0.000 000 000 000 000 000 008 559 9 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 000 000 008 559 9(10) to 64 bit double precision IEEE 754 binary floating point representation standard (1 bit for sign, 11 bits for exponent, 52 bits for mantissa)

What are the steps to convert decimal number
0.000 000 000 000 000 000 008 559 9(10) to 64 bit double precision IEEE 754 binary floating point representation (1 bit for sign, 11 bits for exponent, 52 bits for mantissa)

1. 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: 0.000 000 000 000 000 000 008 559 9.

Multiply it repeatedly by 2.

Keep track of each integer part of the results.

Stop when we get a fractional part that is equal to zero.


  • #) multiplying = integer + fractional part;
  • 1) 0.000 000 000 000 000 000 008 559 9 × 2 = 0 + 0.000 000 000 000 000 000 017 119 8;
  • 2) 0.000 000 000 000 000 000 017 119 8 × 2 = 0 + 0.000 000 000 000 000 000 034 239 6;
  • 3) 0.000 000 000 000 000 000 034 239 6 × 2 = 0 + 0.000 000 000 000 000 000 068 479 2;
  • 4) 0.000 000 000 000 000 000 068 479 2 × 2 = 0 + 0.000 000 000 000 000 000 136 958 4;
  • 5) 0.000 000 000 000 000 000 136 958 4 × 2 = 0 + 0.000 000 000 000 000 000 273 916 8;
  • 6) 0.000 000 000 000 000 000 273 916 8 × 2 = 0 + 0.000 000 000 000 000 000 547 833 6;
  • 7) 0.000 000 000 000 000 000 547 833 6 × 2 = 0 + 0.000 000 000 000 000 001 095 667 2;
  • 8) 0.000 000 000 000 000 001 095 667 2 × 2 = 0 + 0.000 000 000 000 000 002 191 334 4;
  • 9) 0.000 000 000 000 000 002 191 334 4 × 2 = 0 + 0.000 000 000 000 000 004 382 668 8;
  • 10) 0.000 000 000 000 000 004 382 668 8 × 2 = 0 + 0.000 000 000 000 000 008 765 337 6;
  • 11) 0.000 000 000 000 000 008 765 337 6 × 2 = 0 + 0.000 000 000 000 000 017 530 675 2;
  • 12) 0.000 000 000 000 000 017 530 675 2 × 2 = 0 + 0.000 000 000 000 000 035 061 350 4;
  • 13) 0.000 000 000 000 000 035 061 350 4 × 2 = 0 + 0.000 000 000 000 000 070 122 700 8;
  • 14) 0.000 000 000 000 000 070 122 700 8 × 2 = 0 + 0.000 000 000 000 000 140 245 401 6;
  • 15) 0.000 000 000 000 000 140 245 401 6 × 2 = 0 + 0.000 000 000 000 000 280 490 803 2;
  • 16) 0.000 000 000 000 000 280 490 803 2 × 2 = 0 + 0.000 000 000 000 000 560 981 606 4;
  • 17) 0.000 000 000 000 000 560 981 606 4 × 2 = 0 + 0.000 000 000 000 001 121 963 212 8;
  • 18) 0.000 000 000 000 001 121 963 212 8 × 2 = 0 + 0.000 000 000 000 002 243 926 425 6;
  • 19) 0.000 000 000 000 002 243 926 425 6 × 2 = 0 + 0.000 000 000 000 004 487 852 851 2;
  • 20) 0.000 000 000 000 004 487 852 851 2 × 2 = 0 + 0.000 000 000 000 008 975 705 702 4;
  • 21) 0.000 000 000 000 008 975 705 702 4 × 2 = 0 + 0.000 000 000 000 017 951 411 404 8;
  • 22) 0.000 000 000 000 017 951 411 404 8 × 2 = 0 + 0.000 000 000 000 035 902 822 809 6;
  • 23) 0.000 000 000 000 035 902 822 809 6 × 2 = 0 + 0.000 000 000 000 071 805 645 619 2;
  • 24) 0.000 000 000 000 071 805 645 619 2 × 2 = 0 + 0.000 000 000 000 143 611 291 238 4;
  • 25) 0.000 000 000 000 143 611 291 238 4 × 2 = 0 + 0.000 000 000 000 287 222 582 476 8;
  • 26) 0.000 000 000 000 287 222 582 476 8 × 2 = 0 + 0.000 000 000 000 574 445 164 953 6;
  • 27) 0.000 000 000 000 574 445 164 953 6 × 2 = 0 + 0.000 000 000 001 148 890 329 907 2;
  • 28) 0.000 000 000 001 148 890 329 907 2 × 2 = 0 + 0.000 000 000 002 297 780 659 814 4;
  • 29) 0.000 000 000 002 297 780 659 814 4 × 2 = 0 + 0.000 000 000 004 595 561 319 628 8;
  • 30) 0.000 000 000 004 595 561 319 628 8 × 2 = 0 + 0.000 000 000 009 191 122 639 257 6;
  • 31) 0.000 000 000 009 191 122 639 257 6 × 2 = 0 + 0.000 000 000 018 382 245 278 515 2;
  • 32) 0.000 000 000 018 382 245 278 515 2 × 2 = 0 + 0.000 000 000 036 764 490 557 030 4;
  • 33) 0.000 000 000 036 764 490 557 030 4 × 2 = 0 + 0.000 000 000 073 528 981 114 060 8;
  • 34) 0.000 000 000 073 528 981 114 060 8 × 2 = 0 + 0.000 000 000 147 057 962 228 121 6;
  • 35) 0.000 000 000 147 057 962 228 121 6 × 2 = 0 + 0.000 000 000 294 115 924 456 243 2;
  • 36) 0.000 000 000 294 115 924 456 243 2 × 2 = 0 + 0.000 000 000 588 231 848 912 486 4;
  • 37) 0.000 000 000 588 231 848 912 486 4 × 2 = 0 + 0.000 000 001 176 463 697 824 972 8;
  • 38) 0.000 000 001 176 463 697 824 972 8 × 2 = 0 + 0.000 000 002 352 927 395 649 945 6;
  • 39) 0.000 000 002 352 927 395 649 945 6 × 2 = 0 + 0.000 000 004 705 854 791 299 891 2;
  • 40) 0.000 000 004 705 854 791 299 891 2 × 2 = 0 + 0.000 000 009 411 709 582 599 782 4;
  • 41) 0.000 000 009 411 709 582 599 782 4 × 2 = 0 + 0.000 000 018 823 419 165 199 564 8;
  • 42) 0.000 000 018 823 419 165 199 564 8 × 2 = 0 + 0.000 000 037 646 838 330 399 129 6;
  • 43) 0.000 000 037 646 838 330 399 129 6 × 2 = 0 + 0.000 000 075 293 676 660 798 259 2;
  • 44) 0.000 000 075 293 676 660 798 259 2 × 2 = 0 + 0.000 000 150 587 353 321 596 518 4;
  • 45) 0.000 000 150 587 353 321 596 518 4 × 2 = 0 + 0.000 000 301 174 706 643 193 036 8;
  • 46) 0.000 000 301 174 706 643 193 036 8 × 2 = 0 + 0.000 000 602 349 413 286 386 073 6;
  • 47) 0.000 000 602 349 413 286 386 073 6 × 2 = 0 + 0.000 001 204 698 826 572 772 147 2;
  • 48) 0.000 001 204 698 826 572 772 147 2 × 2 = 0 + 0.000 002 409 397 653 145 544 294 4;
  • 49) 0.000 002 409 397 653 145 544 294 4 × 2 = 0 + 0.000 004 818 795 306 291 088 588 8;
  • 50) 0.000 004 818 795 306 291 088 588 8 × 2 = 0 + 0.000 009 637 590 612 582 177 177 6;
  • 51) 0.000 009 637 590 612 582 177 177 6 × 2 = 0 + 0.000 019 275 181 225 164 354 355 2;
  • 52) 0.000 019 275 181 225 164 354 355 2 × 2 = 0 + 0.000 038 550 362 450 328 708 710 4;
  • 53) 0.000 038 550 362 450 328 708 710 4 × 2 = 0 + 0.000 077 100 724 900 657 417 420 8;
  • 54) 0.000 077 100 724 900 657 417 420 8 × 2 = 0 + 0.000 154 201 449 801 314 834 841 6;
  • 55) 0.000 154 201 449 801 314 834 841 6 × 2 = 0 + 0.000 308 402 899 602 629 669 683 2;
  • 56) 0.000 308 402 899 602 629 669 683 2 × 2 = 0 + 0.000 616 805 799 205 259 339 366 4;
  • 57) 0.000 616 805 799 205 259 339 366 4 × 2 = 0 + 0.001 233 611 598 410 518 678 732 8;
  • 58) 0.001 233 611 598 410 518 678 732 8 × 2 = 0 + 0.002 467 223 196 821 037 357 465 6;
  • 59) 0.002 467 223 196 821 037 357 465 6 × 2 = 0 + 0.004 934 446 393 642 074 714 931 2;
  • 60) 0.004 934 446 393 642 074 714 931 2 × 2 = 0 + 0.009 868 892 787 284 149 429 862 4;
  • 61) 0.009 868 892 787 284 149 429 862 4 × 2 = 0 + 0.019 737 785 574 568 298 859 724 8;
  • 62) 0.019 737 785 574 568 298 859 724 8 × 2 = 0 + 0.039 475 571 149 136 597 719 449 6;
  • 63) 0.039 475 571 149 136 597 719 449 6 × 2 = 0 + 0.078 951 142 298 273 195 438 899 2;
  • 64) 0.078 951 142 298 273 195 438 899 2 × 2 = 0 + 0.157 902 284 596 546 390 877 798 4;
  • 65) 0.157 902 284 596 546 390 877 798 4 × 2 = 0 + 0.315 804 569 193 092 781 755 596 8;
  • 66) 0.315 804 569 193 092 781 755 596 8 × 2 = 0 + 0.631 609 138 386 185 563 511 193 6;
  • 67) 0.631 609 138 386 185 563 511 193 6 × 2 = 1 + 0.263 218 276 772 371 127 022 387 2;
  • 68) 0.263 218 276 772 371 127 022 387 2 × 2 = 0 + 0.526 436 553 544 742 254 044 774 4;
  • 69) 0.526 436 553 544 742 254 044 774 4 × 2 = 1 + 0.052 873 107 089 484 508 089 548 8;
  • 70) 0.052 873 107 089 484 508 089 548 8 × 2 = 0 + 0.105 746 214 178 969 016 179 097 6;
  • 71) 0.105 746 214 178 969 016 179 097 6 × 2 = 0 + 0.211 492 428 357 938 032 358 195 2;
  • 72) 0.211 492 428 357 938 032 358 195 2 × 2 = 0 + 0.422 984 856 715 876 064 716 390 4;
  • 73) 0.422 984 856 715 876 064 716 390 4 × 2 = 0 + 0.845 969 713 431 752 129 432 780 8;
  • 74) 0.845 969 713 431 752 129 432 780 8 × 2 = 1 + 0.691 939 426 863 504 258 865 561 6;
  • 75) 0.691 939 426 863 504 258 865 561 6 × 2 = 1 + 0.383 878 853 727 008 517 731 123 2;
  • 76) 0.383 878 853 727 008 517 731 123 2 × 2 = 0 + 0.767 757 707 454 017 035 462 246 4;
  • 77) 0.767 757 707 454 017 035 462 246 4 × 2 = 1 + 0.535 515 414 908 034 070 924 492 8;
  • 78) 0.535 515 414 908 034 070 924 492 8 × 2 = 1 + 0.071 030 829 816 068 141 848 985 6;
  • 79) 0.071 030 829 816 068 141 848 985 6 × 2 = 0 + 0.142 061 659 632 136 283 697 971 2;
  • 80) 0.142 061 659 632 136 283 697 971 2 × 2 = 0 + 0.284 123 319 264 272 567 395 942 4;
  • 81) 0.284 123 319 264 272 567 395 942 4 × 2 = 0 + 0.568 246 638 528 545 134 791 884 8;
  • 82) 0.568 246 638 528 545 134 791 884 8 × 2 = 1 + 0.136 493 277 057 090 269 583 769 6;
  • 83) 0.136 493 277 057 090 269 583 769 6 × 2 = 0 + 0.272 986 554 114 180 539 167 539 2;
  • 84) 0.272 986 554 114 180 539 167 539 2 × 2 = 0 + 0.545 973 108 228 361 078 335 078 4;
  • 85) 0.545 973 108 228 361 078 335 078 4 × 2 = 1 + 0.091 946 216 456 722 156 670 156 8;
  • 86) 0.091 946 216 456 722 156 670 156 8 × 2 = 0 + 0.183 892 432 913 444 313 340 313 6;
  • 87) 0.183 892 432 913 444 313 340 313 6 × 2 = 0 + 0.367 784 865 826 888 626 680 627 2;
  • 88) 0.367 784 865 826 888 626 680 627 2 × 2 = 0 + 0.735 569 731 653 777 253 361 254 4;
  • 89) 0.735 569 731 653 777 253 361 254 4 × 2 = 1 + 0.471 139 463 307 554 506 722 508 8;
  • 90) 0.471 139 463 307 554 506 722 508 8 × 2 = 0 + 0.942 278 926 615 109 013 445 017 6;
  • 91) 0.942 278 926 615 109 013 445 017 6 × 2 = 1 + 0.884 557 853 230 218 026 890 035 2;
  • 92) 0.884 557 853 230 218 026 890 035 2 × 2 = 1 + 0.769 115 706 460 436 053 780 070 4;
  • 93) 0.769 115 706 460 436 053 780 070 4 × 2 = 1 + 0.538 231 412 920 872 107 560 140 8;
  • 94) 0.538 231 412 920 872 107 560 140 8 × 2 = 1 + 0.076 462 825 841 744 215 120 281 6;
  • 95) 0.076 462 825 841 744 215 120 281 6 × 2 = 0 + 0.152 925 651 683 488 430 240 563 2;
  • 96) 0.152 925 651 683 488 430 240 563 2 × 2 = 0 + 0.305 851 303 366 976 860 481 126 4;
  • 97) 0.305 851 303 366 976 860 481 126 4 × 2 = 0 + 0.611 702 606 733 953 720 962 252 8;
  • 98) 0.611 702 606 733 953 720 962 252 8 × 2 = 1 + 0.223 405 213 467 907 441 924 505 6;
  • 99) 0.223 405 213 467 907 441 924 505 6 × 2 = 0 + 0.446 810 426 935 814 883 849 011 2;
  • 100) 0.446 810 426 935 814 883 849 011 2 × 2 = 0 + 0.893 620 853 871 629 767 698 022 4;
  • 101) 0.893 620 853 871 629 767 698 022 4 × 2 = 1 + 0.787 241 707 743 259 535 396 044 8;
  • 102) 0.787 241 707 743 259 535 396 044 8 × 2 = 1 + 0.574 483 415 486 519 070 792 089 6;
  • 103) 0.574 483 415 486 519 070 792 089 6 × 2 = 1 + 0.148 966 830 973 038 141 584 179 2;
  • 104) 0.148 966 830 973 038 141 584 179 2 × 2 = 0 + 0.297 933 661 946 076 283 168 358 4;
  • 105) 0.297 933 661 946 076 283 168 358 4 × 2 = 0 + 0.595 867 323 892 152 566 336 716 8;
  • 106) 0.595 867 323 892 152 566 336 716 8 × 2 = 1 + 0.191 734 647 784 305 132 673 433 6;
  • 107) 0.191 734 647 784 305 132 673 433 6 × 2 = 0 + 0.383 469 295 568 610 265 346 867 2;
  • 108) 0.383 469 295 568 610 265 346 867 2 × 2 = 0 + 0.766 938 591 137 220 530 693 734 4;
  • 109) 0.766 938 591 137 220 530 693 734 4 × 2 = 1 + 0.533 877 182 274 441 061 387 468 8;
  • 110) 0.533 877 182 274 441 061 387 468 8 × 2 = 1 + 0.067 754 364 548 882 122 774 937 6;
  • 111) 0.067 754 364 548 882 122 774 937 6 × 2 = 0 + 0.135 508 729 097 764 245 549 875 2;
  • 112) 0.135 508 729 097 764 245 549 875 2 × 2 = 0 + 0.271 017 458 195 528 491 099 750 4;
  • 113) 0.271 017 458 195 528 491 099 750 4 × 2 = 0 + 0.542 034 916 391 056 982 199 500 8;
  • 114) 0.542 034 916 391 056 982 199 500 8 × 2 = 1 + 0.084 069 832 782 113 964 399 001 6;
  • 115) 0.084 069 832 782 113 964 399 001 6 × 2 = 0 + 0.168 139 665 564 227 928 798 003 2;
  • 116) 0.168 139 665 564 227 928 798 003 2 × 2 = 0 + 0.336 279 331 128 455 857 596 006 4;
  • 117) 0.336 279 331 128 455 857 596 006 4 × 2 = 0 + 0.672 558 662 256 911 715 192 012 8;
  • 118) 0.672 558 662 256 911 715 192 012 8 × 2 = 1 + 0.345 117 324 513 823 430 384 025 6;
  • 119) 0.345 117 324 513 823 430 384 025 6 × 2 = 0 + 0.690 234 649 027 646 860 768 051 2;

We didn't get any fractional part that was equal to zero. But we had enough iterations (over Mantissa limit) and at least one integer that was different from zero => FULL STOP (Losing precision - the converted number we get in the end will be just a very good approximation of the initial one).


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:


0.000 000 000 000 000 000 008 559 9(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0110 1100 0100 1000 1011 1100 0100 1110 0100 1100 0100 010(2)

5. Positive number before normalization:

0.000 000 000 000 000 000 008 559 9(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0110 1100 0100 1000 1011 1100 0100 1110 0100 1100 0100 010(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:


0.000 000 000 000 000 000 008 559 9(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0110 1100 0100 1000 1011 1100 0100 1110 0100 1100 0100 010(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0110 1100 0100 1000 1011 1100 0100 1110 0100 1100 0100 010(2) × 20 =

1.0100 0011 0110 0010 0100 0101 1110 0010 0111 0010 0110 0010 0010(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 0110 0010 0100 0101 1110 0010 0111 0010 0110 0010 0010


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 0110 0010 0100 0101 1110 0010 0111 0010 0110 0010 0010 =

0100 0011 0110 0010 0100 0101 1110 0010 0111 0010 0110 0010 0010


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 0110 0010 0100 0101 1110 0010 0111 0010 0110 0010 0010


Decimal number 0.000 000 000 000 000 000 008 559 9 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1011 1100 - 0100 0011 0110 0010 0100 0101 1110 0010 0111 0010 0110 0010 0010

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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