0.000 000 000 000 000 000 008 561 8 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 000 000 008 561 8(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
0.000 000 000 000 000 000 008 561 8(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: 0.000 000 000 000 000 000 008 561 8.

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) 0.000 000 000 000 000 000 008 561 8 × 2 = 0 + 0.000 000 000 000 000 000 017 123 6;
  • 2) 0.000 000 000 000 000 000 017 123 6 × 2 = 0 + 0.000 000 000 000 000 000 034 247 2;
  • 3) 0.000 000 000 000 000 000 034 247 2 × 2 = 0 + 0.000 000 000 000 000 000 068 494 4;
  • 4) 0.000 000 000 000 000 000 068 494 4 × 2 = 0 + 0.000 000 000 000 000 000 136 988 8;
  • 5) 0.000 000 000 000 000 000 136 988 8 × 2 = 0 + 0.000 000 000 000 000 000 273 977 6;
  • 6) 0.000 000 000 000 000 000 273 977 6 × 2 = 0 + 0.000 000 000 000 000 000 547 955 2;
  • 7) 0.000 000 000 000 000 000 547 955 2 × 2 = 0 + 0.000 000 000 000 000 001 095 910 4;
  • 8) 0.000 000 000 000 000 001 095 910 4 × 2 = 0 + 0.000 000 000 000 000 002 191 820 8;
  • 9) 0.000 000 000 000 000 002 191 820 8 × 2 = 0 + 0.000 000 000 000 000 004 383 641 6;
  • 10) 0.000 000 000 000 000 004 383 641 6 × 2 = 0 + 0.000 000 000 000 000 008 767 283 2;
  • 11) 0.000 000 000 000 000 008 767 283 2 × 2 = 0 + 0.000 000 000 000 000 017 534 566 4;
  • 12) 0.000 000 000 000 000 017 534 566 4 × 2 = 0 + 0.000 000 000 000 000 035 069 132 8;
  • 13) 0.000 000 000 000 000 035 069 132 8 × 2 = 0 + 0.000 000 000 000 000 070 138 265 6;
  • 14) 0.000 000 000 000 000 070 138 265 6 × 2 = 0 + 0.000 000 000 000 000 140 276 531 2;
  • 15) 0.000 000 000 000 000 140 276 531 2 × 2 = 0 + 0.000 000 000 000 000 280 553 062 4;
  • 16) 0.000 000 000 000 000 280 553 062 4 × 2 = 0 + 0.000 000 000 000 000 561 106 124 8;
  • 17) 0.000 000 000 000 000 561 106 124 8 × 2 = 0 + 0.000 000 000 000 001 122 212 249 6;
  • 18) 0.000 000 000 000 001 122 212 249 6 × 2 = 0 + 0.000 000 000 000 002 244 424 499 2;
  • 19) 0.000 000 000 000 002 244 424 499 2 × 2 = 0 + 0.000 000 000 000 004 488 848 998 4;
  • 20) 0.000 000 000 000 004 488 848 998 4 × 2 = 0 + 0.000 000 000 000 008 977 697 996 8;
  • 21) 0.000 000 000 000 008 977 697 996 8 × 2 = 0 + 0.000 000 000 000 017 955 395 993 6;
  • 22) 0.000 000 000 000 017 955 395 993 6 × 2 = 0 + 0.000 000 000 000 035 910 791 987 2;
  • 23) 0.000 000 000 000 035 910 791 987 2 × 2 = 0 + 0.000 000 000 000 071 821 583 974 4;
  • 24) 0.000 000 000 000 071 821 583 974 4 × 2 = 0 + 0.000 000 000 000 143 643 167 948 8;
  • 25) 0.000 000 000 000 143 643 167 948 8 × 2 = 0 + 0.000 000 000 000 287 286 335 897 6;
  • 26) 0.000 000 000 000 287 286 335 897 6 × 2 = 0 + 0.000 000 000 000 574 572 671 795 2;
  • 27) 0.000 000 000 000 574 572 671 795 2 × 2 = 0 + 0.000 000 000 001 149 145 343 590 4;
  • 28) 0.000 000 000 001 149 145 343 590 4 × 2 = 0 + 0.000 000 000 002 298 290 687 180 8;
  • 29) 0.000 000 000 002 298 290 687 180 8 × 2 = 0 + 0.000 000 000 004 596 581 374 361 6;
  • 30) 0.000 000 000 004 596 581 374 361 6 × 2 = 0 + 0.000 000 000 009 193 162 748 723 2;
  • 31) 0.000 000 000 009 193 162 748 723 2 × 2 = 0 + 0.000 000 000 018 386 325 497 446 4;
  • 32) 0.000 000 000 018 386 325 497 446 4 × 2 = 0 + 0.000 000 000 036 772 650 994 892 8;
  • 33) 0.000 000 000 036 772 650 994 892 8 × 2 = 0 + 0.000 000 000 073 545 301 989 785 6;
  • 34) 0.000 000 000 073 545 301 989 785 6 × 2 = 0 + 0.000 000 000 147 090 603 979 571 2;
  • 35) 0.000 000 000 147 090 603 979 571 2 × 2 = 0 + 0.000 000 000 294 181 207 959 142 4;
  • 36) 0.000 000 000 294 181 207 959 142 4 × 2 = 0 + 0.000 000 000 588 362 415 918 284 8;
  • 37) 0.000 000 000 588 362 415 918 284 8 × 2 = 0 + 0.000 000 001 176 724 831 836 569 6;
  • 38) 0.000 000 001 176 724 831 836 569 6 × 2 = 0 + 0.000 000 002 353 449 663 673 139 2;
  • 39) 0.000 000 002 353 449 663 673 139 2 × 2 = 0 + 0.000 000 004 706 899 327 346 278 4;
  • 40) 0.000 000 004 706 899 327 346 278 4 × 2 = 0 + 0.000 000 009 413 798 654 692 556 8;
  • 41) 0.000 000 009 413 798 654 692 556 8 × 2 = 0 + 0.000 000 018 827 597 309 385 113 6;
  • 42) 0.000 000 018 827 597 309 385 113 6 × 2 = 0 + 0.000 000 037 655 194 618 770 227 2;
  • 43) 0.000 000 037 655 194 618 770 227 2 × 2 = 0 + 0.000 000 075 310 389 237 540 454 4;
  • 44) 0.000 000 075 310 389 237 540 454 4 × 2 = 0 + 0.000 000 150 620 778 475 080 908 8;
  • 45) 0.000 000 150 620 778 475 080 908 8 × 2 = 0 + 0.000 000 301 241 556 950 161 817 6;
  • 46) 0.000 000 301 241 556 950 161 817 6 × 2 = 0 + 0.000 000 602 483 113 900 323 635 2;
  • 47) 0.000 000 602 483 113 900 323 635 2 × 2 = 0 + 0.000 001 204 966 227 800 647 270 4;
  • 48) 0.000 001 204 966 227 800 647 270 4 × 2 = 0 + 0.000 002 409 932 455 601 294 540 8;
  • 49) 0.000 002 409 932 455 601 294 540 8 × 2 = 0 + 0.000 004 819 864 911 202 589 081 6;
  • 50) 0.000 004 819 864 911 202 589 081 6 × 2 = 0 + 0.000 009 639 729 822 405 178 163 2;
  • 51) 0.000 009 639 729 822 405 178 163 2 × 2 = 0 + 0.000 019 279 459 644 810 356 326 4;
  • 52) 0.000 019 279 459 644 810 356 326 4 × 2 = 0 + 0.000 038 558 919 289 620 712 652 8;
  • 53) 0.000 038 558 919 289 620 712 652 8 × 2 = 0 + 0.000 077 117 838 579 241 425 305 6;
  • 54) 0.000 077 117 838 579 241 425 305 6 × 2 = 0 + 0.000 154 235 677 158 482 850 611 2;
  • 55) 0.000 154 235 677 158 482 850 611 2 × 2 = 0 + 0.000 308 471 354 316 965 701 222 4;
  • 56) 0.000 308 471 354 316 965 701 222 4 × 2 = 0 + 0.000 616 942 708 633 931 402 444 8;
  • 57) 0.000 616 942 708 633 931 402 444 8 × 2 = 0 + 0.001 233 885 417 267 862 804 889 6;
  • 58) 0.001 233 885 417 267 862 804 889 6 × 2 = 0 + 0.002 467 770 834 535 725 609 779 2;
  • 59) 0.002 467 770 834 535 725 609 779 2 × 2 = 0 + 0.004 935 541 669 071 451 219 558 4;
  • 60) 0.004 935 541 669 071 451 219 558 4 × 2 = 0 + 0.009 871 083 338 142 902 439 116 8;
  • 61) 0.009 871 083 338 142 902 439 116 8 × 2 = 0 + 0.019 742 166 676 285 804 878 233 6;
  • 62) 0.019 742 166 676 285 804 878 233 6 × 2 = 0 + 0.039 484 333 352 571 609 756 467 2;
  • 63) 0.039 484 333 352 571 609 756 467 2 × 2 = 0 + 0.078 968 666 705 143 219 512 934 4;
  • 64) 0.078 968 666 705 143 219 512 934 4 × 2 = 0 + 0.157 937 333 410 286 439 025 868 8;
  • 65) 0.157 937 333 410 286 439 025 868 8 × 2 = 0 + 0.315 874 666 820 572 878 051 737 6;
  • 66) 0.315 874 666 820 572 878 051 737 6 × 2 = 0 + 0.631 749 333 641 145 756 103 475 2;
  • 67) 0.631 749 333 641 145 756 103 475 2 × 2 = 1 + 0.263 498 667 282 291 512 206 950 4;
  • 68) 0.263 498 667 282 291 512 206 950 4 × 2 = 0 + 0.526 997 334 564 583 024 413 900 8;
  • 69) 0.526 997 334 564 583 024 413 900 8 × 2 = 1 + 0.053 994 669 129 166 048 827 801 6;
  • 70) 0.053 994 669 129 166 048 827 801 6 × 2 = 0 + 0.107 989 338 258 332 097 655 603 2;
  • 71) 0.107 989 338 258 332 097 655 603 2 × 2 = 0 + 0.215 978 676 516 664 195 311 206 4;
  • 72) 0.215 978 676 516 664 195 311 206 4 × 2 = 0 + 0.431 957 353 033 328 390 622 412 8;
  • 73) 0.431 957 353 033 328 390 622 412 8 × 2 = 0 + 0.863 914 706 066 656 781 244 825 6;
  • 74) 0.863 914 706 066 656 781 244 825 6 × 2 = 1 + 0.727 829 412 133 313 562 489 651 2;
  • 75) 0.727 829 412 133 313 562 489 651 2 × 2 = 1 + 0.455 658 824 266 627 124 979 302 4;
  • 76) 0.455 658 824 266 627 124 979 302 4 × 2 = 0 + 0.911 317 648 533 254 249 958 604 8;
  • 77) 0.911 317 648 533 254 249 958 604 8 × 2 = 1 + 0.822 635 297 066 508 499 917 209 6;
  • 78) 0.822 635 297 066 508 499 917 209 6 × 2 = 1 + 0.645 270 594 133 016 999 834 419 2;
  • 79) 0.645 270 594 133 016 999 834 419 2 × 2 = 1 + 0.290 541 188 266 033 999 668 838 4;
  • 80) 0.290 541 188 266 033 999 668 838 4 × 2 = 0 + 0.581 082 376 532 067 999 337 676 8;
  • 81) 0.581 082 376 532 067 999 337 676 8 × 2 = 1 + 0.162 164 753 064 135 998 675 353 6;
  • 82) 0.162 164 753 064 135 998 675 353 6 × 2 = 0 + 0.324 329 506 128 271 997 350 707 2;
  • 83) 0.324 329 506 128 271 997 350 707 2 × 2 = 0 + 0.648 659 012 256 543 994 701 414 4;
  • 84) 0.648 659 012 256 543 994 701 414 4 × 2 = 1 + 0.297 318 024 513 087 989 402 828 8;
  • 85) 0.297 318 024 513 087 989 402 828 8 × 2 = 0 + 0.594 636 049 026 175 978 805 657 6;
  • 86) 0.594 636 049 026 175 978 805 657 6 × 2 = 1 + 0.189 272 098 052 351 957 611 315 2;
  • 87) 0.189 272 098 052 351 957 611 315 2 × 2 = 0 + 0.378 544 196 104 703 915 222 630 4;
  • 88) 0.378 544 196 104 703 915 222 630 4 × 2 = 0 + 0.757 088 392 209 407 830 445 260 8;
  • 89) 0.757 088 392 209 407 830 445 260 8 × 2 = 1 + 0.514 176 784 418 815 660 890 521 6;
  • 90) 0.514 176 784 418 815 660 890 521 6 × 2 = 1 + 0.028 353 568 837 631 321 781 043 2;
  • 91) 0.028 353 568 837 631 321 781 043 2 × 2 = 0 + 0.056 707 137 675 262 643 562 086 4;
  • 92) 0.056 707 137 675 262 643 562 086 4 × 2 = 0 + 0.113 414 275 350 525 287 124 172 8;
  • 93) 0.113 414 275 350 525 287 124 172 8 × 2 = 0 + 0.226 828 550 701 050 574 248 345 6;
  • 94) 0.226 828 550 701 050 574 248 345 6 × 2 = 0 + 0.453 657 101 402 101 148 496 691 2;
  • 95) 0.453 657 101 402 101 148 496 691 2 × 2 = 0 + 0.907 314 202 804 202 296 993 382 4;
  • 96) 0.907 314 202 804 202 296 993 382 4 × 2 = 1 + 0.814 628 405 608 404 593 986 764 8;
  • 97) 0.814 628 405 608 404 593 986 764 8 × 2 = 1 + 0.629 256 811 216 809 187 973 529 6;
  • 98) 0.629 256 811 216 809 187 973 529 6 × 2 = 1 + 0.258 513 622 433 618 375 947 059 2;
  • 99) 0.258 513 622 433 618 375 947 059 2 × 2 = 0 + 0.517 027 244 867 236 751 894 118 4;
  • 100) 0.517 027 244 867 236 751 894 118 4 × 2 = 1 + 0.034 054 489 734 473 503 788 236 8;
  • 101) 0.034 054 489 734 473 503 788 236 8 × 2 = 0 + 0.068 108 979 468 947 007 576 473 6;
  • 102) 0.068 108 979 468 947 007 576 473 6 × 2 = 0 + 0.136 217 958 937 894 015 152 947 2;
  • 103) 0.136 217 958 937 894 015 152 947 2 × 2 = 0 + 0.272 435 917 875 788 030 305 894 4;
  • 104) 0.272 435 917 875 788 030 305 894 4 × 2 = 0 + 0.544 871 835 751 576 060 611 788 8;
  • 105) 0.544 871 835 751 576 060 611 788 8 × 2 = 1 + 0.089 743 671 503 152 121 223 577 6;
  • 106) 0.089 743 671 503 152 121 223 577 6 × 2 = 0 + 0.179 487 343 006 304 242 447 155 2;
  • 107) 0.179 487 343 006 304 242 447 155 2 × 2 = 0 + 0.358 974 686 012 608 484 894 310 4;
  • 108) 0.358 974 686 012 608 484 894 310 4 × 2 = 0 + 0.717 949 372 025 216 969 788 620 8;
  • 109) 0.717 949 372 025 216 969 788 620 8 × 2 = 1 + 0.435 898 744 050 433 939 577 241 6;
  • 110) 0.435 898 744 050 433 939 577 241 6 × 2 = 0 + 0.871 797 488 100 867 879 154 483 2;
  • 111) 0.871 797 488 100 867 879 154 483 2 × 2 = 1 + 0.743 594 976 201 735 758 308 966 4;
  • 112) 0.743 594 976 201 735 758 308 966 4 × 2 = 1 + 0.487 189 952 403 471 516 617 932 8;
  • 113) 0.487 189 952 403 471 516 617 932 8 × 2 = 0 + 0.974 379 904 806 943 033 235 865 6;
  • 114) 0.974 379 904 806 943 033 235 865 6 × 2 = 1 + 0.948 759 809 613 886 066 471 731 2;
  • 115) 0.948 759 809 613 886 066 471 731 2 × 2 = 1 + 0.897 519 619 227 772 132 943 462 4;
  • 116) 0.897 519 619 227 772 132 943 462 4 × 2 = 1 + 0.795 039 238 455 544 265 886 924 8;
  • 117) 0.795 039 238 455 544 265 886 924 8 × 2 = 1 + 0.590 078 476 911 088 531 773 849 6;
  • 118) 0.590 078 476 911 088 531 773 849 6 × 2 = 1 + 0.180 156 953 822 177 063 547 699 2;
  • 119) 0.180 156 953 822 177 063 547 699 2 × 2 = 0 + 0.360 313 907 644 354 127 095 398 4;

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:


0.000 000 000 000 000 000 008 561 8(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0110 1110 1001 0100 1100 0001 1101 0000 1000 1011 0111 110(2)

5. Positive number before normalization:

0.000 000 000 000 000 000 008 561 8(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0110 1110 1001 0100 1100 0001 1101 0000 1000 1011 0111 110(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:


0.000 000 000 000 000 000 008 561 8(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0110 1110 1001 0100 1100 0001 1101 0000 1000 1011 0111 110(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0110 1110 1001 0100 1100 0001 1101 0000 1000 1011 0111 110(2) × 20 =

1.0100 0011 0111 0100 1010 0110 0000 1110 1000 0100 0101 1011 1110(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 0111 0100 1010 0110 0000 1110 1000 0100 0101 1011 1110


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 0111 0100 1010 0110 0000 1110 1000 0100 0101 1011 1110 =

0100 0011 0111 0100 1010 0110 0000 1110 1000 0100 0101 1011 1110


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 0111 0100 1010 0110 0000 1110 1000 0100 0101 1011 1110


Decimal number 0.000 000 000 000 000 000 008 561 8 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1011 1100 - 0100 0011 0111 0100 1010 0110 0000 1110 1000 0100 0101 1011 1110

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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