-0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 554 013 86 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 554 013 86(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
-0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 554 013 86(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:

|-0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 554 013 86| = 0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 554 013 86


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: 0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 554 013 86.

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) 0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 554 013 86 × 2 = 0 + 0.001 612 529 247 170 725 028 127 308 313 712 210 068 685 108 027 72;
  • 2) 0.001 612 529 247 170 725 028 127 308 313 712 210 068 685 108 027 72 × 2 = 0 + 0.003 225 058 494 341 450 056 254 616 627 424 420 137 370 216 055 44;
  • 3) 0.003 225 058 494 341 450 056 254 616 627 424 420 137 370 216 055 44 × 2 = 0 + 0.006 450 116 988 682 900 112 509 233 254 848 840 274 740 432 110 88;
  • 4) 0.006 450 116 988 682 900 112 509 233 254 848 840 274 740 432 110 88 × 2 = 0 + 0.012 900 233 977 365 800 225 018 466 509 697 680 549 480 864 221 76;
  • 5) 0.012 900 233 977 365 800 225 018 466 509 697 680 549 480 864 221 76 × 2 = 0 + 0.025 800 467 954 731 600 450 036 933 019 395 361 098 961 728 443 52;
  • 6) 0.025 800 467 954 731 600 450 036 933 019 395 361 098 961 728 443 52 × 2 = 0 + 0.051 600 935 909 463 200 900 073 866 038 790 722 197 923 456 887 04;
  • 7) 0.051 600 935 909 463 200 900 073 866 038 790 722 197 923 456 887 04 × 2 = 0 + 0.103 201 871 818 926 401 800 147 732 077 581 444 395 846 913 774 08;
  • 8) 0.103 201 871 818 926 401 800 147 732 077 581 444 395 846 913 774 08 × 2 = 0 + 0.206 403 743 637 852 803 600 295 464 155 162 888 791 693 827 548 16;
  • 9) 0.206 403 743 637 852 803 600 295 464 155 162 888 791 693 827 548 16 × 2 = 0 + 0.412 807 487 275 705 607 200 590 928 310 325 777 583 387 655 096 32;
  • 10) 0.412 807 487 275 705 607 200 590 928 310 325 777 583 387 655 096 32 × 2 = 0 + 0.825 614 974 551 411 214 401 181 856 620 651 555 166 775 310 192 64;
  • 11) 0.825 614 974 551 411 214 401 181 856 620 651 555 166 775 310 192 64 × 2 = 1 + 0.651 229 949 102 822 428 802 363 713 241 303 110 333 550 620 385 28;
  • 12) 0.651 229 949 102 822 428 802 363 713 241 303 110 333 550 620 385 28 × 2 = 1 + 0.302 459 898 205 644 857 604 727 426 482 606 220 667 101 240 770 56;
  • 13) 0.302 459 898 205 644 857 604 727 426 482 606 220 667 101 240 770 56 × 2 = 0 + 0.604 919 796 411 289 715 209 454 852 965 212 441 334 202 481 541 12;
  • 14) 0.604 919 796 411 289 715 209 454 852 965 212 441 334 202 481 541 12 × 2 = 1 + 0.209 839 592 822 579 430 418 909 705 930 424 882 668 404 963 082 24;
  • 15) 0.209 839 592 822 579 430 418 909 705 930 424 882 668 404 963 082 24 × 2 = 0 + 0.419 679 185 645 158 860 837 819 411 860 849 765 336 809 926 164 48;
  • 16) 0.419 679 185 645 158 860 837 819 411 860 849 765 336 809 926 164 48 × 2 = 0 + 0.839 358 371 290 317 721 675 638 823 721 699 530 673 619 852 328 96;
  • 17) 0.839 358 371 290 317 721 675 638 823 721 699 530 673 619 852 328 96 × 2 = 1 + 0.678 716 742 580 635 443 351 277 647 443 399 061 347 239 704 657 92;
  • 18) 0.678 716 742 580 635 443 351 277 647 443 399 061 347 239 704 657 92 × 2 = 1 + 0.357 433 485 161 270 886 702 555 294 886 798 122 694 479 409 315 84;
  • 19) 0.357 433 485 161 270 886 702 555 294 886 798 122 694 479 409 315 84 × 2 = 0 + 0.714 866 970 322 541 773 405 110 589 773 596 245 388 958 818 631 68;
  • 20) 0.714 866 970 322 541 773 405 110 589 773 596 245 388 958 818 631 68 × 2 = 1 + 0.429 733 940 645 083 546 810 221 179 547 192 490 777 917 637 263 36;
  • 21) 0.429 733 940 645 083 546 810 221 179 547 192 490 777 917 637 263 36 × 2 = 0 + 0.859 467 881 290 167 093 620 442 359 094 384 981 555 835 274 526 72;
  • 22) 0.859 467 881 290 167 093 620 442 359 094 384 981 555 835 274 526 72 × 2 = 1 + 0.718 935 762 580 334 187 240 884 718 188 769 963 111 670 549 053 44;
  • 23) 0.718 935 762 580 334 187 240 884 718 188 769 963 111 670 549 053 44 × 2 = 1 + 0.437 871 525 160 668 374 481 769 436 377 539 926 223 341 098 106 88;
  • 24) 0.437 871 525 160 668 374 481 769 436 377 539 926 223 341 098 106 88 × 2 = 0 + 0.875 743 050 321 336 748 963 538 872 755 079 852 446 682 196 213 76;
  • 25) 0.875 743 050 321 336 748 963 538 872 755 079 852 446 682 196 213 76 × 2 = 1 + 0.751 486 100 642 673 497 927 077 745 510 159 704 893 364 392 427 52;
  • 26) 0.751 486 100 642 673 497 927 077 745 510 159 704 893 364 392 427 52 × 2 = 1 + 0.502 972 201 285 346 995 854 155 491 020 319 409 786 728 784 855 04;
  • 27) 0.502 972 201 285 346 995 854 155 491 020 319 409 786 728 784 855 04 × 2 = 1 + 0.005 944 402 570 693 991 708 310 982 040 638 819 573 457 569 710 08;
  • 28) 0.005 944 402 570 693 991 708 310 982 040 638 819 573 457 569 710 08 × 2 = 0 + 0.011 888 805 141 387 983 416 621 964 081 277 639 146 915 139 420 16;
  • 29) 0.011 888 805 141 387 983 416 621 964 081 277 639 146 915 139 420 16 × 2 = 0 + 0.023 777 610 282 775 966 833 243 928 162 555 278 293 830 278 840 32;
  • 30) 0.023 777 610 282 775 966 833 243 928 162 555 278 293 830 278 840 32 × 2 = 0 + 0.047 555 220 565 551 933 666 487 856 325 110 556 587 660 557 680 64;
  • 31) 0.047 555 220 565 551 933 666 487 856 325 110 556 587 660 557 680 64 × 2 = 0 + 0.095 110 441 131 103 867 332 975 712 650 221 113 175 321 115 361 28;
  • 32) 0.095 110 441 131 103 867 332 975 712 650 221 113 175 321 115 361 28 × 2 = 0 + 0.190 220 882 262 207 734 665 951 425 300 442 226 350 642 230 722 56;
  • 33) 0.190 220 882 262 207 734 665 951 425 300 442 226 350 642 230 722 56 × 2 = 0 + 0.380 441 764 524 415 469 331 902 850 600 884 452 701 284 461 445 12;
  • 34) 0.380 441 764 524 415 469 331 902 850 600 884 452 701 284 461 445 12 × 2 = 0 + 0.760 883 529 048 830 938 663 805 701 201 768 905 402 568 922 890 24;
  • 35) 0.760 883 529 048 830 938 663 805 701 201 768 905 402 568 922 890 24 × 2 = 1 + 0.521 767 058 097 661 877 327 611 402 403 537 810 805 137 845 780 48;
  • 36) 0.521 767 058 097 661 877 327 611 402 403 537 810 805 137 845 780 48 × 2 = 1 + 0.043 534 116 195 323 754 655 222 804 807 075 621 610 275 691 560 96;
  • 37) 0.043 534 116 195 323 754 655 222 804 807 075 621 610 275 691 560 96 × 2 = 0 + 0.087 068 232 390 647 509 310 445 609 614 151 243 220 551 383 121 92;
  • 38) 0.087 068 232 390 647 509 310 445 609 614 151 243 220 551 383 121 92 × 2 = 0 + 0.174 136 464 781 295 018 620 891 219 228 302 486 441 102 766 243 84;
  • 39) 0.174 136 464 781 295 018 620 891 219 228 302 486 441 102 766 243 84 × 2 = 0 + 0.348 272 929 562 590 037 241 782 438 456 604 972 882 205 532 487 68;
  • 40) 0.348 272 929 562 590 037 241 782 438 456 604 972 882 205 532 487 68 × 2 = 0 + 0.696 545 859 125 180 074 483 564 876 913 209 945 764 411 064 975 36;
  • 41) 0.696 545 859 125 180 074 483 564 876 913 209 945 764 411 064 975 36 × 2 = 1 + 0.393 091 718 250 360 148 967 129 753 826 419 891 528 822 129 950 72;
  • 42) 0.393 091 718 250 360 148 967 129 753 826 419 891 528 822 129 950 72 × 2 = 0 + 0.786 183 436 500 720 297 934 259 507 652 839 783 057 644 259 901 44;
  • 43) 0.786 183 436 500 720 297 934 259 507 652 839 783 057 644 259 901 44 × 2 = 1 + 0.572 366 873 001 440 595 868 519 015 305 679 566 115 288 519 802 88;
  • 44) 0.572 366 873 001 440 595 868 519 015 305 679 566 115 288 519 802 88 × 2 = 1 + 0.144 733 746 002 881 191 737 038 030 611 359 132 230 577 039 605 76;
  • 45) 0.144 733 746 002 881 191 737 038 030 611 359 132 230 577 039 605 76 × 2 = 0 + 0.289 467 492 005 762 383 474 076 061 222 718 264 461 154 079 211 52;
  • 46) 0.289 467 492 005 762 383 474 076 061 222 718 264 461 154 079 211 52 × 2 = 0 + 0.578 934 984 011 524 766 948 152 122 445 436 528 922 308 158 423 04;
  • 47) 0.578 934 984 011 524 766 948 152 122 445 436 528 922 308 158 423 04 × 2 = 1 + 0.157 869 968 023 049 533 896 304 244 890 873 057 844 616 316 846 08;
  • 48) 0.157 869 968 023 049 533 896 304 244 890 873 057 844 616 316 846 08 × 2 = 0 + 0.315 739 936 046 099 067 792 608 489 781 746 115 689 232 633 692 16;
  • 49) 0.315 739 936 046 099 067 792 608 489 781 746 115 689 232 633 692 16 × 2 = 0 + 0.631 479 872 092 198 135 585 216 979 563 492 231 378 465 267 384 32;
  • 50) 0.631 479 872 092 198 135 585 216 979 563 492 231 378 465 267 384 32 × 2 = 1 + 0.262 959 744 184 396 271 170 433 959 126 984 462 756 930 534 768 64;
  • 51) 0.262 959 744 184 396 271 170 433 959 126 984 462 756 930 534 768 64 × 2 = 0 + 0.525 919 488 368 792 542 340 867 918 253 968 925 513 861 069 537 28;
  • 52) 0.525 919 488 368 792 542 340 867 918 253 968 925 513 861 069 537 28 × 2 = 1 + 0.051 838 976 737 585 084 681 735 836 507 937 851 027 722 139 074 56;
  • 53) 0.051 838 976 737 585 084 681 735 836 507 937 851 027 722 139 074 56 × 2 = 0 + 0.103 677 953 475 170 169 363 471 673 015 875 702 055 444 278 149 12;
  • 54) 0.103 677 953 475 170 169 363 471 673 015 875 702 055 444 278 149 12 × 2 = 0 + 0.207 355 906 950 340 338 726 943 346 031 751 404 110 888 556 298 24;
  • 55) 0.207 355 906 950 340 338 726 943 346 031 751 404 110 888 556 298 24 × 2 = 0 + 0.414 711 813 900 680 677 453 886 692 063 502 808 221 777 112 596 48;
  • 56) 0.414 711 813 900 680 677 453 886 692 063 502 808 221 777 112 596 48 × 2 = 0 + 0.829 423 627 801 361 354 907 773 384 127 005 616 443 554 225 192 96;
  • 57) 0.829 423 627 801 361 354 907 773 384 127 005 616 443 554 225 192 96 × 2 = 1 + 0.658 847 255 602 722 709 815 546 768 254 011 232 887 108 450 385 92;
  • 58) 0.658 847 255 602 722 709 815 546 768 254 011 232 887 108 450 385 92 × 2 = 1 + 0.317 694 511 205 445 419 631 093 536 508 022 465 774 216 900 771 84;
  • 59) 0.317 694 511 205 445 419 631 093 536 508 022 465 774 216 900 771 84 × 2 = 0 + 0.635 389 022 410 890 839 262 187 073 016 044 931 548 433 801 543 68;
  • 60) 0.635 389 022 410 890 839 262 187 073 016 044 931 548 433 801 543 68 × 2 = 1 + 0.270 778 044 821 781 678 524 374 146 032 089 863 096 867 603 087 36;
  • 61) 0.270 778 044 821 781 678 524 374 146 032 089 863 096 867 603 087 36 × 2 = 0 + 0.541 556 089 643 563 357 048 748 292 064 179 726 193 735 206 174 72;
  • 62) 0.541 556 089 643 563 357 048 748 292 064 179 726 193 735 206 174 72 × 2 = 1 + 0.083 112 179 287 126 714 097 496 584 128 359 452 387 470 412 349 44;
  • 63) 0.083 112 179 287 126 714 097 496 584 128 359 452 387 470 412 349 44 × 2 = 0 + 0.166 224 358 574 253 428 194 993 168 256 718 904 774 940 824 698 88;

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:


0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 554 013 86(10) =

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

6. Positive number before normalization:

0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 554 013 86(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:


0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 554 013 86(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 -0.000 806 264 623 585 362 514 063 654 156 856 105 034 342 554 013 86 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