0.000 000 000 000 000 000 008 538 18 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 000 000 008 538 18(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
0.000 000 000 000 000 000 008 538 18(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: 0.000 000 000 000 000 000 008 538 18.

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) 0.000 000 000 000 000 000 008 538 18 × 2 = 0 + 0.000 000 000 000 000 000 017 076 36;
  • 2) 0.000 000 000 000 000 000 017 076 36 × 2 = 0 + 0.000 000 000 000 000 000 034 152 72;
  • 3) 0.000 000 000 000 000 000 034 152 72 × 2 = 0 + 0.000 000 000 000 000 000 068 305 44;
  • 4) 0.000 000 000 000 000 000 068 305 44 × 2 = 0 + 0.000 000 000 000 000 000 136 610 88;
  • 5) 0.000 000 000 000 000 000 136 610 88 × 2 = 0 + 0.000 000 000 000 000 000 273 221 76;
  • 6) 0.000 000 000 000 000 000 273 221 76 × 2 = 0 + 0.000 000 000 000 000 000 546 443 52;
  • 7) 0.000 000 000 000 000 000 546 443 52 × 2 = 0 + 0.000 000 000 000 000 001 092 887 04;
  • 8) 0.000 000 000 000 000 001 092 887 04 × 2 = 0 + 0.000 000 000 000 000 002 185 774 08;
  • 9) 0.000 000 000 000 000 002 185 774 08 × 2 = 0 + 0.000 000 000 000 000 004 371 548 16;
  • 10) 0.000 000 000 000 000 004 371 548 16 × 2 = 0 + 0.000 000 000 000 000 008 743 096 32;
  • 11) 0.000 000 000 000 000 008 743 096 32 × 2 = 0 + 0.000 000 000 000 000 017 486 192 64;
  • 12) 0.000 000 000 000 000 017 486 192 64 × 2 = 0 + 0.000 000 000 000 000 034 972 385 28;
  • 13) 0.000 000 000 000 000 034 972 385 28 × 2 = 0 + 0.000 000 000 000 000 069 944 770 56;
  • 14) 0.000 000 000 000 000 069 944 770 56 × 2 = 0 + 0.000 000 000 000 000 139 889 541 12;
  • 15) 0.000 000 000 000 000 139 889 541 12 × 2 = 0 + 0.000 000 000 000 000 279 779 082 24;
  • 16) 0.000 000 000 000 000 279 779 082 24 × 2 = 0 + 0.000 000 000 000 000 559 558 164 48;
  • 17) 0.000 000 000 000 000 559 558 164 48 × 2 = 0 + 0.000 000 000 000 001 119 116 328 96;
  • 18) 0.000 000 000 000 001 119 116 328 96 × 2 = 0 + 0.000 000 000 000 002 238 232 657 92;
  • 19) 0.000 000 000 000 002 238 232 657 92 × 2 = 0 + 0.000 000 000 000 004 476 465 315 84;
  • 20) 0.000 000 000 000 004 476 465 315 84 × 2 = 0 + 0.000 000 000 000 008 952 930 631 68;
  • 21) 0.000 000 000 000 008 952 930 631 68 × 2 = 0 + 0.000 000 000 000 017 905 861 263 36;
  • 22) 0.000 000 000 000 017 905 861 263 36 × 2 = 0 + 0.000 000 000 000 035 811 722 526 72;
  • 23) 0.000 000 000 000 035 811 722 526 72 × 2 = 0 + 0.000 000 000 000 071 623 445 053 44;
  • 24) 0.000 000 000 000 071 623 445 053 44 × 2 = 0 + 0.000 000 000 000 143 246 890 106 88;
  • 25) 0.000 000 000 000 143 246 890 106 88 × 2 = 0 + 0.000 000 000 000 286 493 780 213 76;
  • 26) 0.000 000 000 000 286 493 780 213 76 × 2 = 0 + 0.000 000 000 000 572 987 560 427 52;
  • 27) 0.000 000 000 000 572 987 560 427 52 × 2 = 0 + 0.000 000 000 001 145 975 120 855 04;
  • 28) 0.000 000 000 001 145 975 120 855 04 × 2 = 0 + 0.000 000 000 002 291 950 241 710 08;
  • 29) 0.000 000 000 002 291 950 241 710 08 × 2 = 0 + 0.000 000 000 004 583 900 483 420 16;
  • 30) 0.000 000 000 004 583 900 483 420 16 × 2 = 0 + 0.000 000 000 009 167 800 966 840 32;
  • 31) 0.000 000 000 009 167 800 966 840 32 × 2 = 0 + 0.000 000 000 018 335 601 933 680 64;
  • 32) 0.000 000 000 018 335 601 933 680 64 × 2 = 0 + 0.000 000 000 036 671 203 867 361 28;
  • 33) 0.000 000 000 036 671 203 867 361 28 × 2 = 0 + 0.000 000 000 073 342 407 734 722 56;
  • 34) 0.000 000 000 073 342 407 734 722 56 × 2 = 0 + 0.000 000 000 146 684 815 469 445 12;
  • 35) 0.000 000 000 146 684 815 469 445 12 × 2 = 0 + 0.000 000 000 293 369 630 938 890 24;
  • 36) 0.000 000 000 293 369 630 938 890 24 × 2 = 0 + 0.000 000 000 586 739 261 877 780 48;
  • 37) 0.000 000 000 586 739 261 877 780 48 × 2 = 0 + 0.000 000 001 173 478 523 755 560 96;
  • 38) 0.000 000 001 173 478 523 755 560 96 × 2 = 0 + 0.000 000 002 346 957 047 511 121 92;
  • 39) 0.000 000 002 346 957 047 511 121 92 × 2 = 0 + 0.000 000 004 693 914 095 022 243 84;
  • 40) 0.000 000 004 693 914 095 022 243 84 × 2 = 0 + 0.000 000 009 387 828 190 044 487 68;
  • 41) 0.000 000 009 387 828 190 044 487 68 × 2 = 0 + 0.000 000 018 775 656 380 088 975 36;
  • 42) 0.000 000 018 775 656 380 088 975 36 × 2 = 0 + 0.000 000 037 551 312 760 177 950 72;
  • 43) 0.000 000 037 551 312 760 177 950 72 × 2 = 0 + 0.000 000 075 102 625 520 355 901 44;
  • 44) 0.000 000 075 102 625 520 355 901 44 × 2 = 0 + 0.000 000 150 205 251 040 711 802 88;
  • 45) 0.000 000 150 205 251 040 711 802 88 × 2 = 0 + 0.000 000 300 410 502 081 423 605 76;
  • 46) 0.000 000 300 410 502 081 423 605 76 × 2 = 0 + 0.000 000 600 821 004 162 847 211 52;
  • 47) 0.000 000 600 821 004 162 847 211 52 × 2 = 0 + 0.000 001 201 642 008 325 694 423 04;
  • 48) 0.000 001 201 642 008 325 694 423 04 × 2 = 0 + 0.000 002 403 284 016 651 388 846 08;
  • 49) 0.000 002 403 284 016 651 388 846 08 × 2 = 0 + 0.000 004 806 568 033 302 777 692 16;
  • 50) 0.000 004 806 568 033 302 777 692 16 × 2 = 0 + 0.000 009 613 136 066 605 555 384 32;
  • 51) 0.000 009 613 136 066 605 555 384 32 × 2 = 0 + 0.000 019 226 272 133 211 110 768 64;
  • 52) 0.000 019 226 272 133 211 110 768 64 × 2 = 0 + 0.000 038 452 544 266 422 221 537 28;
  • 53) 0.000 038 452 544 266 422 221 537 28 × 2 = 0 + 0.000 076 905 088 532 844 443 074 56;
  • 54) 0.000 076 905 088 532 844 443 074 56 × 2 = 0 + 0.000 153 810 177 065 688 886 149 12;
  • 55) 0.000 153 810 177 065 688 886 149 12 × 2 = 0 + 0.000 307 620 354 131 377 772 298 24;
  • 56) 0.000 307 620 354 131 377 772 298 24 × 2 = 0 + 0.000 615 240 708 262 755 544 596 48;
  • 57) 0.000 615 240 708 262 755 544 596 48 × 2 = 0 + 0.001 230 481 416 525 511 089 192 96;
  • 58) 0.001 230 481 416 525 511 089 192 96 × 2 = 0 + 0.002 460 962 833 051 022 178 385 92;
  • 59) 0.002 460 962 833 051 022 178 385 92 × 2 = 0 + 0.004 921 925 666 102 044 356 771 84;
  • 60) 0.004 921 925 666 102 044 356 771 84 × 2 = 0 + 0.009 843 851 332 204 088 713 543 68;
  • 61) 0.009 843 851 332 204 088 713 543 68 × 2 = 0 + 0.019 687 702 664 408 177 427 087 36;
  • 62) 0.019 687 702 664 408 177 427 087 36 × 2 = 0 + 0.039 375 405 328 816 354 854 174 72;
  • 63) 0.039 375 405 328 816 354 854 174 72 × 2 = 0 + 0.078 750 810 657 632 709 708 349 44;
  • 64) 0.078 750 810 657 632 709 708 349 44 × 2 = 0 + 0.157 501 621 315 265 419 416 698 88;
  • 65) 0.157 501 621 315 265 419 416 698 88 × 2 = 0 + 0.315 003 242 630 530 838 833 397 76;
  • 66) 0.315 003 242 630 530 838 833 397 76 × 2 = 0 + 0.630 006 485 261 061 677 666 795 52;
  • 67) 0.630 006 485 261 061 677 666 795 52 × 2 = 1 + 0.260 012 970 522 123 355 333 591 04;
  • 68) 0.260 012 970 522 123 355 333 591 04 × 2 = 0 + 0.520 025 941 044 246 710 667 182 08;
  • 69) 0.520 025 941 044 246 710 667 182 08 × 2 = 1 + 0.040 051 882 088 493 421 334 364 16;
  • 70) 0.040 051 882 088 493 421 334 364 16 × 2 = 0 + 0.080 103 764 176 986 842 668 728 32;
  • 71) 0.080 103 764 176 986 842 668 728 32 × 2 = 0 + 0.160 207 528 353 973 685 337 456 64;
  • 72) 0.160 207 528 353 973 685 337 456 64 × 2 = 0 + 0.320 415 056 707 947 370 674 913 28;
  • 73) 0.320 415 056 707 947 370 674 913 28 × 2 = 0 + 0.640 830 113 415 894 741 349 826 56;
  • 74) 0.640 830 113 415 894 741 349 826 56 × 2 = 1 + 0.281 660 226 831 789 482 699 653 12;
  • 75) 0.281 660 226 831 789 482 699 653 12 × 2 = 0 + 0.563 320 453 663 578 965 399 306 24;
  • 76) 0.563 320 453 663 578 965 399 306 24 × 2 = 1 + 0.126 640 907 327 157 930 798 612 48;
  • 77) 0.126 640 907 327 157 930 798 612 48 × 2 = 0 + 0.253 281 814 654 315 861 597 224 96;
  • 78) 0.253 281 814 654 315 861 597 224 96 × 2 = 0 + 0.506 563 629 308 631 723 194 449 92;
  • 79) 0.506 563 629 308 631 723 194 449 92 × 2 = 1 + 0.013 127 258 617 263 446 388 899 84;
  • 80) 0.013 127 258 617 263 446 388 899 84 × 2 = 0 + 0.026 254 517 234 526 892 777 799 68;
  • 81) 0.026 254 517 234 526 892 777 799 68 × 2 = 0 + 0.052 509 034 469 053 785 555 599 36;
  • 82) 0.052 509 034 469 053 785 555 599 36 × 2 = 0 + 0.105 018 068 938 107 571 111 198 72;
  • 83) 0.105 018 068 938 107 571 111 198 72 × 2 = 0 + 0.210 036 137 876 215 142 222 397 44;
  • 84) 0.210 036 137 876 215 142 222 397 44 × 2 = 0 + 0.420 072 275 752 430 284 444 794 88;
  • 85) 0.420 072 275 752 430 284 444 794 88 × 2 = 0 + 0.840 144 551 504 860 568 889 589 76;
  • 86) 0.840 144 551 504 860 568 889 589 76 × 2 = 1 + 0.680 289 103 009 721 137 779 179 52;
  • 87) 0.680 289 103 009 721 137 779 179 52 × 2 = 1 + 0.360 578 206 019 442 275 558 359 04;
  • 88) 0.360 578 206 019 442 275 558 359 04 × 2 = 0 + 0.721 156 412 038 884 551 116 718 08;
  • 89) 0.721 156 412 038 884 551 116 718 08 × 2 = 1 + 0.442 312 824 077 769 102 233 436 16;
  • 90) 0.442 312 824 077 769 102 233 436 16 × 2 = 0 + 0.884 625 648 155 538 204 466 872 32;
  • 91) 0.884 625 648 155 538 204 466 872 32 × 2 = 1 + 0.769 251 296 311 076 408 933 744 64;
  • 92) 0.769 251 296 311 076 408 933 744 64 × 2 = 1 + 0.538 502 592 622 152 817 867 489 28;
  • 93) 0.538 502 592 622 152 817 867 489 28 × 2 = 1 + 0.077 005 185 244 305 635 734 978 56;
  • 94) 0.077 005 185 244 305 635 734 978 56 × 2 = 0 + 0.154 010 370 488 611 271 469 957 12;
  • 95) 0.154 010 370 488 611 271 469 957 12 × 2 = 0 + 0.308 020 740 977 222 542 939 914 24;
  • 96) 0.308 020 740 977 222 542 939 914 24 × 2 = 0 + 0.616 041 481 954 445 085 879 828 48;
  • 97) 0.616 041 481 954 445 085 879 828 48 × 2 = 1 + 0.232 082 963 908 890 171 759 656 96;
  • 98) 0.232 082 963 908 890 171 759 656 96 × 2 = 0 + 0.464 165 927 817 780 343 519 313 92;
  • 99) 0.464 165 927 817 780 343 519 313 92 × 2 = 0 + 0.928 331 855 635 560 687 038 627 84;
  • 100) 0.928 331 855 635 560 687 038 627 84 × 2 = 1 + 0.856 663 711 271 121 374 077 255 68;
  • 101) 0.856 663 711 271 121 374 077 255 68 × 2 = 1 + 0.713 327 422 542 242 748 154 511 36;
  • 102) 0.713 327 422 542 242 748 154 511 36 × 2 = 1 + 0.426 654 845 084 485 496 309 022 72;
  • 103) 0.426 654 845 084 485 496 309 022 72 × 2 = 0 + 0.853 309 690 168 970 992 618 045 44;
  • 104) 0.853 309 690 168 970 992 618 045 44 × 2 = 1 + 0.706 619 380 337 941 985 236 090 88;
  • 105) 0.706 619 380 337 941 985 236 090 88 × 2 = 1 + 0.413 238 760 675 883 970 472 181 76;
  • 106) 0.413 238 760 675 883 970 472 181 76 × 2 = 0 + 0.826 477 521 351 767 940 944 363 52;
  • 107) 0.826 477 521 351 767 940 944 363 52 × 2 = 1 + 0.652 955 042 703 535 881 888 727 04;
  • 108) 0.652 955 042 703 535 881 888 727 04 × 2 = 1 + 0.305 910 085 407 071 763 777 454 08;
  • 109) 0.305 910 085 407 071 763 777 454 08 × 2 = 0 + 0.611 820 170 814 143 527 554 908 16;
  • 110) 0.611 820 170 814 143 527 554 908 16 × 2 = 1 + 0.223 640 341 628 287 055 109 816 32;
  • 111) 0.223 640 341 628 287 055 109 816 32 × 2 = 0 + 0.447 280 683 256 574 110 219 632 64;
  • 112) 0.447 280 683 256 574 110 219 632 64 × 2 = 0 + 0.894 561 366 513 148 220 439 265 28;
  • 113) 0.894 561 366 513 148 220 439 265 28 × 2 = 1 + 0.789 122 733 026 296 440 878 530 56;
  • 114) 0.789 122 733 026 296 440 878 530 56 × 2 = 1 + 0.578 245 466 052 592 881 757 061 12;
  • 115) 0.578 245 466 052 592 881 757 061 12 × 2 = 1 + 0.156 490 932 105 185 763 514 122 24;
  • 116) 0.156 490 932 105 185 763 514 122 24 × 2 = 0 + 0.312 981 864 210 371 527 028 244 48;
  • 117) 0.312 981 864 210 371 527 028 244 48 × 2 = 0 + 0.625 963 728 420 743 054 056 488 96;
  • 118) 0.625 963 728 420 743 054 056 488 96 × 2 = 1 + 0.251 927 456 841 486 108 112 977 92;
  • 119) 0.251 927 456 841 486 108 112 977 92 × 2 = 0 + 0.503 854 913 682 972 216 225 955 84;

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:


0.000 000 000 000 000 000 008 538 18(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0101 0010 0000 0110 1011 1000 1001 1101 1011 0100 1110 010(2)

5. Positive number before normalization:

0.000 000 000 000 000 000 008 538 18(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0101 0010 0000 0110 1011 1000 1001 1101 1011 0100 1110 010(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 67 positions to the right, so that only one non zero digit remains to the left of it:


0.000 000 000 000 000 000 008 538 18(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0101 0010 0000 0110 1011 1000 1001 1101 1011 0100 1110 010(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0101 0010 0000 0110 1011 1000 1001 1101 1011 0100 1110 010(2) × 20 =

1.0100 0010 1001 0000 0011 0101 1100 0100 1110 1101 1010 0111 0010(2) × 2-67


7. Up to this moment, there are the following elements that would feed into the 64 bit double precision IEEE 754 binary floating point representation:

Sign 0 (a positive number)

Exponent (unadjusted): -67

Mantissa (not normalized):
1.0100 0010 1001 0000 0011 0101 1100 0100 1110 1101 1010 0111 0010


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

Exponent (unadjusted) + 2(11-1) - 1 =

-67 + 2(11-1) - 1 =

(-67 + 1 023)(10) =

956(10)


9. Convert the adjusted exponent from the decimal (base 10) to 11 bit binary.

Use the same technique of repeatedly dividing by 2:


  • division = quotient + remainder;
  • 956 ÷ 2 = 478 + 0;
  • 478 ÷ 2 = 239 + 0;
  • 239 ÷ 2 = 119 + 1;
  • 119 ÷ 2 = 59 + 1;
  • 59 ÷ 2 = 29 + 1;
  • 29 ÷ 2 = 14 + 1;
  • 14 ÷ 2 = 7 + 0;
  • 7 ÷ 2 = 3 + 1;
  • 3 ÷ 2 = 1 + 1;
  • 1 ÷ 2 = 0 + 1;

10. Construct the base 2 representation of the adjusted exponent.

Take all the remainders starting from the bottom of the list constructed above.


Exponent (adjusted) =

956(10) =

011 1011 1100(2)


11. Normalize the mantissa.

a) Remove the leading (the leftmost) bit, since it's allways 1, and the decimal point, if the case.

b) Adjust its length to 52 bits, only if necessary (not the case here).


Mantissa (normalized) =

1. 0100 0010 1001 0000 0011 0101 1100 0100 1110 1101 1010 0111 0010 =

0100 0010 1001 0000 0011 0101 1100 0100 1110 1101 1010 0111 0010


12. The three elements that make up the number's 64 bit double precision IEEE 754 binary floating point representation:

Sign (1 bit) =
0 (a positive number)

Exponent (11 bits) =
011 1011 1100

Mantissa (52 bits) =
0100 0010 1001 0000 0011 0101 1100 0100 1110 1101 1010 0111 0010


Decimal number 0.000 000 000 000 000 000 008 538 18 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1011 1100 - 0100 0010 1001 0000 0011 0101 1100 0100 1110 1101 1010 0111 0010

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How to convert numbers from the decimal system (base ten) to 64 bit double precision IEEE 754 binary floating point standard

Follow the steps below to convert a base 10 decimal number to 64 bit double precision IEEE 754 binary floating point:

  • 1. If the number to be converted is negative, start with its the positive version.
  • 2. First convert the integer part. Divide repeatedly by 2 the positive representation of the integer number that is to be converted to binary, until we get a quotient that is equal to zero, keeping track of each remainder.
  • 3. Construct the base 2 representation of the positive integer part of the number, by taking all the remainders from the previous operations, starting from the bottom of the list constructed above. Thus, the last remainder of the divisions becomes the first symbol (the leftmost) of the base two number, while the first remainder becomes the last symbol (the rightmost).
  • 4. Then convert the fractional part. Multiply the number repeatedly by 2, until we get a fractional part that is equal to zero, keeping track of each integer part of the results.
  • 5. Construct the base 2 representation of the fractional part of the number, by taking all the integer parts of the multiplying operations, starting from the top of the list constructed above (they should appear in the binary representation, from left to right, in the order they have been calculated).
  • 6. Normalize the binary representation of the number, shifting the decimal mark (the decimal point) "n" positions either to the left, or to the right, so that only one non zero digit remains to the left of the decimal mark.
  • 7. Adjust the exponent in 11 bit excess/bias notation and then convert it from decimal (base 10) to 11 bit binary, by using the same technique of repeatedly dividing by 2, as shown above:
    Exponent (adjusted) = Exponent (unadjusted) + 2(11-1) - 1
  • 8. Normalize mantissa, remove the leading (leftmost) bit, since it's allways '1' (and the decimal mark, if the case) and adjust its length to 52 bits, either by removing the excess bits from the right (losing precision...) or by adding extra bits set on '0' to the right.
  • 9. Sign (it takes 1 bit) is either 1 for a negative or 0 for a positive number.

Example: convert the negative number -31.640 215 from the decimal system (base ten) to 64 bit double precision IEEE 754 binary floating point:

  • 1. Start with the positive version of the number:

    |-31.640 215| = 31.640 215

  • 2. First convert the integer part, 31. Divide it repeatedly by 2, keeping track of each remainder, until we get a quotient that is equal to zero:
    • division = quotient + remainder;
    • 31 ÷ 2 = 15 + 1;
    • 15 ÷ 2 = 7 + 1;
    • 7 ÷ 2 = 3 + 1;
    • 3 ÷ 2 = 1 + 1;
    • 1 ÷ 2 = 0 + 1;
    • We have encountered a quotient that is ZERO => FULL STOP
  • 3. Construct the base 2 representation of the integer part of the number by taking all the remainders of the previous dividing operations, starting from the bottom of the list constructed above:

    31(10) = 1 1111(2)

  • 4. Then, convert the fractional part, 0.640 215. Multiply repeatedly by 2, keeping track of each integer part of the results, until we get a fractional part that is equal to zero:
    • #) multiplying = integer + fractional part;
    • 1) 0.640 215 × 2 = 1 + 0.280 43;
    • 2) 0.280 43 × 2 = 0 + 0.560 86;
    • 3) 0.560 86 × 2 = 1 + 0.121 72;
    • 4) 0.121 72 × 2 = 0 + 0.243 44;
    • 5) 0.243 44 × 2 = 0 + 0.486 88;
    • 6) 0.486 88 × 2 = 0 + 0.973 76;
    • 7) 0.973 76 × 2 = 1 + 0.947 52;
    • 8) 0.947 52 × 2 = 1 + 0.895 04;
    • 9) 0.895 04 × 2 = 1 + 0.790 08;
    • 10) 0.790 08 × 2 = 1 + 0.580 16;
    • 11) 0.580 16 × 2 = 1 + 0.160 32;
    • 12) 0.160 32 × 2 = 0 + 0.320 64;
    • 13) 0.320 64 × 2 = 0 + 0.641 28;
    • 14) 0.641 28 × 2 = 1 + 0.282 56;
    • 15) 0.282 56 × 2 = 0 + 0.565 12;
    • 16) 0.565 12 × 2 = 1 + 0.130 24;
    • 17) 0.130 24 × 2 = 0 + 0.260 48;
    • 18) 0.260 48 × 2 = 0 + 0.520 96;
    • 19) 0.520 96 × 2 = 1 + 0.041 92;
    • 20) 0.041 92 × 2 = 0 + 0.083 84;
    • 21) 0.083 84 × 2 = 0 + 0.167 68;
    • 22) 0.167 68 × 2 = 0 + 0.335 36;
    • 23) 0.335 36 × 2 = 0 + 0.670 72;
    • 24) 0.670 72 × 2 = 1 + 0.341 44;
    • 25) 0.341 44 × 2 = 0 + 0.682 88;
    • 26) 0.682 88 × 2 = 1 + 0.365 76;
    • 27) 0.365 76 × 2 = 0 + 0.731 52;
    • 28) 0.731 52 × 2 = 1 + 0.463 04;
    • 29) 0.463 04 × 2 = 0 + 0.926 08;
    • 30) 0.926 08 × 2 = 1 + 0.852 16;
    • 31) 0.852 16 × 2 = 1 + 0.704 32;
    • 32) 0.704 32 × 2 = 1 + 0.408 64;
    • 33) 0.408 64 × 2 = 0 + 0.817 28;
    • 34) 0.817 28 × 2 = 1 + 0.634 56;
    • 35) 0.634 56 × 2 = 1 + 0.269 12;
    • 36) 0.269 12 × 2 = 0 + 0.538 24;
    • 37) 0.538 24 × 2 = 1 + 0.076 48;
    • 38) 0.076 48 × 2 = 0 + 0.152 96;
    • 39) 0.152 96 × 2 = 0 + 0.305 92;
    • 40) 0.305 92 × 2 = 0 + 0.611 84;
    • 41) 0.611 84 × 2 = 1 + 0.223 68;
    • 42) 0.223 68 × 2 = 0 + 0.447 36;
    • 43) 0.447 36 × 2 = 0 + 0.894 72;
    • 44) 0.894 72 × 2 = 1 + 0.789 44;
    • 45) 0.789 44 × 2 = 1 + 0.578 88;
    • 46) 0.578 88 × 2 = 1 + 0.157 76;
    • 47) 0.157 76 × 2 = 0 + 0.315 52;
    • 48) 0.315 52 × 2 = 0 + 0.631 04;
    • 49) 0.631 04 × 2 = 1 + 0.262 08;
    • 50) 0.262 08 × 2 = 0 + 0.524 16;
    • 51) 0.524 16 × 2 = 1 + 0.048 32;
    • 52) 0.048 32 × 2 = 0 + 0.096 64;
    • 53) 0.096 64 × 2 = 0 + 0.193 28;
    • We didn't get any fractional part that was equal to zero. But we had enough iterations (over Mantissa limit = 52) and at least one integer part that was different from zero => FULL STOP (losing precision...).
  • 5. Construct the base 2 representation of the fractional part of the number, by taking all the integer parts of the previous multiplying operations, starting from the top of the constructed list above:

    0.640 215(10) = 0.1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2)

  • 6. Summarizing - the positive number before normalization:

    31.640 215(10) = 1 1111.1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2)

  • 7. Normalize the binary representation of the number, shifting the decimal mark 4 positions to the left so that only one non-zero digit stays to the left of the decimal mark:

    31.640 215(10) =
    1 1111.1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2) =
    1 1111.1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2) × 20 =
    1.1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2) × 24

  • 8. Up to this moment, there are the following elements that would feed into the 64 bit double precision IEEE 754 binary floating point representation:

    Sign: 1 (a negative number)

    Exponent (unadjusted): 4

    Mantissa (not-normalized): 1.1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0

  • 9. Adjust the exponent in 11 bit excess/bias notation and then convert it from decimal (base 10) to 11 bit binary (base 2), by using the same technique of repeatedly dividing it by 2, as shown above:

    Exponent (adjusted) = Exponent (unadjusted) + 2(11-1) - 1 = (4 + 1023)(10) = 1027(10) =
    100 0000 0011(2)

  • 10. Normalize mantissa, remove the leading (leftmost) bit, since it's allways '1' (and the decimal sign) and adjust its length to 52 bits, by removing the excess bits, from the right (losing precision...):

    Mantissa (not-normalized): 1.1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0

    Mantissa (normalized): 1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100

  • Conclusion:

    Sign (1 bit) = 1 (a negative number)

    Exponent (8 bits) = 100 0000 0011

    Mantissa (52 bits) = 1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100

  • Number -31.640 215, converted from decimal system (base 10) to 64 bit double precision IEEE 754 binary floating point =
    1 - 100 0000 0011 - 1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100