065 310 863 36;
  • 49) 0.315 739 936 046 099 067 792 608 489 781 745 116 453 065 310 863 36 × 2 = 0 + 0.631 479 872 092 198 135 585 216 979 563 490 232 906 130 621 726 72;
  • 50) 0.631 479 872 092 198 135 585 216 979 563 490 232 906 130 621 726 72 × 2 = 1 + 0.262 959 744 184 396 271 170 433 959 126 980 465 812 261 243 453 44;
  • 51) 0.262 959 744 184 396 271 170 433 959 126 980 465 812 261 243 453 44 × 2 = 0 + 0.525 919 488 368 792 542 340 867 918 253 960 931 624 522 486 906 88;
  • 52) 0.525 919 488 368 792 542 340 867 918 253 960 931 624 522 486 906 88 × 2 = 1 + 0.051 838 976 737 585 084 681 735 836 507 921 863 249 044 973 813 76;
  • 53) 0.051 838 976 737 585 084 681 735 836 507 921 863 249 044 973 813 76 × 2 = 0 + 0.103 677 953 475 170 169 363 471 673 015 843 726 498 089 947 627 52;
  • 54) 0.103 677 953 475 170 169 363 471 673 015 843 726 498 089 947 627 52 × 2 = 0 + 0.207 355 906 950 340 338 726 943 346 031 687 452 996 179 895 255 04;
  • 55) 0.207 355 906 950 340 338 726 943 346 031 687 452 996 179 895 255 04 × 2 = 0 + 0.414 711 813 900 680 677 453 886 692 063 374 905 992 359 790 510 08;
  • 56) 0.414 711 813 900 680 677 453 886 692 063 374 905 992 359 790 510 08 × 2 = 0 + 0.829 423 627 801 361 354 907 773 384 126 749 811 984 719 581 020 16;
  • 57) 0.829 423 627 801 361 354 907 773 384 126 749 811 984 719 581 020 16 × 2 = 1 + 0.658 847 255 602 722 709 815 546 768 253 499 623 969 439 162 040 32;
  • 58) 0.658 847 255 602 722 709 815 546 768 253 499 623 969 439 162 040 32 × 2 = 1 + 0.317 694 511 205 445 419 631 093 536 506 999 247 938 878 324 080 64;
  • 59) 0.317 694 511 205 445 419 631 093 536 506 999 247 938 878 324 080 64 × 2 = 0 + 0.635 389 022 410 890 839 262 187 073 013 998 495 877 756 648 161 28;
  • 60) 0.635 389 022 410 890 839 262 187 073 013 998 495 877 756 648 161 28 × 2 = 1 + 0.270 778 044 821 781 678 524 374 146 027 996 991 755 513 296 322 56;
  • 61) 0.270 778 044 821 781 678 524 374 146 027 996 991 755 513 296 322 56 × 2 = 0 + 0.541 556 089 643 563 357 048 748 292 055 993 983 511 026 592 645 12;
  • 62) 0.541 556 089 643 563 357 048 748 292 055 993 983 511 026 592 645 12 × 2 = 1 + 0.083 112 179 287 126 714 097 496 584 111 987 967 022 053 185 290 24;
  • 63) 0.083 112 179 287 126 714 097 496 584 111 987 967 022 053 185 290 24 × 2 = 0 + 0.166 224 358 574 253 428 194 993 168 223 975 934 044 106 370 580 48;

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:


0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 554 010 31(10) =

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

6. Positive number before normalization:

0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 554 010 31(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:


0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 554 010 31(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 -0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 554 010 31 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