0.000 000 000 000 000 000 008 535 3 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 000 000 008 535 3(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
0.000 000 000 000 000 000 008 535 3(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: 0.000 000 000 000 000 000 008 535 3.

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) 0.000 000 000 000 000 000 008 535 3 × 2 = 0 + 0.000 000 000 000 000 000 017 070 6;
  • 2) 0.000 000 000 000 000 000 017 070 6 × 2 = 0 + 0.000 000 000 000 000 000 034 141 2;
  • 3) 0.000 000 000 000 000 000 034 141 2 × 2 = 0 + 0.000 000 000 000 000 000 068 282 4;
  • 4) 0.000 000 000 000 000 000 068 282 4 × 2 = 0 + 0.000 000 000 000 000 000 136 564 8;
  • 5) 0.000 000 000 000 000 000 136 564 8 × 2 = 0 + 0.000 000 000 000 000 000 273 129 6;
  • 6) 0.000 000 000 000 000 000 273 129 6 × 2 = 0 + 0.000 000 000 000 000 000 546 259 2;
  • 7) 0.000 000 000 000 000 000 546 259 2 × 2 = 0 + 0.000 000 000 000 000 001 092 518 4;
  • 8) 0.000 000 000 000 000 001 092 518 4 × 2 = 0 + 0.000 000 000 000 000 002 185 036 8;
  • 9) 0.000 000 000 000 000 002 185 036 8 × 2 = 0 + 0.000 000 000 000 000 004 370 073 6;
  • 10) 0.000 000 000 000 000 004 370 073 6 × 2 = 0 + 0.000 000 000 000 000 008 740 147 2;
  • 11) 0.000 000 000 000 000 008 740 147 2 × 2 = 0 + 0.000 000 000 000 000 017 480 294 4;
  • 12) 0.000 000 000 000 000 017 480 294 4 × 2 = 0 + 0.000 000 000 000 000 034 960 588 8;
  • 13) 0.000 000 000 000 000 034 960 588 8 × 2 = 0 + 0.000 000 000 000 000 069 921 177 6;
  • 14) 0.000 000 000 000 000 069 921 177 6 × 2 = 0 + 0.000 000 000 000 000 139 842 355 2;
  • 15) 0.000 000 000 000 000 139 842 355 2 × 2 = 0 + 0.000 000 000 000 000 279 684 710 4;
  • 16) 0.000 000 000 000 000 279 684 710 4 × 2 = 0 + 0.000 000 000 000 000 559 369 420 8;
  • 17) 0.000 000 000 000 000 559 369 420 8 × 2 = 0 + 0.000 000 000 000 001 118 738 841 6;
  • 18) 0.000 000 000 000 001 118 738 841 6 × 2 = 0 + 0.000 000 000 000 002 237 477 683 2;
  • 19) 0.000 000 000 000 002 237 477 683 2 × 2 = 0 + 0.000 000 000 000 004 474 955 366 4;
  • 20) 0.000 000 000 000 004 474 955 366 4 × 2 = 0 + 0.000 000 000 000 008 949 910 732 8;
  • 21) 0.000 000 000 000 008 949 910 732 8 × 2 = 0 + 0.000 000 000 000 017 899 821 465 6;
  • 22) 0.000 000 000 000 017 899 821 465 6 × 2 = 0 + 0.000 000 000 000 035 799 642 931 2;
  • 23) 0.000 000 000 000 035 799 642 931 2 × 2 = 0 + 0.000 000 000 000 071 599 285 862 4;
  • 24) 0.000 000 000 000 071 599 285 862 4 × 2 = 0 + 0.000 000 000 000 143 198 571 724 8;
  • 25) 0.000 000 000 000 143 198 571 724 8 × 2 = 0 + 0.000 000 000 000 286 397 143 449 6;
  • 26) 0.000 000 000 000 286 397 143 449 6 × 2 = 0 + 0.000 000 000 000 572 794 286 899 2;
  • 27) 0.000 000 000 000 572 794 286 899 2 × 2 = 0 + 0.000 000 000 001 145 588 573 798 4;
  • 28) 0.000 000 000 001 145 588 573 798 4 × 2 = 0 + 0.000 000 000 002 291 177 147 596 8;
  • 29) 0.000 000 000 002 291 177 147 596 8 × 2 = 0 + 0.000 000 000 004 582 354 295 193 6;
  • 30) 0.000 000 000 004 582 354 295 193 6 × 2 = 0 + 0.000 000 000 009 164 708 590 387 2;
  • 31) 0.000 000 000 009 164 708 590 387 2 × 2 = 0 + 0.000 000 000 018 329 417 180 774 4;
  • 32) 0.000 000 000 018 329 417 180 774 4 × 2 = 0 + 0.000 000 000 036 658 834 361 548 8;
  • 33) 0.000 000 000 036 658 834 361 548 8 × 2 = 0 + 0.000 000 000 073 317 668 723 097 6;
  • 34) 0.000 000 000 073 317 668 723 097 6 × 2 = 0 + 0.000 000 000 146 635 337 446 195 2;
  • 35) 0.000 000 000 146 635 337 446 195 2 × 2 = 0 + 0.000 000 000 293 270 674 892 390 4;
  • 36) 0.000 000 000 293 270 674 892 390 4 × 2 = 0 + 0.000 000 000 586 541 349 784 780 8;
  • 37) 0.000 000 000 586 541 349 784 780 8 × 2 = 0 + 0.000 000 001 173 082 699 569 561 6;
  • 38) 0.000 000 001 173 082 699 569 561 6 × 2 = 0 + 0.000 000 002 346 165 399 139 123 2;
  • 39) 0.000 000 002 346 165 399 139 123 2 × 2 = 0 + 0.000 000 004 692 330 798 278 246 4;
  • 40) 0.000 000 004 692 330 798 278 246 4 × 2 = 0 + 0.000 000 009 384 661 596 556 492 8;
  • 41) 0.000 000 009 384 661 596 556 492 8 × 2 = 0 + 0.000 000 018 769 323 193 112 985 6;
  • 42) 0.000 000 018 769 323 193 112 985 6 × 2 = 0 + 0.000 000 037 538 646 386 225 971 2;
  • 43) 0.000 000 037 538 646 386 225 971 2 × 2 = 0 + 0.000 000 075 077 292 772 451 942 4;
  • 44) 0.000 000 075 077 292 772 451 942 4 × 2 = 0 + 0.000 000 150 154 585 544 903 884 8;
  • 45) 0.000 000 150 154 585 544 903 884 8 × 2 = 0 + 0.000 000 300 309 171 089 807 769 6;
  • 46) 0.000 000 300 309 171 089 807 769 6 × 2 = 0 + 0.000 000 600 618 342 179 615 539 2;
  • 47) 0.000 000 600 618 342 179 615 539 2 × 2 = 0 + 0.000 001 201 236 684 359 231 078 4;
  • 48) 0.000 001 201 236 684 359 231 078 4 × 2 = 0 + 0.000 002 402 473 368 718 462 156 8;
  • 49) 0.000 002 402 473 368 718 462 156 8 × 2 = 0 + 0.000 004 804 946 737 436 924 313 6;
  • 50) 0.000 004 804 946 737 436 924 313 6 × 2 = 0 + 0.000 009 609 893 474 873 848 627 2;
  • 51) 0.000 009 609 893 474 873 848 627 2 × 2 = 0 + 0.000 019 219 786 949 747 697 254 4;
  • 52) 0.000 019 219 786 949 747 697 254 4 × 2 = 0 + 0.000 038 439 573 899 495 394 508 8;
  • 53) 0.000 038 439 573 899 495 394 508 8 × 2 = 0 + 0.000 076 879 147 798 990 789 017 6;
  • 54) 0.000 076 879 147 798 990 789 017 6 × 2 = 0 + 0.000 153 758 295 597 981 578 035 2;
  • 55) 0.000 153 758 295 597 981 578 035 2 × 2 = 0 + 0.000 307 516 591 195 963 156 070 4;
  • 56) 0.000 307 516 591 195 963 156 070 4 × 2 = 0 + 0.000 615 033 182 391 926 312 140 8;
  • 57) 0.000 615 033 182 391 926 312 140 8 × 2 = 0 + 0.001 230 066 364 783 852 624 281 6;
  • 58) 0.001 230 066 364 783 852 624 281 6 × 2 = 0 + 0.002 460 132 729 567 705 248 563 2;
  • 59) 0.002 460 132 729 567 705 248 563 2 × 2 = 0 + 0.004 920 265 459 135 410 497 126 4;
  • 60) 0.004 920 265 459 135 410 497 126 4 × 2 = 0 + 0.009 840 530 918 270 820 994 252 8;
  • 61) 0.009 840 530 918 270 820 994 252 8 × 2 = 0 + 0.019 681 061 836 541 641 988 505 6;
  • 62) 0.019 681 061 836 541 641 988 505 6 × 2 = 0 + 0.039 362 123 673 083 283 977 011 2;
  • 63) 0.039 362 123 673 083 283 977 011 2 × 2 = 0 + 0.078 724 247 346 166 567 954 022 4;
  • 64) 0.078 724 247 346 166 567 954 022 4 × 2 = 0 + 0.157 448 494 692 333 135 908 044 8;
  • 65) 0.157 448 494 692 333 135 908 044 8 × 2 = 0 + 0.314 896 989 384 666 271 816 089 6;
  • 66) 0.314 896 989 384 666 271 816 089 6 × 2 = 0 + 0.629 793 978 769 332 543 632 179 2;
  • 67) 0.629 793 978 769 332 543 632 179 2 × 2 = 1 + 0.259 587 957 538 665 087 264 358 4;
  • 68) 0.259 587 957 538 665 087 264 358 4 × 2 = 0 + 0.519 175 915 077 330 174 528 716 8;
  • 69) 0.519 175 915 077 330 174 528 716 8 × 2 = 1 + 0.038 351 830 154 660 349 057 433 6;
  • 70) 0.038 351 830 154 660 349 057 433 6 × 2 = 0 + 0.076 703 660 309 320 698 114 867 2;
  • 71) 0.076 703 660 309 320 698 114 867 2 × 2 = 0 + 0.153 407 320 618 641 396 229 734 4;
  • 72) 0.153 407 320 618 641 396 229 734 4 × 2 = 0 + 0.306 814 641 237 282 792 459 468 8;
  • 73) 0.306 814 641 237 282 792 459 468 8 × 2 = 0 + 0.613 629 282 474 565 584 918 937 6;
  • 74) 0.613 629 282 474 565 584 918 937 6 × 2 = 1 + 0.227 258 564 949 131 169 837 875 2;
  • 75) 0.227 258 564 949 131 169 837 875 2 × 2 = 0 + 0.454 517 129 898 262 339 675 750 4;
  • 76) 0.454 517 129 898 262 339 675 750 4 × 2 = 0 + 0.909 034 259 796 524 679 351 500 8;
  • 77) 0.909 034 259 796 524 679 351 500 8 × 2 = 1 + 0.818 068 519 593 049 358 703 001 6;
  • 78) 0.818 068 519 593 049 358 703 001 6 × 2 = 1 + 0.636 137 039 186 098 717 406 003 2;
  • 79) 0.636 137 039 186 098 717 406 003 2 × 2 = 1 + 0.272 274 078 372 197 434 812 006 4;
  • 80) 0.272 274 078 372 197 434 812 006 4 × 2 = 0 + 0.544 548 156 744 394 869 624 012 8;
  • 81) 0.544 548 156 744 394 869 624 012 8 × 2 = 1 + 0.089 096 313 488 789 739 248 025 6;
  • 82) 0.089 096 313 488 789 739 248 025 6 × 2 = 0 + 0.178 192 626 977 579 478 496 051 2;
  • 83) 0.178 192 626 977 579 478 496 051 2 × 2 = 0 + 0.356 385 253 955 158 956 992 102 4;
  • 84) 0.356 385 253 955 158 956 992 102 4 × 2 = 0 + 0.712 770 507 910 317 913 984 204 8;
  • 85) 0.712 770 507 910 317 913 984 204 8 × 2 = 1 + 0.425 541 015 820 635 827 968 409 6;
  • 86) 0.425 541 015 820 635 827 968 409 6 × 2 = 0 + 0.851 082 031 641 271 655 936 819 2;
  • 87) 0.851 082 031 641 271 655 936 819 2 × 2 = 1 + 0.702 164 063 282 543 311 873 638 4;
  • 88) 0.702 164 063 282 543 311 873 638 4 × 2 = 1 + 0.404 328 126 565 086 623 747 276 8;
  • 89) 0.404 328 126 565 086 623 747 276 8 × 2 = 0 + 0.808 656 253 130 173 247 494 553 6;
  • 90) 0.808 656 253 130 173 247 494 553 6 × 2 = 1 + 0.617 312 506 260 346 494 989 107 2;
  • 91) 0.617 312 506 260 346 494 989 107 2 × 2 = 1 + 0.234 625 012 520 692 989 978 214 4;
  • 92) 0.234 625 012 520 692 989 978 214 4 × 2 = 0 + 0.469 250 025 041 385 979 956 428 8;
  • 93) 0.469 250 025 041 385 979 956 428 8 × 2 = 0 + 0.938 500 050 082 771 959 912 857 6;
  • 94) 0.938 500 050 082 771 959 912 857 6 × 2 = 1 + 0.877 000 100 165 543 919 825 715 2;
  • 95) 0.877 000 100 165 543 919 825 715 2 × 2 = 1 + 0.754 000 200 331 087 839 651 430 4;
  • 96) 0.754 000 200 331 087 839 651 430 4 × 2 = 1 + 0.508 000 400 662 175 679 302 860 8;
  • 97) 0.508 000 400 662 175 679 302 860 8 × 2 = 1 + 0.016 000 801 324 351 358 605 721 6;
  • 98) 0.016 000 801 324 351 358 605 721 6 × 2 = 0 + 0.032 001 602 648 702 717 211 443 2;
  • 99) 0.032 001 602 648 702 717 211 443 2 × 2 = 0 + 0.064 003 205 297 405 434 422 886 4;
  • 100) 0.064 003 205 297 405 434 422 886 4 × 2 = 0 + 0.128 006 410 594 810 868 845 772 8;
  • 101) 0.128 006 410 594 810 868 845 772 8 × 2 = 0 + 0.256 012 821 189 621 737 691 545 6;
  • 102) 0.256 012 821 189 621 737 691 545 6 × 2 = 0 + 0.512 025 642 379 243 475 383 091 2;
  • 103) 0.512 025 642 379 243 475 383 091 2 × 2 = 1 + 0.024 051 284 758 486 950 766 182 4;
  • 104) 0.024 051 284 758 486 950 766 182 4 × 2 = 0 + 0.048 102 569 516 973 901 532 364 8;
  • 105) 0.048 102 569 516 973 901 532 364 8 × 2 = 0 + 0.096 205 139 033 947 803 064 729 6;
  • 106) 0.096 205 139 033 947 803 064 729 6 × 2 = 0 + 0.192 410 278 067 895 606 129 459 2;
  • 107) 0.192 410 278 067 895 606 129 459 2 × 2 = 0 + 0.384 820 556 135 791 212 258 918 4;
  • 108) 0.384 820 556 135 791 212 258 918 4 × 2 = 0 + 0.769 641 112 271 582 424 517 836 8;
  • 109) 0.769 641 112 271 582 424 517 836 8 × 2 = 1 + 0.539 282 224 543 164 849 035 673 6;
  • 110) 0.539 282 224 543 164 849 035 673 6 × 2 = 1 + 0.078 564 449 086 329 698 071 347 2;
  • 111) 0.078 564 449 086 329 698 071 347 2 × 2 = 0 + 0.157 128 898 172 659 396 142 694 4;
  • 112) 0.157 128 898 172 659 396 142 694 4 × 2 = 0 + 0.314 257 796 345 318 792 285 388 8;
  • 113) 0.314 257 796 345 318 792 285 388 8 × 2 = 0 + 0.628 515 592 690 637 584 570 777 6;
  • 114) 0.628 515 592 690 637 584 570 777 6 × 2 = 1 + 0.257 031 185 381 275 169 141 555 2;
  • 115) 0.257 031 185 381 275 169 141 555 2 × 2 = 0 + 0.514 062 370 762 550 338 283 110 4;
  • 116) 0.514 062 370 762 550 338 283 110 4 × 2 = 1 + 0.028 124 741 525 100 676 566 220 8;
  • 117) 0.028 124 741 525 100 676 566 220 8 × 2 = 0 + 0.056 249 483 050 201 353 132 441 6;
  • 118) 0.056 249 483 050 201 353 132 441 6 × 2 = 0 + 0.112 498 966 100 402 706 264 883 2;
  • 119) 0.112 498 966 100 402 706 264 883 2 × 2 = 0 + 0.224 997 932 200 805 412 529 766 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:


0.000 000 000 000 000 000 008 535 3(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1110 1000 1011 0110 0111 1000 0010 0000 1100 0101 000(2)

5. Positive number before normalization:

0.000 000 000 000 000 000 008 535 3(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1110 1000 1011 0110 0111 1000 0010 0000 1100 0101 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:


0.000 000 000 000 000 000 008 535 3(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1110 1000 1011 0110 0111 1000 0010 0000 1100 0101 000(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1110 1000 1011 0110 0111 1000 0010 0000 1100 0101 000(2) × 20 =

1.0100 0010 0111 0100 0101 1011 0011 1100 0001 0000 0110 0010 1000(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 0111 0100 0101 1011 0011 1100 0001 0000 0110 0010 1000


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 0111 0100 0101 1011 0011 1100 0001 0000 0110 0010 1000 =

0100 0010 0111 0100 0101 1011 0011 1100 0001 0000 0110 0010 1000


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 0111 0100 0101 1011 0011 1100 0001 0000 0110 0010 1000


Decimal number 0.000 000 000 000 000 000 008 535 3 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1011 1100 - 0100 0010 0111 0100 0101 1011 0011 1100 0001 0000 0110 0010 1000

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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