0.000 000 000 000 000 000 008 544 1 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 000 000 008 544 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
0.000 000 000 000 000 000 008 544 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: 0.000 000 000 000 000 000 008 544 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) 0.000 000 000 000 000 000 008 544 1 × 2 = 0 + 0.000 000 000 000 000 000 017 088 2;
  • 2) 0.000 000 000 000 000 000 017 088 2 × 2 = 0 + 0.000 000 000 000 000 000 034 176 4;
  • 3) 0.000 000 000 000 000 000 034 176 4 × 2 = 0 + 0.000 000 000 000 000 000 068 352 8;
  • 4) 0.000 000 000 000 000 000 068 352 8 × 2 = 0 + 0.000 000 000 000 000 000 136 705 6;
  • 5) 0.000 000 000 000 000 000 136 705 6 × 2 = 0 + 0.000 000 000 000 000 000 273 411 2;
  • 6) 0.000 000 000 000 000 000 273 411 2 × 2 = 0 + 0.000 000 000 000 000 000 546 822 4;
  • 7) 0.000 000 000 000 000 000 546 822 4 × 2 = 0 + 0.000 000 000 000 000 001 093 644 8;
  • 8) 0.000 000 000 000 000 001 093 644 8 × 2 = 0 + 0.000 000 000 000 000 002 187 289 6;
  • 9) 0.000 000 000 000 000 002 187 289 6 × 2 = 0 + 0.000 000 000 000 000 004 374 579 2;
  • 10) 0.000 000 000 000 000 004 374 579 2 × 2 = 0 + 0.000 000 000 000 000 008 749 158 4;
  • 11) 0.000 000 000 000 000 008 749 158 4 × 2 = 0 + 0.000 000 000 000 000 017 498 316 8;
  • 12) 0.000 000 000 000 000 017 498 316 8 × 2 = 0 + 0.000 000 000 000 000 034 996 633 6;
  • 13) 0.000 000 000 000 000 034 996 633 6 × 2 = 0 + 0.000 000 000 000 000 069 993 267 2;
  • 14) 0.000 000 000 000 000 069 993 267 2 × 2 = 0 + 0.000 000 000 000 000 139 986 534 4;
  • 15) 0.000 000 000 000 000 139 986 534 4 × 2 = 0 + 0.000 000 000 000 000 279 973 068 8;
  • 16) 0.000 000 000 000 000 279 973 068 8 × 2 = 0 + 0.000 000 000 000 000 559 946 137 6;
  • 17) 0.000 000 000 000 000 559 946 137 6 × 2 = 0 + 0.000 000 000 000 001 119 892 275 2;
  • 18) 0.000 000 000 000 001 119 892 275 2 × 2 = 0 + 0.000 000 000 000 002 239 784 550 4;
  • 19) 0.000 000 000 000 002 239 784 550 4 × 2 = 0 + 0.000 000 000 000 004 479 569 100 8;
  • 20) 0.000 000 000 000 004 479 569 100 8 × 2 = 0 + 0.000 000 000 000 008 959 138 201 6;
  • 21) 0.000 000 000 000 008 959 138 201 6 × 2 = 0 + 0.000 000 000 000 017 918 276 403 2;
  • 22) 0.000 000 000 000 017 918 276 403 2 × 2 = 0 + 0.000 000 000 000 035 836 552 806 4;
  • 23) 0.000 000 000 000 035 836 552 806 4 × 2 = 0 + 0.000 000 000 000 071 673 105 612 8;
  • 24) 0.000 000 000 000 071 673 105 612 8 × 2 = 0 + 0.000 000 000 000 143 346 211 225 6;
  • 25) 0.000 000 000 000 143 346 211 225 6 × 2 = 0 + 0.000 000 000 000 286 692 422 451 2;
  • 26) 0.000 000 000 000 286 692 422 451 2 × 2 = 0 + 0.000 000 000 000 573 384 844 902 4;
  • 27) 0.000 000 000 000 573 384 844 902 4 × 2 = 0 + 0.000 000 000 001 146 769 689 804 8;
  • 28) 0.000 000 000 001 146 769 689 804 8 × 2 = 0 + 0.000 000 000 002 293 539 379 609 6;
  • 29) 0.000 000 000 002 293 539 379 609 6 × 2 = 0 + 0.000 000 000 004 587 078 759 219 2;
  • 30) 0.000 000 000 004 587 078 759 219 2 × 2 = 0 + 0.000 000 000 009 174 157 518 438 4;
  • 31) 0.000 000 000 009 174 157 518 438 4 × 2 = 0 + 0.000 000 000 018 348 315 036 876 8;
  • 32) 0.000 000 000 018 348 315 036 876 8 × 2 = 0 + 0.000 000 000 036 696 630 073 753 6;
  • 33) 0.000 000 000 036 696 630 073 753 6 × 2 = 0 + 0.000 000 000 073 393 260 147 507 2;
  • 34) 0.000 000 000 073 393 260 147 507 2 × 2 = 0 + 0.000 000 000 146 786 520 295 014 4;
  • 35) 0.000 000 000 146 786 520 295 014 4 × 2 = 0 + 0.000 000 000 293 573 040 590 028 8;
  • 36) 0.000 000 000 293 573 040 590 028 8 × 2 = 0 + 0.000 000 000 587 146 081 180 057 6;
  • 37) 0.000 000 000 587 146 081 180 057 6 × 2 = 0 + 0.000 000 001 174 292 162 360 115 2;
  • 38) 0.000 000 001 174 292 162 360 115 2 × 2 = 0 + 0.000 000 002 348 584 324 720 230 4;
  • 39) 0.000 000 002 348 584 324 720 230 4 × 2 = 0 + 0.000 000 004 697 168 649 440 460 8;
  • 40) 0.000 000 004 697 168 649 440 460 8 × 2 = 0 + 0.000 000 009 394 337 298 880 921 6;
  • 41) 0.000 000 009 394 337 298 880 921 6 × 2 = 0 + 0.000 000 018 788 674 597 761 843 2;
  • 42) 0.000 000 018 788 674 597 761 843 2 × 2 = 0 + 0.000 000 037 577 349 195 523 686 4;
  • 43) 0.000 000 037 577 349 195 523 686 4 × 2 = 0 + 0.000 000 075 154 698 391 047 372 8;
  • 44) 0.000 000 075 154 698 391 047 372 8 × 2 = 0 + 0.000 000 150 309 396 782 094 745 6;
  • 45) 0.000 000 150 309 396 782 094 745 6 × 2 = 0 + 0.000 000 300 618 793 564 189 491 2;
  • 46) 0.000 000 300 618 793 564 189 491 2 × 2 = 0 + 0.000 000 601 237 587 128 378 982 4;
  • 47) 0.000 000 601 237 587 128 378 982 4 × 2 = 0 + 0.000 001 202 475 174 256 757 964 8;
  • 48) 0.000 001 202 475 174 256 757 964 8 × 2 = 0 + 0.000 002 404 950 348 513 515 929 6;
  • 49) 0.000 002 404 950 348 513 515 929 6 × 2 = 0 + 0.000 004 809 900 697 027 031 859 2;
  • 50) 0.000 004 809 900 697 027 031 859 2 × 2 = 0 + 0.000 009 619 801 394 054 063 718 4;
  • 51) 0.000 009 619 801 394 054 063 718 4 × 2 = 0 + 0.000 019 239 602 788 108 127 436 8;
  • 52) 0.000 019 239 602 788 108 127 436 8 × 2 = 0 + 0.000 038 479 205 576 216 254 873 6;
  • 53) 0.000 038 479 205 576 216 254 873 6 × 2 = 0 + 0.000 076 958 411 152 432 509 747 2;
  • 54) 0.000 076 958 411 152 432 509 747 2 × 2 = 0 + 0.000 153 916 822 304 865 019 494 4;
  • 55) 0.000 153 916 822 304 865 019 494 4 × 2 = 0 + 0.000 307 833 644 609 730 038 988 8;
  • 56) 0.000 307 833 644 609 730 038 988 8 × 2 = 0 + 0.000 615 667 289 219 460 077 977 6;
  • 57) 0.000 615 667 289 219 460 077 977 6 × 2 = 0 + 0.001 231 334 578 438 920 155 955 2;
  • 58) 0.001 231 334 578 438 920 155 955 2 × 2 = 0 + 0.002 462 669 156 877 840 311 910 4;
  • 59) 0.002 462 669 156 877 840 311 910 4 × 2 = 0 + 0.004 925 338 313 755 680 623 820 8;
  • 60) 0.004 925 338 313 755 680 623 820 8 × 2 = 0 + 0.009 850 676 627 511 361 247 641 6;
  • 61) 0.009 850 676 627 511 361 247 641 6 × 2 = 0 + 0.019 701 353 255 022 722 495 283 2;
  • 62) 0.019 701 353 255 022 722 495 283 2 × 2 = 0 + 0.039 402 706 510 045 444 990 566 4;
  • 63) 0.039 402 706 510 045 444 990 566 4 × 2 = 0 + 0.078 805 413 020 090 889 981 132 8;
  • 64) 0.078 805 413 020 090 889 981 132 8 × 2 = 0 + 0.157 610 826 040 181 779 962 265 6;
  • 65) 0.157 610 826 040 181 779 962 265 6 × 2 = 0 + 0.315 221 652 080 363 559 924 531 2;
  • 66) 0.315 221 652 080 363 559 924 531 2 × 2 = 0 + 0.630 443 304 160 727 119 849 062 4;
  • 67) 0.630 443 304 160 727 119 849 062 4 × 2 = 1 + 0.260 886 608 321 454 239 698 124 8;
  • 68) 0.260 886 608 321 454 239 698 124 8 × 2 = 0 + 0.521 773 216 642 908 479 396 249 6;
  • 69) 0.521 773 216 642 908 479 396 249 6 × 2 = 1 + 0.043 546 433 285 816 958 792 499 2;
  • 70) 0.043 546 433 285 816 958 792 499 2 × 2 = 0 + 0.087 092 866 571 633 917 584 998 4;
  • 71) 0.087 092 866 571 633 917 584 998 4 × 2 = 0 + 0.174 185 733 143 267 835 169 996 8;
  • 72) 0.174 185 733 143 267 835 169 996 8 × 2 = 0 + 0.348 371 466 286 535 670 339 993 6;
  • 73) 0.348 371 466 286 535 670 339 993 6 × 2 = 0 + 0.696 742 932 573 071 340 679 987 2;
  • 74) 0.696 742 932 573 071 340 679 987 2 × 2 = 1 + 0.393 485 865 146 142 681 359 974 4;
  • 75) 0.393 485 865 146 142 681 359 974 4 × 2 = 0 + 0.786 971 730 292 285 362 719 948 8;
  • 76) 0.786 971 730 292 285 362 719 948 8 × 2 = 1 + 0.573 943 460 584 570 725 439 897 6;
  • 77) 0.573 943 460 584 570 725 439 897 6 × 2 = 1 + 0.147 886 921 169 141 450 879 795 2;
  • 78) 0.147 886 921 169 141 450 879 795 2 × 2 = 0 + 0.295 773 842 338 282 901 759 590 4;
  • 79) 0.295 773 842 338 282 901 759 590 4 × 2 = 0 + 0.591 547 684 676 565 803 519 180 8;
  • 80) 0.591 547 684 676 565 803 519 180 8 × 2 = 1 + 0.183 095 369 353 131 607 038 361 6;
  • 81) 0.183 095 369 353 131 607 038 361 6 × 2 = 0 + 0.366 190 738 706 263 214 076 723 2;
  • 82) 0.366 190 738 706 263 214 076 723 2 × 2 = 0 + 0.732 381 477 412 526 428 153 446 4;
  • 83) 0.732 381 477 412 526 428 153 446 4 × 2 = 1 + 0.464 762 954 825 052 856 306 892 8;
  • 84) 0.464 762 954 825 052 856 306 892 8 × 2 = 0 + 0.929 525 909 650 105 712 613 785 6;
  • 85) 0.929 525 909 650 105 712 613 785 6 × 2 = 1 + 0.859 051 819 300 211 425 227 571 2;
  • 86) 0.859 051 819 300 211 425 227 571 2 × 2 = 1 + 0.718 103 638 600 422 850 455 142 4;
  • 87) 0.718 103 638 600 422 850 455 142 4 × 2 = 1 + 0.436 207 277 200 845 700 910 284 8;
  • 88) 0.436 207 277 200 845 700 910 284 8 × 2 = 0 + 0.872 414 554 401 691 401 820 569 6;
  • 89) 0.872 414 554 401 691 401 820 569 6 × 2 = 1 + 0.744 829 108 803 382 803 641 139 2;
  • 90) 0.744 829 108 803 382 803 641 139 2 × 2 = 1 + 0.489 658 217 606 765 607 282 278 4;
  • 91) 0.489 658 217 606 765 607 282 278 4 × 2 = 0 + 0.979 316 435 213 531 214 564 556 8;
  • 92) 0.979 316 435 213 531 214 564 556 8 × 2 = 1 + 0.958 632 870 427 062 429 129 113 6;
  • 93) 0.958 632 870 427 062 429 129 113 6 × 2 = 1 + 0.917 265 740 854 124 858 258 227 2;
  • 94) 0.917 265 740 854 124 858 258 227 2 × 2 = 1 + 0.834 531 481 708 249 716 516 454 4;
  • 95) 0.834 531 481 708 249 716 516 454 4 × 2 = 1 + 0.669 062 963 416 499 433 032 908 8;
  • 96) 0.669 062 963 416 499 433 032 908 8 × 2 = 1 + 0.338 125 926 832 998 866 065 817 6;
  • 97) 0.338 125 926 832 998 866 065 817 6 × 2 = 0 + 0.676 251 853 665 997 732 131 635 2;
  • 98) 0.676 251 853 665 997 732 131 635 2 × 2 = 1 + 0.352 503 707 331 995 464 263 270 4;
  • 99) 0.352 503 707 331 995 464 263 270 4 × 2 = 0 + 0.705 007 414 663 990 928 526 540 8;
  • 100) 0.705 007 414 663 990 928 526 540 8 × 2 = 1 + 0.410 014 829 327 981 857 053 081 6;
  • 101) 0.410 014 829 327 981 857 053 081 6 × 2 = 0 + 0.820 029 658 655 963 714 106 163 2;
  • 102) 0.820 029 658 655 963 714 106 163 2 × 2 = 1 + 0.640 059 317 311 927 428 212 326 4;
  • 103) 0.640 059 317 311 927 428 212 326 4 × 2 = 1 + 0.280 118 634 623 854 856 424 652 8;
  • 104) 0.280 118 634 623 854 856 424 652 8 × 2 = 0 + 0.560 237 269 247 709 712 849 305 6;
  • 105) 0.560 237 269 247 709 712 849 305 6 × 2 = 1 + 0.120 474 538 495 419 425 698 611 2;
  • 106) 0.120 474 538 495 419 425 698 611 2 × 2 = 0 + 0.240 949 076 990 838 851 397 222 4;
  • 107) 0.240 949 076 990 838 851 397 222 4 × 2 = 0 + 0.481 898 153 981 677 702 794 444 8;
  • 108) 0.481 898 153 981 677 702 794 444 8 × 2 = 0 + 0.963 796 307 963 355 405 588 889 6;
  • 109) 0.963 796 307 963 355 405 588 889 6 × 2 = 1 + 0.927 592 615 926 710 811 177 779 2;
  • 110) 0.927 592 615 926 710 811 177 779 2 × 2 = 1 + 0.855 185 231 853 421 622 355 558 4;
  • 111) 0.855 185 231 853 421 622 355 558 4 × 2 = 1 + 0.710 370 463 706 843 244 711 116 8;
  • 112) 0.710 370 463 706 843 244 711 116 8 × 2 = 1 + 0.420 740 927 413 686 489 422 233 6;
  • 113) 0.420 740 927 413 686 489 422 233 6 × 2 = 0 + 0.841 481 854 827 372 978 844 467 2;
  • 114) 0.841 481 854 827 372 978 844 467 2 × 2 = 1 + 0.682 963 709 654 745 957 688 934 4;
  • 115) 0.682 963 709 654 745 957 688 934 4 × 2 = 1 + 0.365 927 419 309 491 915 377 868 8;
  • 116) 0.365 927 419 309 491 915 377 868 8 × 2 = 0 + 0.731 854 838 618 983 830 755 737 6;
  • 117) 0.731 854 838 618 983 830 755 737 6 × 2 = 1 + 0.463 709 677 237 967 661 511 475 2;
  • 118) 0.463 709 677 237 967 661 511 475 2 × 2 = 0 + 0.927 419 354 475 935 323 022 950 4;
  • 119) 0.927 419 354 475 935 323 022 950 4 × 2 = 1 + 0.854 838 708 951 870 646 045 900 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:


0.000 000 000 000 000 000 008 544 1(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0101 1001 0010 1110 1101 1111 0101 0110 1000 1111 0110 101(2)

5. Positive number before normalization:

0.000 000 000 000 000 000 008 544 1(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0101 1001 0010 1110 1101 1111 0101 0110 1000 1111 0110 101(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:


0.000 000 000 000 000 000 008 544 1(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0101 1001 0010 1110 1101 1111 0101 0110 1000 1111 0110 101(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0101 1001 0010 1110 1101 1111 0101 0110 1000 1111 0110 101(2) × 20 =

1.0100 0010 1100 1001 0111 0110 1111 1010 1011 0100 0111 1011 0101(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 0010 1100 1001 0111 0110 1111 1010 1011 0100 0111 1011 0101


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 0010 1100 1001 0111 0110 1111 1010 1011 0100 0111 1011 0101 =

0100 0010 1100 1001 0111 0110 1111 1010 1011 0100 0111 1011 0101


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 0010 1100 1001 0111 0110 1111 1010 1011 0100 0111 1011 0101


Decimal number 0.000 000 000 000 000 000 008 544 1 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1011 1100 - 0100 0010 1100 1001 0111 0110 1111 1010 1011 0100 0111 1011 0101

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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