0.000 000 000 000 000 000 008 550 1 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 000 000 008 550 1(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
0.000 000 000 000 000 000 008 550 1(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. 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;

2. 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)


3. Convert to binary (base 2) the fractional part: 0.000 000 000 000 000 000 008 550 1.

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) 0.000 000 000 000 000 000 008 550 1 × 2 = 0 + 0.000 000 000 000 000 000 017 100 2;
  • 2) 0.000 000 000 000 000 000 017 100 2 × 2 = 0 + 0.000 000 000 000 000 000 034 200 4;
  • 3) 0.000 000 000 000 000 000 034 200 4 × 2 = 0 + 0.000 000 000 000 000 000 068 400 8;
  • 4) 0.000 000 000 000 000 000 068 400 8 × 2 = 0 + 0.000 000 000 000 000 000 136 801 6;
  • 5) 0.000 000 000 000 000 000 136 801 6 × 2 = 0 + 0.000 000 000 000 000 000 273 603 2;
  • 6) 0.000 000 000 000 000 000 273 603 2 × 2 = 0 + 0.000 000 000 000 000 000 547 206 4;
  • 7) 0.000 000 000 000 000 000 547 206 4 × 2 = 0 + 0.000 000 000 000 000 001 094 412 8;
  • 8) 0.000 000 000 000 000 001 094 412 8 × 2 = 0 + 0.000 000 000 000 000 002 188 825 6;
  • 9) 0.000 000 000 000 000 002 188 825 6 × 2 = 0 + 0.000 000 000 000 000 004 377 651 2;
  • 10) 0.000 000 000 000 000 004 377 651 2 × 2 = 0 + 0.000 000 000 000 000 008 755 302 4;
  • 11) 0.000 000 000 000 000 008 755 302 4 × 2 = 0 + 0.000 000 000 000 000 017 510 604 8;
  • 12) 0.000 000 000 000 000 017 510 604 8 × 2 = 0 + 0.000 000 000 000 000 035 021 209 6;
  • 13) 0.000 000 000 000 000 035 021 209 6 × 2 = 0 + 0.000 000 000 000 000 070 042 419 2;
  • 14) 0.000 000 000 000 000 070 042 419 2 × 2 = 0 + 0.000 000 000 000 000 140 084 838 4;
  • 15) 0.000 000 000 000 000 140 084 838 4 × 2 = 0 + 0.000 000 000 000 000 280 169 676 8;
  • 16) 0.000 000 000 000 000 280 169 676 8 × 2 = 0 + 0.000 000 000 000 000 560 339 353 6;
  • 17) 0.000 000 000 000 000 560 339 353 6 × 2 = 0 + 0.000 000 000 000 001 120 678 707 2;
  • 18) 0.000 000 000 000 001 120 678 707 2 × 2 = 0 + 0.000 000 000 000 002 241 357 414 4;
  • 19) 0.000 000 000 000 002 241 357 414 4 × 2 = 0 + 0.000 000 000 000 004 482 714 828 8;
  • 20) 0.000 000 000 000 004 482 714 828 8 × 2 = 0 + 0.000 000 000 000 008 965 429 657 6;
  • 21) 0.000 000 000 000 008 965 429 657 6 × 2 = 0 + 0.000 000 000 000 017 930 859 315 2;
  • 22) 0.000 000 000 000 017 930 859 315 2 × 2 = 0 + 0.000 000 000 000 035 861 718 630 4;
  • 23) 0.000 000 000 000 035 861 718 630 4 × 2 = 0 + 0.000 000 000 000 071 723 437 260 8;
  • 24) 0.000 000 000 000 071 723 437 260 8 × 2 = 0 + 0.000 000 000 000 143 446 874 521 6;
  • 25) 0.000 000 000 000 143 446 874 521 6 × 2 = 0 + 0.000 000 000 000 286 893 749 043 2;
  • 26) 0.000 000 000 000 286 893 749 043 2 × 2 = 0 + 0.000 000 000 000 573 787 498 086 4;
  • 27) 0.000 000 000 000 573 787 498 086 4 × 2 = 0 + 0.000 000 000 001 147 574 996 172 8;
  • 28) 0.000 000 000 001 147 574 996 172 8 × 2 = 0 + 0.000 000 000 002 295 149 992 345 6;
  • 29) 0.000 000 000 002 295 149 992 345 6 × 2 = 0 + 0.000 000 000 004 590 299 984 691 2;
  • 30) 0.000 000 000 004 590 299 984 691 2 × 2 = 0 + 0.000 000 000 009 180 599 969 382 4;
  • 31) 0.000 000 000 009 180 599 969 382 4 × 2 = 0 + 0.000 000 000 018 361 199 938 764 8;
  • 32) 0.000 000 000 018 361 199 938 764 8 × 2 = 0 + 0.000 000 000 036 722 399 877 529 6;
  • 33) 0.000 000 000 036 722 399 877 529 6 × 2 = 0 + 0.000 000 000 073 444 799 755 059 2;
  • 34) 0.000 000 000 073 444 799 755 059 2 × 2 = 0 + 0.000 000 000 146 889 599 510 118 4;
  • 35) 0.000 000 000 146 889 599 510 118 4 × 2 = 0 + 0.000 000 000 293 779 199 020 236 8;
  • 36) 0.000 000 000 293 779 199 020 236 8 × 2 = 0 + 0.000 000 000 587 558 398 040 473 6;
  • 37) 0.000 000 000 587 558 398 040 473 6 × 2 = 0 + 0.000 000 001 175 116 796 080 947 2;
  • 38) 0.000 000 001 175 116 796 080 947 2 × 2 = 0 + 0.000 000 002 350 233 592 161 894 4;
  • 39) 0.000 000 002 350 233 592 161 894 4 × 2 = 0 + 0.000 000 004 700 467 184 323 788 8;
  • 40) 0.000 000 004 700 467 184 323 788 8 × 2 = 0 + 0.000 000 009 400 934 368 647 577 6;
  • 41) 0.000 000 009 400 934 368 647 577 6 × 2 = 0 + 0.000 000 018 801 868 737 295 155 2;
  • 42) 0.000 000 018 801 868 737 295 155 2 × 2 = 0 + 0.000 000 037 603 737 474 590 310 4;
  • 43) 0.000 000 037 603 737 474 590 310 4 × 2 = 0 + 0.000 000 075 207 474 949 180 620 8;
  • 44) 0.000 000 075 207 474 949 180 620 8 × 2 = 0 + 0.000 000 150 414 949 898 361 241 6;
  • 45) 0.000 000 150 414 949 898 361 241 6 × 2 = 0 + 0.000 000 300 829 899 796 722 483 2;
  • 46) 0.000 000 300 829 899 796 722 483 2 × 2 = 0 + 0.000 000 601 659 799 593 444 966 4;
  • 47) 0.000 000 601 659 799 593 444 966 4 × 2 = 0 + 0.000 001 203 319 599 186 889 932 8;
  • 48) 0.000 001 203 319 599 186 889 932 8 × 2 = 0 + 0.000 002 406 639 198 373 779 865 6;
  • 49) 0.000 002 406 639 198 373 779 865 6 × 2 = 0 + 0.000 004 813 278 396 747 559 731 2;
  • 50) 0.000 004 813 278 396 747 559 731 2 × 2 = 0 + 0.000 009 626 556 793 495 119 462 4;
  • 51) 0.000 009 626 556 793 495 119 462 4 × 2 = 0 + 0.000 019 253 113 586 990 238 924 8;
  • 52) 0.000 019 253 113 586 990 238 924 8 × 2 = 0 + 0.000 038 506 227 173 980 477 849 6;
  • 53) 0.000 038 506 227 173 980 477 849 6 × 2 = 0 + 0.000 077 012 454 347 960 955 699 2;
  • 54) 0.000 077 012 454 347 960 955 699 2 × 2 = 0 + 0.000 154 024 908 695 921 911 398 4;
  • 55) 0.000 154 024 908 695 921 911 398 4 × 2 = 0 + 0.000 308 049 817 391 843 822 796 8;
  • 56) 0.000 308 049 817 391 843 822 796 8 × 2 = 0 + 0.000 616 099 634 783 687 645 593 6;
  • 57) 0.000 616 099 634 783 687 645 593 6 × 2 = 0 + 0.001 232 199 269 567 375 291 187 2;
  • 58) 0.001 232 199 269 567 375 291 187 2 × 2 = 0 + 0.002 464 398 539 134 750 582 374 4;
  • 59) 0.002 464 398 539 134 750 582 374 4 × 2 = 0 + 0.004 928 797 078 269 501 164 748 8;
  • 60) 0.004 928 797 078 269 501 164 748 8 × 2 = 0 + 0.009 857 594 156 539 002 329 497 6;
  • 61) 0.009 857 594 156 539 002 329 497 6 × 2 = 0 + 0.019 715 188 313 078 004 658 995 2;
  • 62) 0.019 715 188 313 078 004 658 995 2 × 2 = 0 + 0.039 430 376 626 156 009 317 990 4;
  • 63) 0.039 430 376 626 156 009 317 990 4 × 2 = 0 + 0.078 860 753 252 312 018 635 980 8;
  • 64) 0.078 860 753 252 312 018 635 980 8 × 2 = 0 + 0.157 721 506 504 624 037 271 961 6;
  • 65) 0.157 721 506 504 624 037 271 961 6 × 2 = 0 + 0.315 443 013 009 248 074 543 923 2;
  • 66) 0.315 443 013 009 248 074 543 923 2 × 2 = 0 + 0.630 886 026 018 496 149 087 846 4;
  • 67) 0.630 886 026 018 496 149 087 846 4 × 2 = 1 + 0.261 772 052 036 992 298 175 692 8;
  • 68) 0.261 772 052 036 992 298 175 692 8 × 2 = 0 + 0.523 544 104 073 984 596 351 385 6;
  • 69) 0.523 544 104 073 984 596 351 385 6 × 2 = 1 + 0.047 088 208 147 969 192 702 771 2;
  • 70) 0.047 088 208 147 969 192 702 771 2 × 2 = 0 + 0.094 176 416 295 938 385 405 542 4;
  • 71) 0.094 176 416 295 938 385 405 542 4 × 2 = 0 + 0.188 352 832 591 876 770 811 084 8;
  • 72) 0.188 352 832 591 876 770 811 084 8 × 2 = 0 + 0.376 705 665 183 753 541 622 169 6;
  • 73) 0.376 705 665 183 753 541 622 169 6 × 2 = 0 + 0.753 411 330 367 507 083 244 339 2;
  • 74) 0.753 411 330 367 507 083 244 339 2 × 2 = 1 + 0.506 822 660 735 014 166 488 678 4;
  • 75) 0.506 822 660 735 014 166 488 678 4 × 2 = 1 + 0.013 645 321 470 028 332 977 356 8;
  • 76) 0.013 645 321 470 028 332 977 356 8 × 2 = 0 + 0.027 290 642 940 056 665 954 713 6;
  • 77) 0.027 290 642 940 056 665 954 713 6 × 2 = 0 + 0.054 581 285 880 113 331 909 427 2;
  • 78) 0.054 581 285 880 113 331 909 427 2 × 2 = 0 + 0.109 162 571 760 226 663 818 854 4;
  • 79) 0.109 162 571 760 226 663 818 854 4 × 2 = 0 + 0.218 325 143 520 453 327 637 708 8;
  • 80) 0.218 325 143 520 453 327 637 708 8 × 2 = 0 + 0.436 650 287 040 906 655 275 417 6;
  • 81) 0.436 650 287 040 906 655 275 417 6 × 2 = 0 + 0.873 300 574 081 813 310 550 835 2;
  • 82) 0.873 300 574 081 813 310 550 835 2 × 2 = 1 + 0.746 601 148 163 626 621 101 670 4;
  • 83) 0.746 601 148 163 626 621 101 670 4 × 2 = 1 + 0.493 202 296 327 253 242 203 340 8;
  • 84) 0.493 202 296 327 253 242 203 340 8 × 2 = 0 + 0.986 404 592 654 506 484 406 681 6;
  • 85) 0.986 404 592 654 506 484 406 681 6 × 2 = 1 + 0.972 809 185 309 012 968 813 363 2;
  • 86) 0.972 809 185 309 012 968 813 363 2 × 2 = 1 + 0.945 618 370 618 025 937 626 726 4;
  • 87) 0.945 618 370 618 025 937 626 726 4 × 2 = 1 + 0.891 236 741 236 051 875 253 452 8;
  • 88) 0.891 236 741 236 051 875 253 452 8 × 2 = 1 + 0.782 473 482 472 103 750 506 905 6;
  • 89) 0.782 473 482 472 103 750 506 905 6 × 2 = 1 + 0.564 946 964 944 207 501 013 811 2;
  • 90) 0.564 946 964 944 207 501 013 811 2 × 2 = 1 + 0.129 893 929 888 415 002 027 622 4;
  • 91) 0.129 893 929 888 415 002 027 622 4 × 2 = 0 + 0.259 787 859 776 830 004 055 244 8;
  • 92) 0.259 787 859 776 830 004 055 244 8 × 2 = 0 + 0.519 575 719 553 660 008 110 489 6;
  • 93) 0.519 575 719 553 660 008 110 489 6 × 2 = 1 + 0.039 151 439 107 320 016 220 979 2;
  • 94) 0.039 151 439 107 320 016 220 979 2 × 2 = 0 + 0.078 302 878 214 640 032 441 958 4;
  • 95) 0.078 302 878 214 640 032 441 958 4 × 2 = 0 + 0.156 605 756 429 280 064 883 916 8;
  • 96) 0.156 605 756 429 280 064 883 916 8 × 2 = 0 + 0.313 211 512 858 560 129 767 833 6;
  • 97) 0.313 211 512 858 560 129 767 833 6 × 2 = 0 + 0.626 423 025 717 120 259 535 667 2;
  • 98) 0.626 423 025 717 120 259 535 667 2 × 2 = 1 + 0.252 846 051 434 240 519 071 334 4;
  • 99) 0.252 846 051 434 240 519 071 334 4 × 2 = 0 + 0.505 692 102 868 481 038 142 668 8;
  • 100) 0.505 692 102 868 481 038 142 668 8 × 2 = 1 + 0.011 384 205 736 962 076 285 337 6;
  • 101) 0.011 384 205 736 962 076 285 337 6 × 2 = 0 + 0.022 768 411 473 924 152 570 675 2;
  • 102) 0.022 768 411 473 924 152 570 675 2 × 2 = 0 + 0.045 536 822 947 848 305 141 350 4;
  • 103) 0.045 536 822 947 848 305 141 350 4 × 2 = 0 + 0.091 073 645 895 696 610 282 700 8;
  • 104) 0.091 073 645 895 696 610 282 700 8 × 2 = 0 + 0.182 147 291 791 393 220 565 401 6;
  • 105) 0.182 147 291 791 393 220 565 401 6 × 2 = 0 + 0.364 294 583 582 786 441 130 803 2;
  • 106) 0.364 294 583 582 786 441 130 803 2 × 2 = 0 + 0.728 589 167 165 572 882 261 606 4;
  • 107) 0.728 589 167 165 572 882 261 606 4 × 2 = 1 + 0.457 178 334 331 145 764 523 212 8;
  • 108) 0.457 178 334 331 145 764 523 212 8 × 2 = 0 + 0.914 356 668 662 291 529 046 425 6;
  • 109) 0.914 356 668 662 291 529 046 425 6 × 2 = 1 + 0.828 713 337 324 583 058 092 851 2;
  • 110) 0.828 713 337 324 583 058 092 851 2 × 2 = 1 + 0.657 426 674 649 166 116 185 702 4;
  • 111) 0.657 426 674 649 166 116 185 702 4 × 2 = 1 + 0.314 853 349 298 332 232 371 404 8;
  • 112) 0.314 853 349 298 332 232 371 404 8 × 2 = 0 + 0.629 706 698 596 664 464 742 809 6;
  • 113) 0.629 706 698 596 664 464 742 809 6 × 2 = 1 + 0.259 413 397 193 328 929 485 619 2;
  • 114) 0.259 413 397 193 328 929 485 619 2 × 2 = 0 + 0.518 826 794 386 657 858 971 238 4;
  • 115) 0.518 826 794 386 657 858 971 238 4 × 2 = 1 + 0.037 653 588 773 315 717 942 476 8;
  • 116) 0.037 653 588 773 315 717 942 476 8 × 2 = 0 + 0.075 307 177 546 631 435 884 953 6;
  • 117) 0.075 307 177 546 631 435 884 953 6 × 2 = 0 + 0.150 614 355 093 262 871 769 907 2;
  • 118) 0.150 614 355 093 262 871 769 907 2 × 2 = 0 + 0.301 228 710 186 525 743 539 814 4;
  • 119) 0.301 228 710 186 525 743 539 814 4 × 2 = 0 + 0.602 457 420 373 051 487 079 628 8;

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).


4. 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:


0.000 000 000 000 000 000 008 550 1(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0110 0000 0110 1111 1100 1000 0101 0000 0010 1110 1010 000(2)

5. Positive number before normalization:

0.000 000 000 000 000 000 008 550 1(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0110 0000 0110 1111 1100 1000 0101 0000 0010 1110 1010 000(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 67 positions to the right, so that only one non zero digit remains to the left of it:


0.000 000 000 000 000 000 008 550 1(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0110 0000 0110 1111 1100 1000 0101 0000 0010 1110 1010 000(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0110 0000 0110 1111 1100 1000 0101 0000 0010 1110 1010 000(2) × 20 =

1.0100 0011 0000 0011 0111 1110 0100 0010 1000 0001 0111 0101 0000(2) × 2-67


7. 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 0 (a positive number)

Exponent (unadjusted): -67

Mantissa (not normalized):
1.0100 0011 0000 0011 0111 1110 0100 0010 1000 0001 0111 0101 0000


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

Exponent (unadjusted) + 2(11-1) - 1 =

-67 + 2(11-1) - 1 =

(-67 + 1 023)(10) =

956(10)


9. 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;
  • 956 ÷ 2 = 478 + 0;
  • 478 ÷ 2 = 239 + 0;
  • 239 ÷ 2 = 119 + 1;
  • 119 ÷ 2 = 59 + 1;
  • 59 ÷ 2 = 29 + 1;
  • 29 ÷ 2 = 14 + 1;
  • 14 ÷ 2 = 7 + 0;
  • 7 ÷ 2 = 3 + 1;
  • 3 ÷ 2 = 1 + 1;
  • 1 ÷ 2 = 0 + 1;

10. Construct the base 2 representation of the adjusted exponent.

Take all the remainders starting from the bottom of the list constructed above.


Exponent (adjusted) =

956(10) =

011 1011 1100(2)


11. 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. 0100 0011 0000 0011 0111 1110 0100 0010 1000 0001 0111 0101 0000 =

0100 0011 0000 0011 0111 1110 0100 0010 1000 0001 0111 0101 0000


12. The three elements that make up the number's 64 bit double precision IEEE 754 binary floating point representation:

Sign (1 bit) =
0 (a positive number)

Exponent (11 bits) =
011 1011 1100

Mantissa (52 bits) =
0100 0011 0000 0011 0111 1110 0100 0010 1000 0001 0111 0101 0000


Decimal number 0.000 000 000 000 000 000 008 550 1 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1011 1100 - 0100 0011 0000 0011 0111 1110 0100 0010 1000 0001 0111 0101 0000

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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