0.000 000 000 000 000 000 008 535 36 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 000 000 008 535 36(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 36(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 36.

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 36 × 2 = 0 + 0.000 000 000 000 000 000 017 070 72;
  • 2) 0.000 000 000 000 000 000 017 070 72 × 2 = 0 + 0.000 000 000 000 000 000 034 141 44;
  • 3) 0.000 000 000 000 000 000 034 141 44 × 2 = 0 + 0.000 000 000 000 000 000 068 282 88;
  • 4) 0.000 000 000 000 000 000 068 282 88 × 2 = 0 + 0.000 000 000 000 000 000 136 565 76;
  • 5) 0.000 000 000 000 000 000 136 565 76 × 2 = 0 + 0.000 000 000 000 000 000 273 131 52;
  • 6) 0.000 000 000 000 000 000 273 131 52 × 2 = 0 + 0.000 000 000 000 000 000 546 263 04;
  • 7) 0.000 000 000 000 000 000 546 263 04 × 2 = 0 + 0.000 000 000 000 000 001 092 526 08;
  • 8) 0.000 000 000 000 000 001 092 526 08 × 2 = 0 + 0.000 000 000 000 000 002 185 052 16;
  • 9) 0.000 000 000 000 000 002 185 052 16 × 2 = 0 + 0.000 000 000 000 000 004 370 104 32;
  • 10) 0.000 000 000 000 000 004 370 104 32 × 2 = 0 + 0.000 000 000 000 000 008 740 208 64;
  • 11) 0.000 000 000 000 000 008 740 208 64 × 2 = 0 + 0.000 000 000 000 000 017 480 417 28;
  • 12) 0.000 000 000 000 000 017 480 417 28 × 2 = 0 + 0.000 000 000 000 000 034 960 834 56;
  • 13) 0.000 000 000 000 000 034 960 834 56 × 2 = 0 + 0.000 000 000 000 000 069 921 669 12;
  • 14) 0.000 000 000 000 000 069 921 669 12 × 2 = 0 + 0.000 000 000 000 000 139 843 338 24;
  • 15) 0.000 000 000 000 000 139 843 338 24 × 2 = 0 + 0.000 000 000 000 000 279 686 676 48;
  • 16) 0.000 000 000 000 000 279 686 676 48 × 2 = 0 + 0.000 000 000 000 000 559 373 352 96;
  • 17) 0.000 000 000 000 000 559 373 352 96 × 2 = 0 + 0.000 000 000 000 001 118 746 705 92;
  • 18) 0.000 000 000 000 001 118 746 705 92 × 2 = 0 + 0.000 000 000 000 002 237 493 411 84;
  • 19) 0.000 000 000 000 002 237 493 411 84 × 2 = 0 + 0.000 000 000 000 004 474 986 823 68;
  • 20) 0.000 000 000 000 004 474 986 823 68 × 2 = 0 + 0.000 000 000 000 008 949 973 647 36;
  • 21) 0.000 000 000 000 008 949 973 647 36 × 2 = 0 + 0.000 000 000 000 017 899 947 294 72;
  • 22) 0.000 000 000 000 017 899 947 294 72 × 2 = 0 + 0.000 000 000 000 035 799 894 589 44;
  • 23) 0.000 000 000 000 035 799 894 589 44 × 2 = 0 + 0.000 000 000 000 071 599 789 178 88;
  • 24) 0.000 000 000 000 071 599 789 178 88 × 2 = 0 + 0.000 000 000 000 143 199 578 357 76;
  • 25) 0.000 000 000 000 143 199 578 357 76 × 2 = 0 + 0.000 000 000 000 286 399 156 715 52;
  • 26) 0.000 000 000 000 286 399 156 715 52 × 2 = 0 + 0.000 000 000 000 572 798 313 431 04;
  • 27) 0.000 000 000 000 572 798 313 431 04 × 2 = 0 + 0.000 000 000 001 145 596 626 862 08;
  • 28) 0.000 000 000 001 145 596 626 862 08 × 2 = 0 + 0.000 000 000 002 291 193 253 724 16;
  • 29) 0.000 000 000 002 291 193 253 724 16 × 2 = 0 + 0.000 000 000 004 582 386 507 448 32;
  • 30) 0.000 000 000 004 582 386 507 448 32 × 2 = 0 + 0.000 000 000 009 164 773 014 896 64;
  • 31) 0.000 000 000 009 164 773 014 896 64 × 2 = 0 + 0.000 000 000 018 329 546 029 793 28;
  • 32) 0.000 000 000 018 329 546 029 793 28 × 2 = 0 + 0.000 000 000 036 659 092 059 586 56;
  • 33) 0.000 000 000 036 659 092 059 586 56 × 2 = 0 + 0.000 000 000 073 318 184 119 173 12;
  • 34) 0.000 000 000 073 318 184 119 173 12 × 2 = 0 + 0.000 000 000 146 636 368 238 346 24;
  • 35) 0.000 000 000 146 636 368 238 346 24 × 2 = 0 + 0.000 000 000 293 272 736 476 692 48;
  • 36) 0.000 000 000 293 272 736 476 692 48 × 2 = 0 + 0.000 000 000 586 545 472 953 384 96;
  • 37) 0.000 000 000 586 545 472 953 384 96 × 2 = 0 + 0.000 000 001 173 090 945 906 769 92;
  • 38) 0.000 000 001 173 090 945 906 769 92 × 2 = 0 + 0.000 000 002 346 181 891 813 539 84;
  • 39) 0.000 000 002 346 181 891 813 539 84 × 2 = 0 + 0.000 000 004 692 363 783 627 079 68;
  • 40) 0.000 000 004 692 363 783 627 079 68 × 2 = 0 + 0.000 000 009 384 727 567 254 159 36;
  • 41) 0.000 000 009 384 727 567 254 159 36 × 2 = 0 + 0.000 000 018 769 455 134 508 318 72;
  • 42) 0.000 000 018 769 455 134 508 318 72 × 2 = 0 + 0.000 000 037 538 910 269 016 637 44;
  • 43) 0.000 000 037 538 910 269 016 637 44 × 2 = 0 + 0.000 000 075 077 820 538 033 274 88;
  • 44) 0.000 000 075 077 820 538 033 274 88 × 2 = 0 + 0.000 000 150 155 641 076 066 549 76;
  • 45) 0.000 000 150 155 641 076 066 549 76 × 2 = 0 + 0.000 000 300 311 282 152 133 099 52;
  • 46) 0.000 000 300 311 282 152 133 099 52 × 2 = 0 + 0.000 000 600 622 564 304 266 199 04;
  • 47) 0.000 000 600 622 564 304 266 199 04 × 2 = 0 + 0.000 001 201 245 128 608 532 398 08;
  • 48) 0.000 001 201 245 128 608 532 398 08 × 2 = 0 + 0.000 002 402 490 257 217 064 796 16;
  • 49) 0.000 002 402 490 257 217 064 796 16 × 2 = 0 + 0.000 004 804 980 514 434 129 592 32;
  • 50) 0.000 004 804 980 514 434 129 592 32 × 2 = 0 + 0.000 009 609 961 028 868 259 184 64;
  • 51) 0.000 009 609 961 028 868 259 184 64 × 2 = 0 + 0.000 019 219 922 057 736 518 369 28;
  • 52) 0.000 019 219 922 057 736 518 369 28 × 2 = 0 + 0.000 038 439 844 115 473 036 738 56;
  • 53) 0.000 038 439 844 115 473 036 738 56 × 2 = 0 + 0.000 076 879 688 230 946 073 477 12;
  • 54) 0.000 076 879 688 230 946 073 477 12 × 2 = 0 + 0.000 153 759 376 461 892 146 954 24;
  • 55) 0.000 153 759 376 461 892 146 954 24 × 2 = 0 + 0.000 307 518 752 923 784 293 908 48;
  • 56) 0.000 307 518 752 923 784 293 908 48 × 2 = 0 + 0.000 615 037 505 847 568 587 816 96;
  • 57) 0.000 615 037 505 847 568 587 816 96 × 2 = 0 + 0.001 230 075 011 695 137 175 633 92;
  • 58) 0.001 230 075 011 695 137 175 633 92 × 2 = 0 + 0.002 460 150 023 390 274 351 267 84;
  • 59) 0.002 460 150 023 390 274 351 267 84 × 2 = 0 + 0.004 920 300 046 780 548 702 535 68;
  • 60) 0.004 920 300 046 780 548 702 535 68 × 2 = 0 + 0.009 840 600 093 561 097 405 071 36;
  • 61) 0.009 840 600 093 561 097 405 071 36 × 2 = 0 + 0.019 681 200 187 122 194 810 142 72;
  • 62) 0.019 681 200 187 122 194 810 142 72 × 2 = 0 + 0.039 362 400 374 244 389 620 285 44;
  • 63) 0.039 362 400 374 244 389 620 285 44 × 2 = 0 + 0.078 724 800 748 488 779 240 570 88;
  • 64) 0.078 724 800 748 488 779 240 570 88 × 2 = 0 + 0.157 449 601 496 977 558 481 141 76;
  • 65) 0.157 449 601 496 977 558 481 141 76 × 2 = 0 + 0.314 899 202 993 955 116 962 283 52;
  • 66) 0.314 899 202 993 955 116 962 283 52 × 2 = 0 + 0.629 798 405 987 910 233 924 567 04;
  • 67) 0.629 798 405 987 910 233 924 567 04 × 2 = 1 + 0.259 596 811 975 820 467 849 134 08;
  • 68) 0.259 596 811 975 820 467 849 134 08 × 2 = 0 + 0.519 193 623 951 640 935 698 268 16;
  • 69) 0.519 193 623 951 640 935 698 268 16 × 2 = 1 + 0.038 387 247 903 281 871 396 536 32;
  • 70) 0.038 387 247 903 281 871 396 536 32 × 2 = 0 + 0.076 774 495 806 563 742 793 072 64;
  • 71) 0.076 774 495 806 563 742 793 072 64 × 2 = 0 + 0.153 548 991 613 127 485 586 145 28;
  • 72) 0.153 548 991 613 127 485 586 145 28 × 2 = 0 + 0.307 097 983 226 254 971 172 290 56;
  • 73) 0.307 097 983 226 254 971 172 290 56 × 2 = 0 + 0.614 195 966 452 509 942 344 581 12;
  • 74) 0.614 195 966 452 509 942 344 581 12 × 2 = 1 + 0.228 391 932 905 019 884 689 162 24;
  • 75) 0.228 391 932 905 019 884 689 162 24 × 2 = 0 + 0.456 783 865 810 039 769 378 324 48;
  • 76) 0.456 783 865 810 039 769 378 324 48 × 2 = 0 + 0.913 567 731 620 079 538 756 648 96;
  • 77) 0.913 567 731 620 079 538 756 648 96 × 2 = 1 + 0.827 135 463 240 159 077 513 297 92;
  • 78) 0.827 135 463 240 159 077 513 297 92 × 2 = 1 + 0.654 270 926 480 318 155 026 595 84;
  • 79) 0.654 270 926 480 318 155 026 595 84 × 2 = 1 + 0.308 541 852 960 636 310 053 191 68;
  • 80) 0.308 541 852 960 636 310 053 191 68 × 2 = 0 + 0.617 083 705 921 272 620 106 383 36;
  • 81) 0.617 083 705 921 272 620 106 383 36 × 2 = 1 + 0.234 167 411 842 545 240 212 766 72;
  • 82) 0.234 167 411 842 545 240 212 766 72 × 2 = 0 + 0.468 334 823 685 090 480 425 533 44;
  • 83) 0.468 334 823 685 090 480 425 533 44 × 2 = 0 + 0.936 669 647 370 180 960 851 066 88;
  • 84) 0.936 669 647 370 180 960 851 066 88 × 2 = 1 + 0.873 339 294 740 361 921 702 133 76;
  • 85) 0.873 339 294 740 361 921 702 133 76 × 2 = 1 + 0.746 678 589 480 723 843 404 267 52;
  • 86) 0.746 678 589 480 723 843 404 267 52 × 2 = 1 + 0.493 357 178 961 447 686 808 535 04;
  • 87) 0.493 357 178 961 447 686 808 535 04 × 2 = 0 + 0.986 714 357 922 895 373 617 070 08;
  • 88) 0.986 714 357 922 895 373 617 070 08 × 2 = 1 + 0.973 428 715 845 790 747 234 140 16;
  • 89) 0.973 428 715 845 790 747 234 140 16 × 2 = 1 + 0.946 857 431 691 581 494 468 280 32;
  • 90) 0.946 857 431 691 581 494 468 280 32 × 2 = 1 + 0.893 714 863 383 162 988 936 560 64;
  • 91) 0.893 714 863 383 162 988 936 560 64 × 2 = 1 + 0.787 429 726 766 325 977 873 121 28;
  • 92) 0.787 429 726 766 325 977 873 121 28 × 2 = 1 + 0.574 859 453 532 651 955 746 242 56;
  • 93) 0.574 859 453 532 651 955 746 242 56 × 2 = 1 + 0.149 718 907 065 303 911 492 485 12;
  • 94) 0.149 718 907 065 303 911 492 485 12 × 2 = 0 + 0.299 437 814 130 607 822 984 970 24;
  • 95) 0.299 437 814 130 607 822 984 970 24 × 2 = 0 + 0.598 875 628 261 215 645 969 940 48;
  • 96) 0.598 875 628 261 215 645 969 940 48 × 2 = 1 + 0.197 751 256 522 431 291 939 880 96;
  • 97) 0.197 751 256 522 431 291 939 880 96 × 2 = 0 + 0.395 502 513 044 862 583 879 761 92;
  • 98) 0.395 502 513 044 862 583 879 761 92 × 2 = 0 + 0.791 005 026 089 725 167 759 523 84;
  • 99) 0.791 005 026 089 725 167 759 523 84 × 2 = 1 + 0.582 010 052 179 450 335 519 047 68;
  • 100) 0.582 010 052 179 450 335 519 047 68 × 2 = 1 + 0.164 020 104 358 900 671 038 095 36;
  • 101) 0.164 020 104 358 900 671 038 095 36 × 2 = 0 + 0.328 040 208 717 801 342 076 190 72;
  • 102) 0.328 040 208 717 801 342 076 190 72 × 2 = 0 + 0.656 080 417 435 602 684 152 381 44;
  • 103) 0.656 080 417 435 602 684 152 381 44 × 2 = 1 + 0.312 160 834 871 205 368 304 762 88;
  • 104) 0.312 160 834 871 205 368 304 762 88 × 2 = 0 + 0.624 321 669 742 410 736 609 525 76;
  • 105) 0.624 321 669 742 410 736 609 525 76 × 2 = 1 + 0.248 643 339 484 821 473 219 051 52;
  • 106) 0.248 643 339 484 821 473 219 051 52 × 2 = 0 + 0.497 286 678 969 642 946 438 103 04;
  • 107) 0.497 286 678 969 642 946 438 103 04 × 2 = 0 + 0.994 573 357 939 285 892 876 206 08;
  • 108) 0.994 573 357 939 285 892 876 206 08 × 2 = 1 + 0.989 146 715 878 571 785 752 412 16;
  • 109) 0.989 146 715 878 571 785 752 412 16 × 2 = 1 + 0.978 293 431 757 143 571 504 824 32;
  • 110) 0.978 293 431 757 143 571 504 824 32 × 2 = 1 + 0.956 586 863 514 287 143 009 648 64;
  • 111) 0.956 586 863 514 287 143 009 648 64 × 2 = 1 + 0.913 173 727 028 574 286 019 297 28;
  • 112) 0.913 173 727 028 574 286 019 297 28 × 2 = 1 + 0.826 347 454 057 148 572 038 594 56;
  • 113) 0.826 347 454 057 148 572 038 594 56 × 2 = 1 + 0.652 694 908 114 297 144 077 189 12;
  • 114) 0.652 694 908 114 297 144 077 189 12 × 2 = 1 + 0.305 389 816 228 594 288 154 378 24;
  • 115) 0.305 389 816 228 594 288 154 378 24 × 2 = 0 + 0.610 779 632 457 188 576 308 756 48;
  • 116) 0.610 779 632 457 188 576 308 756 48 × 2 = 1 + 0.221 559 264 914 377 152 617 512 96;
  • 117) 0.221 559 264 914 377 152 617 512 96 × 2 = 0 + 0.443 118 529 828 754 305 235 025 92;
  • 118) 0.443 118 529 828 754 305 235 025 92 × 2 = 0 + 0.886 237 059 657 508 610 470 051 84;
  • 119) 0.886 237 059 657 508 610 470 051 84 × 2 = 1 + 0.772 474 119 315 017 220 940 103 68;

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 36(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1110 1001 1101 1111 1001 0011 0010 1001 1111 1101 001(2)

5. Positive number before normalization:

0.000 000 000 000 000 000 008 535 36(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1110 1001 1101 1111 1001 0011 0010 1001 1111 1101 001(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 36(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1110 1001 1101 1111 1001 0011 0010 1001 1111 1101 001(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1110 1001 1101 1111 1001 0011 0010 1001 1111 1101 001(2) × 20 =

1.0100 0010 0111 0100 1110 1111 1100 1001 1001 0100 1111 1110 1001(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 1110 1111 1100 1001 1001 0100 1111 1110 1001


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 1110 1111 1100 1001 1001 0100 1111 1110 1001 =

0100 0010 0111 0100 1110 1111 1100 1001 1001 0100 1111 1110 1001


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 1110 1111 1100 1001 1001 0100 1111 1110 1001


Decimal number 0.000 000 000 000 000 000 008 535 36 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1011 1100 - 0100 0010 0111 0100 1110 1111 1100 1001 1001 0100 1111 1110 1001

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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