-0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 553 989 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 553 989(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
-0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 553 989(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:

|-0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 553 989| = 0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 553 989


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: 0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 553 989.

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) 0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 553 989 × 2 = 0 + 0.001 612 529 247 170 725 028 127 308 313 712 210 068 685 107 978;
  • 2) 0.001 612 529 247 170 725 028 127 308 313 712 210 068 685 107 978 × 2 = 0 + 0.003 225 058 494 341 450 056 254 616 627 424 420 137 370 215 956;
  • 3) 0.003 225 058 494 341 450 056 254 616 627 424 420 137 370 215 956 × 2 = 0 + 0.006 450 116 988 682 900 112 509 233 254 848 840 274 740 431 912;
  • 4) 0.006 450 116 988 682 900 112 509 233 254 848 840 274 740 431 912 × 2 = 0 + 0.012 900 233 977 365 800 225 018 466 509 697 680 549 480 863 824;
  • 5) 0.012 900 233 977 365 800 225 018 466 509 697 680 549 480 863 824 × 2 = 0 + 0.025 800 467 954 731 600 450 036 933 019 395 361 098 961 727 648;
  • 6) 0.025 800 467 954 731 600 450 036 933 019 395 361 098 961 727 648 × 2 = 0 + 0.051 600 935 909 463 200 900 073 866 038 790 722 197 923 455 296;
  • 7) 0.051 600 935 909 463 200 900 073 866 038 790 722 197 923 455 296 × 2 = 0 + 0.103 201 871 818 926 401 800 147 732 077 581 444 395 846 910 592;
  • 8) 0.103 201 871 818 926 401 800 147 732 077 581 444 395 846 910 592 × 2 = 0 + 0.206 403 743 637 852 803 600 295 464 155 162 888 791 693 821 184;
  • 9) 0.206 403 743 637 852 803 600 295 464 155 162 888 791 693 821 184 × 2 = 0 + 0.412 807 487 275 705 607 200 590 928 310 325 777 583 387 642 368;
  • 10) 0.412 807 487 275 705 607 200 590 928 310 325 777 583 387 642 368 × 2 = 0 + 0.825 614 974 551 411 214 401 181 856 620 651 555 166 775 284 736;
  • 11) 0.825 614 974 551 411 214 401 181 856 620 651 555 166 775 284 736 × 2 = 1 + 0.651 229 949 102 822 428 802 363 713 241 303 110 333 550 569 472;
  • 12) 0.651 229 949 102 822 428 802 363 713 241 303 110 333 550 569 472 × 2 = 1 + 0.302 459 898 205 644 857 604 727 426 482 606 220 667 101 138 944;
  • 13) 0.302 459 898 205 644 857 604 727 426 482 606 220 667 101 138 944 × 2 = 0 + 0.604 919 796 411 289 715 209 454 852 965 212 441 334 202 277 888;
  • 14) 0.604 919 796 411 289 715 209 454 852 965 212 441 334 202 277 888 × 2 = 1 + 0.209 839 592 822 579 430 418 909 705 930 424 882 668 404 555 776;
  • 15) 0.209 839 592 822 579 430 418 909 705 930 424 882 668 404 555 776 × 2 = 0 + 0.419 679 185 645 158 860 837 819 411 860 849 765 336 809 111 552;
  • 16) 0.419 679 185 645 158 860 837 819 411 860 849 765 336 809 111 552 × 2 = 0 + 0.839 358 371 290 317 721 675 638 823 721 699 530 673 618 223 104;
  • 17) 0.839 358 371 290 317 721 675 638 823 721 699 530 673 618 223 104 × 2 = 1 + 0.678 716 742 580 635 443 351 277 647 443 399 061 347 236 446 208;
  • 18) 0.678 716 742 580 635 443 351 277 647 443 399 061 347 236 446 208 × 2 = 1 + 0.357 433 485 161 270 886 702 555 294 886 798 122 694 472 892 416;
  • 19) 0.357 433 485 161 270 886 702 555 294 886 798 122 694 472 892 416 × 2 = 0 + 0.714 866 970 322 541 773 405 110 589 773 596 245 388 945 784 832;
  • 20) 0.714 866 970 322 541 773 405 110 589 773 596 245 388 945 784 832 × 2 = 1 + 0.429 733 940 645 083 546 810 221 179 547 192 490 777 891 569 664;
  • 21) 0.429 733 940 645 083 546 810 221 179 547 192 490 777 891 569 664 × 2 = 0 + 0.859 467 881 290 167 093 620 442 359 094 384 981 555 783 139 328;
  • 22) 0.859 467 881 290 167 093 620 442 359 094 384 981 555 783 139 328 × 2 = 1 + 0.718 935 762 580 334 187 240 884 718 188 769 963 111 566 278 656;
  • 23) 0.718 935 762 580 334 187 240 884 718 188 769 963 111 566 278 656 × 2 = 1 + 0.437 871 525 160 668 374 481 769 436 377 539 926 223 132 557 312;
  • 24) 0.437 871 525 160 668 374 481 769 436 377 539 926 223 132 557 312 × 2 = 0 + 0.875 743 050 321 336 748 963 538 872 755 079 852 446 265 114 624;
  • 25) 0.875 743 050 321 336 748 963 538 872 755 079 852 446 265 114 624 × 2 = 1 + 0.751 486 100 642 673 497 927 077 745 510 159 704 892 530 229 248;
  • 26) 0.751 486 100 642 673 497 927 077 745 510 159 704 892 530 229 248 × 2 = 1 + 0.502 972 201 285 346 995 854 155 491 020 319 409 785 060 458 496;
  • 27) 0.502 972 201 285 346 995 854 155 491 020 319 409 785 060 458 496 × 2 = 1 + 0.005 944 402 570 693 991 708 310 982 040 638 819 570 120 916 992;
  • 28) 0.005 944 402 570 693 991 708 310 982 040 638 819 570 120 916 992 × 2 = 0 + 0.011 888 805 141 387 983 416 621 964 081 277 639 140 241 833 984;
  • 29) 0.011 888 805 141 387 983 416 621 964 081 277 639 140 241 833 984 × 2 = 0 + 0.023 777 610 282 775 966 833 243 928 162 555 278 280 483 667 968;
  • 30) 0.023 777 610 282 775 966 833 243 928 162 555 278 280 483 667 968 × 2 = 0 + 0.047 555 220 565 551 933 666 487 856 325 110 556 560 967 335 936;
  • 31) 0.047 555 220 565 551 933 666 487 856 325 110 556 560 967 335 936 × 2 = 0 + 0.095 110 441 131 103 867 332 975 712 650 221 113 121 934 671 872;
  • 32) 0.095 110 441 131 103 867 332 975 712 650 221 113 121 934 671 872 × 2 = 0 + 0.190 220 882 262 207 734 665 951 425 300 442 226 243 869 343 744;
  • 33) 0.190 220 882 262 207 734 665 951 425 300 442 226 243 869 343 744 × 2 = 0 + 0.380 441 764 524 415 469 331 902 850 600 884 452 487 738 687 488;
  • 34) 0.380 441 764 524 415 469 331 902 850 600 884 452 487 738 687 488 × 2 = 0 + 0.760 883 529 048 830 938 663 805 701 201 768 904 975 477 374 976;
  • 35) 0.760 883 529 048 830 938 663 805 701 201 768 904 975 477 374 976 × 2 = 1 + 0.521 767 058 097 661 877 327 611 402 403 537 809 950 954 749 952;
  • 36) 0.521 767 058 097 661 877 327 611 402 403 537 809 950 954 749 952 × 2 = 1 + 0.043 534 116 195 323 754 655 222 804 807 075 619 901 909 499 904;
  • 37) 0.043 534 116 195 323 754 655 222 804 807 075 619 901 909 499 904 × 2 = 0 + 0.087 068 232 390 647 509 310 445 609 614 151 239 803 818 999 808;
  • 38) 0.087 068 232 390 647 509 310 445 609 614 151 239 803 818 999 808 × 2 = 0 + 0.174 136 464 781 295 018 620 891 219 228 302 479 607 637 999 616;
  • 39) 0.174 136 464 781 295 018 620 891 219 228 302 479 607 637 999 616 × 2 = 0 + 0.348 272 929 562 590 037 241 782 438 456 604 959 215 275 999 232;
  • 40) 0.348 272 929 562 590 037 241 782 438 456 604 959 215 275 999 232 × 2 = 0 + 0.696 545 859 125 180 074 483 564 876 913 209 918 430 551 998 464;
  • 41) 0.696 545 859 125 180 074 483 564 876 913 209 918 430 551 998 464 × 2 = 1 + 0.393 091 718 250 360 148 967 129 753 826 419 836 861 103 996 928;
  • 42) 0.393 091 718 250 360 148 967 129 753 826 419 836 861 103 996 928 × 2 = 0 + 0.786 183 436 500 720 297 934 259 507 652 839 673 722 207 993 856;
  • 43) 0.786 183 436 500 720 297 934 259 507 652 839 673 722 207 993 856 × 2 = 1 + 0.572 366 873 001 440 595 868 519 015 305 679 347 444 415 987 712;
  • 44) 0.572 366 873 001 440 595 868 519 015 305 679 347 444 415 987 712 × 2 = 1 + 0.144 733 746 002 881 191 737 038 030 611 358 694 888 831 975 424;
  • 45) 0.144 733 746 002 881 191 737 038 030 611 358 694 888 831 975 424 × 2 = 0 + 0.289 467 492 005 762 383 474 076 061 222 717 389 777 663 950 848;
  • 46) 0.289 467 492 005 762 383 474 076 061 222 717 389 777 663 950 848 × 2 = 0 + 0.578 934 984 011 524 766 948 152 122 445 434 779 555 327 901 696;
  • 47) 0.578 934 984 011 524 766 948 152 122 445 434 779 555 327 901 696 × 2 = 1 + 0.157 869 968 023 049 533 896 304 244 890 869 559 110 655 803 392;
  • 48) 0.157 869 968 023 049 533 896 304 244 890 869 559 110 655 803 392 × 2 = 0 + 0.315 739 936 046 099 067 792 608 489 781 739 118 221 311 606 784;
  • 49) 0.315 739 936 046 099 067 792 608 489 781 739 118 221 311 606 784 × 2 = 0 + 0.631 479 872 092 198 135 585 216 979 563 478 236 442 623 213 568;
  • 50) 0.631 479 872 092 198 135 585 216 979 563 478 236 442 623 213 568 × 2 = 1 + 0.262 959 744 184 396 271 170 433 959 126 956 472 885 246 427 136;
  • 51) 0.262 959 744 184 396 271 170 433 959 126 956 472 885 246 427 136 × 2 = 0 + 0.525 919 488 368 792 542 340 867 918 253 912 945 770 492 854 272;
  • 52) 0.525 919 488 368 792 542 340 867 918 253 912 945 770 492 854 272 × 2 = 1 + 0.051 838 976 737 585 084 681 735 836 507 825 891 540 985 708 544;
  • 53) 0.051 838 976 737 585 084 681 735 836 507 825 891 540 985 708 544 × 2 = 0 + 0.103 677 953 475 170 169 363 471 673 015 651 783 081 971 417 088;
  • 54) 0.103 677 953 475 170 169 363 471 673 015 651 783 081 971 417 088 × 2 = 0 + 0.207 355 906 950 340 338 726 943 346 031 303 566 163 942 834 176;
  • 55) 0.207 355 906 950 340 338 726 943 346 031 303 566 163 942 834 176 × 2 = 0 + 0.414 711 813 900 680 677 453 886 692 062 607 132 327 885 668 352;
  • 56) 0.414 711 813 900 680 677 453 886 692 062 607 132 327 885 668 352 × 2 = 0 + 0.829 423 627 801 361 354 907 773 384 125 214 264 655 771 336 704;
  • 57) 0.829 423 627 801 361 354 907 773 384 125 214 264 655 771 336 704 × 2 = 1 + 0.658 847 255 602 722 709 815 546 768 250 428 529 311 542 673 408;
  • 58) 0.658 847 255 602 722 709 815 546 768 250 428 529 311 542 673 408 × 2 = 1 + 0.317 694 511 205 445 419 631 093 536 500 857 058 623 085 346 816;
  • 59) 0.317 694 511 205 445 419 631 093 536 500 857 058 623 085 346 816 × 2 = 0 + 0.635 389 022 410 890 839 262 187 073 001 714 117 246 170 693 632;
  • 60) 0.635 389 022 410 890 839 262 187 073 001 714 117 246 170 693 632 × 2 = 1 + 0.270 778 044 821 781 678 524 374 146 003 428 234 492 341 387 264;
  • 61) 0.270 778 044 821 781 678 524 374 146 003 428 234 492 341 387 264 × 2 = 0 + 0.541 556 089 643 563 357 048 748 292 006 856 468 984 682 774 528;
  • 62) 0.541 556 089 643 563 357 048 748 292 006 856 468 984 682 774 528 × 2 = 1 + 0.083 112 179 287 126 714 097 496 584 013 712 937 969 365 549 056;
  • 63) 0.083 112 179 287 126 714 097 496 584 013 712 937 969 365 549 056 × 2 = 0 + 0.166 224 358 574 253 428 194 993 168 027 425 875 938 731 098 112;

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:


0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 553 989(10) =

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

6. Positive number before normalization:

0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 553 989(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:


0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 553 989(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 -0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 553 989 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

Share

» Convert decimal -0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 553 909 to 64 bit double precision IEEE 754 binary floating point representation standard

» Calculations Performed by Our Visitors: Decimal Numbers Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard. Data organized on a Monthly Basis

» Month 06, 2026 [June]: Decimal Numbers Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard. Calculations performed during the month of: June - by our visitors

» Improve your social skills: My Manager Only Says “We” When Referring to “My Work”. NotebookLM YouTube Video of 11.13 minutes

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