0.000 000 000 000 000 000 008 536 1 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 000 000 008 536 1(10) to 64 bit double precision IEEE 754 binary floating point representation standard (1 bit for sign, 11 bits for exponent, 52 bits for mantissa)

What are the steps to convert decimal number
0.000 000 000 000 000 000 008 536 1(10) to 64 bit double precision IEEE 754 binary floating point representation (1 bit for sign, 11 bits for exponent, 52 bits for mantissa)

1. First, convert to binary (in base 2) the integer part: 0.
Divide the number repeatedly by 2.

Keep track of each remainder.

We stop when we get a quotient that is equal to zero.


  • division = quotient + remainder;
  • 0 ÷ 2 = 0 + 0;

2. Construct the base 2 representation of the integer part of the number.

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

0(10) =

0(2)


3. Convert to binary (base 2) the fractional part: 0.000 000 000 000 000 000 008 536 1.

Multiply it repeatedly by 2.

Keep track of each integer part of the results.

Stop when we get a fractional part that is equal to zero.


  • #) multiplying = integer + fractional part;
  • 1) 0.000 000 000 000 000 000 008 536 1 × 2 = 0 + 0.000 000 000 000 000 000 017 072 2;
  • 2) 0.000 000 000 000 000 000 017 072 2 × 2 = 0 + 0.000 000 000 000 000 000 034 144 4;
  • 3) 0.000 000 000 000 000 000 034 144 4 × 2 = 0 + 0.000 000 000 000 000 000 068 288 8;
  • 4) 0.000 000 000 000 000 000 068 288 8 × 2 = 0 + 0.000 000 000 000 000 000 136 577 6;
  • 5) 0.000 000 000 000 000 000 136 577 6 × 2 = 0 + 0.000 000 000 000 000 000 273 155 2;
  • 6) 0.000 000 000 000 000 000 273 155 2 × 2 = 0 + 0.000 000 000 000 000 000 546 310 4;
  • 7) 0.000 000 000 000 000 000 546 310 4 × 2 = 0 + 0.000 000 000 000 000 001 092 620 8;
  • 8) 0.000 000 000 000 000 001 092 620 8 × 2 = 0 + 0.000 000 000 000 000 002 185 241 6;
  • 9) 0.000 000 000 000 000 002 185 241 6 × 2 = 0 + 0.000 000 000 000 000 004 370 483 2;
  • 10) 0.000 000 000 000 000 004 370 483 2 × 2 = 0 + 0.000 000 000 000 000 008 740 966 4;
  • 11) 0.000 000 000 000 000 008 740 966 4 × 2 = 0 + 0.000 000 000 000 000 017 481 932 8;
  • 12) 0.000 000 000 000 000 017 481 932 8 × 2 = 0 + 0.000 000 000 000 000 034 963 865 6;
  • 13) 0.000 000 000 000 000 034 963 865 6 × 2 = 0 + 0.000 000 000 000 000 069 927 731 2;
  • 14) 0.000 000 000 000 000 069 927 731 2 × 2 = 0 + 0.000 000 000 000 000 139 855 462 4;
  • 15) 0.000 000 000 000 000 139 855 462 4 × 2 = 0 + 0.000 000 000 000 000 279 710 924 8;
  • 16) 0.000 000 000 000 000 279 710 924 8 × 2 = 0 + 0.000 000 000 000 000 559 421 849 6;
  • 17) 0.000 000 000 000 000 559 421 849 6 × 2 = 0 + 0.000 000 000 000 001 118 843 699 2;
  • 18) 0.000 000 000 000 001 118 843 699 2 × 2 = 0 + 0.000 000 000 000 002 237 687 398 4;
  • 19) 0.000 000 000 000 002 237 687 398 4 × 2 = 0 + 0.000 000 000 000 004 475 374 796 8;
  • 20) 0.000 000 000 000 004 475 374 796 8 × 2 = 0 + 0.000 000 000 000 008 950 749 593 6;
  • 21) 0.000 000 000 000 008 950 749 593 6 × 2 = 0 + 0.000 000 000 000 017 901 499 187 2;
  • 22) 0.000 000 000 000 017 901 499 187 2 × 2 = 0 + 0.000 000 000 000 035 802 998 374 4;
  • 23) 0.000 000 000 000 035 802 998 374 4 × 2 = 0 + 0.000 000 000 000 071 605 996 748 8;
  • 24) 0.000 000 000 000 071 605 996 748 8 × 2 = 0 + 0.000 000 000 000 143 211 993 497 6;
  • 25) 0.000 000 000 000 143 211 993 497 6 × 2 = 0 + 0.000 000 000 000 286 423 986 995 2;
  • 26) 0.000 000 000 000 286 423 986 995 2 × 2 = 0 + 0.000 000 000 000 572 847 973 990 4;
  • 27) 0.000 000 000 000 572 847 973 990 4 × 2 = 0 + 0.000 000 000 001 145 695 947 980 8;
  • 28) 0.000 000 000 001 145 695 947 980 8 × 2 = 0 + 0.000 000 000 002 291 391 895 961 6;
  • 29) 0.000 000 000 002 291 391 895 961 6 × 2 = 0 + 0.000 000 000 004 582 783 791 923 2;
  • 30) 0.000 000 000 004 582 783 791 923 2 × 2 = 0 + 0.000 000 000 009 165 567 583 846 4;
  • 31) 0.000 000 000 009 165 567 583 846 4 × 2 = 0 + 0.000 000 000 018 331 135 167 692 8;
  • 32) 0.000 000 000 018 331 135 167 692 8 × 2 = 0 + 0.000 000 000 036 662 270 335 385 6;
  • 33) 0.000 000 000 036 662 270 335 385 6 × 2 = 0 + 0.000 000 000 073 324 540 670 771 2;
  • 34) 0.000 000 000 073 324 540 670 771 2 × 2 = 0 + 0.000 000 000 146 649 081 341 542 4;
  • 35) 0.000 000 000 146 649 081 341 542 4 × 2 = 0 + 0.000 000 000 293 298 162 683 084 8;
  • 36) 0.000 000 000 293 298 162 683 084 8 × 2 = 0 + 0.000 000 000 586 596 325 366 169 6;
  • 37) 0.000 000 000 586 596 325 366 169 6 × 2 = 0 + 0.000 000 001 173 192 650 732 339 2;
  • 38) 0.000 000 001 173 192 650 732 339 2 × 2 = 0 + 0.000 000 002 346 385 301 464 678 4;
  • 39) 0.000 000 002 346 385 301 464 678 4 × 2 = 0 + 0.000 000 004 692 770 602 929 356 8;
  • 40) 0.000 000 004 692 770 602 929 356 8 × 2 = 0 + 0.000 000 009 385 541 205 858 713 6;
  • 41) 0.000 000 009 385 541 205 858 713 6 × 2 = 0 + 0.000 000 018 771 082 411 717 427 2;
  • 42) 0.000 000 018 771 082 411 717 427 2 × 2 = 0 + 0.000 000 037 542 164 823 434 854 4;
  • 43) 0.000 000 037 542 164 823 434 854 4 × 2 = 0 + 0.000 000 075 084 329 646 869 708 8;
  • 44) 0.000 000 075 084 329 646 869 708 8 × 2 = 0 + 0.000 000 150 168 659 293 739 417 6;
  • 45) 0.000 000 150 168 659 293 739 417 6 × 2 = 0 + 0.000 000 300 337 318 587 478 835 2;
  • 46) 0.000 000 300 337 318 587 478 835 2 × 2 = 0 + 0.000 000 600 674 637 174 957 670 4;
  • 47) 0.000 000 600 674 637 174 957 670 4 × 2 = 0 + 0.000 001 201 349 274 349 915 340 8;
  • 48) 0.000 001 201 349 274 349 915 340 8 × 2 = 0 + 0.000 002 402 698 548 699 830 681 6;
  • 49) 0.000 002 402 698 548 699 830 681 6 × 2 = 0 + 0.000 004 805 397 097 399 661 363 2;
  • 50) 0.000 004 805 397 097 399 661 363 2 × 2 = 0 + 0.000 009 610 794 194 799 322 726 4;
  • 51) 0.000 009 610 794 194 799 322 726 4 × 2 = 0 + 0.000 019 221 588 389 598 645 452 8;
  • 52) 0.000 019 221 588 389 598 645 452 8 × 2 = 0 + 0.000 038 443 176 779 197 290 905 6;
  • 53) 0.000 038 443 176 779 197 290 905 6 × 2 = 0 + 0.000 076 886 353 558 394 581 811 2;
  • 54) 0.000 076 886 353 558 394 581 811 2 × 2 = 0 + 0.000 153 772 707 116 789 163 622 4;
  • 55) 0.000 153 772 707 116 789 163 622 4 × 2 = 0 + 0.000 307 545 414 233 578 327 244 8;
  • 56) 0.000 307 545 414 233 578 327 244 8 × 2 = 0 + 0.000 615 090 828 467 156 654 489 6;
  • 57) 0.000 615 090 828 467 156 654 489 6 × 2 = 0 + 0.001 230 181 656 934 313 308 979 2;
  • 58) 0.001 230 181 656 934 313 308 979 2 × 2 = 0 + 0.002 460 363 313 868 626 617 958 4;
  • 59) 0.002 460 363 313 868 626 617 958 4 × 2 = 0 + 0.004 920 726 627 737 253 235 916 8;
  • 60) 0.004 920 726 627 737 253 235 916 8 × 2 = 0 + 0.009 841 453 255 474 506 471 833 6;
  • 61) 0.009 841 453 255 474 506 471 833 6 × 2 = 0 + 0.019 682 906 510 949 012 943 667 2;
  • 62) 0.019 682 906 510 949 012 943 667 2 × 2 = 0 + 0.039 365 813 021 898 025 887 334 4;
  • 63) 0.039 365 813 021 898 025 887 334 4 × 2 = 0 + 0.078 731 626 043 796 051 774 668 8;
  • 64) 0.078 731 626 043 796 051 774 668 8 × 2 = 0 + 0.157 463 252 087 592 103 549 337 6;
  • 65) 0.157 463 252 087 592 103 549 337 6 × 2 = 0 + 0.314 926 504 175 184 207 098 675 2;
  • 66) 0.314 926 504 175 184 207 098 675 2 × 2 = 0 + 0.629 853 008 350 368 414 197 350 4;
  • 67) 0.629 853 008 350 368 414 197 350 4 × 2 = 1 + 0.259 706 016 700 736 828 394 700 8;
  • 68) 0.259 706 016 700 736 828 394 700 8 × 2 = 0 + 0.519 412 033 401 473 656 789 401 6;
  • 69) 0.519 412 033 401 473 656 789 401 6 × 2 = 1 + 0.038 824 066 802 947 313 578 803 2;
  • 70) 0.038 824 066 802 947 313 578 803 2 × 2 = 0 + 0.077 648 133 605 894 627 157 606 4;
  • 71) 0.077 648 133 605 894 627 157 606 4 × 2 = 0 + 0.155 296 267 211 789 254 315 212 8;
  • 72) 0.155 296 267 211 789 254 315 212 8 × 2 = 0 + 0.310 592 534 423 578 508 630 425 6;
  • 73) 0.310 592 534 423 578 508 630 425 6 × 2 = 0 + 0.621 185 068 847 157 017 260 851 2;
  • 74) 0.621 185 068 847 157 017 260 851 2 × 2 = 1 + 0.242 370 137 694 314 034 521 702 4;
  • 75) 0.242 370 137 694 314 034 521 702 4 × 2 = 0 + 0.484 740 275 388 628 069 043 404 8;
  • 76) 0.484 740 275 388 628 069 043 404 8 × 2 = 0 + 0.969 480 550 777 256 138 086 809 6;
  • 77) 0.969 480 550 777 256 138 086 809 6 × 2 = 1 + 0.938 961 101 554 512 276 173 619 2;
  • 78) 0.938 961 101 554 512 276 173 619 2 × 2 = 1 + 0.877 922 203 109 024 552 347 238 4;
  • 79) 0.877 922 203 109 024 552 347 238 4 × 2 = 1 + 0.755 844 406 218 049 104 694 476 8;
  • 80) 0.755 844 406 218 049 104 694 476 8 × 2 = 1 + 0.511 688 812 436 098 209 388 953 6;
  • 81) 0.511 688 812 436 098 209 388 953 6 × 2 = 1 + 0.023 377 624 872 196 418 777 907 2;
  • 82) 0.023 377 624 872 196 418 777 907 2 × 2 = 0 + 0.046 755 249 744 392 837 555 814 4;
  • 83) 0.046 755 249 744 392 837 555 814 4 × 2 = 0 + 0.093 510 499 488 785 675 111 628 8;
  • 84) 0.093 510 499 488 785 675 111 628 8 × 2 = 0 + 0.187 020 998 977 571 350 223 257 6;
  • 85) 0.187 020 998 977 571 350 223 257 6 × 2 = 0 + 0.374 041 997 955 142 700 446 515 2;
  • 86) 0.374 041 997 955 142 700 446 515 2 × 2 = 0 + 0.748 083 995 910 285 400 893 030 4;
  • 87) 0.748 083 995 910 285 400 893 030 4 × 2 = 1 + 0.496 167 991 820 570 801 786 060 8;
  • 88) 0.496 167 991 820 570 801 786 060 8 × 2 = 0 + 0.992 335 983 641 141 603 572 121 6;
  • 89) 0.992 335 983 641 141 603 572 121 6 × 2 = 1 + 0.984 671 967 282 283 207 144 243 2;
  • 90) 0.984 671 967 282 283 207 144 243 2 × 2 = 1 + 0.969 343 934 564 566 414 288 486 4;
  • 91) 0.969 343 934 564 566 414 288 486 4 × 2 = 1 + 0.938 687 869 129 132 828 576 972 8;
  • 92) 0.938 687 869 129 132 828 576 972 8 × 2 = 1 + 0.877 375 738 258 265 657 153 945 6;
  • 93) 0.877 375 738 258 265 657 153 945 6 × 2 = 1 + 0.754 751 476 516 531 314 307 891 2;
  • 94) 0.754 751 476 516 531 314 307 891 2 × 2 = 1 + 0.509 502 953 033 062 628 615 782 4;
  • 95) 0.509 502 953 033 062 628 615 782 4 × 2 = 1 + 0.019 005 906 066 125 257 231 564 8;
  • 96) 0.019 005 906 066 125 257 231 564 8 × 2 = 0 + 0.038 011 812 132 250 514 463 129 6;
  • 97) 0.038 011 812 132 250 514 463 129 6 × 2 = 0 + 0.076 023 624 264 501 028 926 259 2;
  • 98) 0.076 023 624 264 501 028 926 259 2 × 2 = 0 + 0.152 047 248 529 002 057 852 518 4;
  • 99) 0.152 047 248 529 002 057 852 518 4 × 2 = 0 + 0.304 094 497 058 004 115 705 036 8;
  • 100) 0.304 094 497 058 004 115 705 036 8 × 2 = 0 + 0.608 188 994 116 008 231 410 073 6;
  • 101) 0.608 188 994 116 008 231 410 073 6 × 2 = 1 + 0.216 377 988 232 016 462 820 147 2;
  • 102) 0.216 377 988 232 016 462 820 147 2 × 2 = 0 + 0.432 755 976 464 032 925 640 294 4;
  • 103) 0.432 755 976 464 032 925 640 294 4 × 2 = 0 + 0.865 511 952 928 065 851 280 588 8;
  • 104) 0.865 511 952 928 065 851 280 588 8 × 2 = 1 + 0.731 023 905 856 131 702 561 177 6;
  • 105) 0.731 023 905 856 131 702 561 177 6 × 2 = 1 + 0.462 047 811 712 263 405 122 355 2;
  • 106) 0.462 047 811 712 263 405 122 355 2 × 2 = 0 + 0.924 095 623 424 526 810 244 710 4;
  • 107) 0.924 095 623 424 526 810 244 710 4 × 2 = 1 + 0.848 191 246 849 053 620 489 420 8;
  • 108) 0.848 191 246 849 053 620 489 420 8 × 2 = 1 + 0.696 382 493 698 107 240 978 841 6;
  • 109) 0.696 382 493 698 107 240 978 841 6 × 2 = 1 + 0.392 764 987 396 214 481 957 683 2;
  • 110) 0.392 764 987 396 214 481 957 683 2 × 2 = 0 + 0.785 529 974 792 428 963 915 366 4;
  • 111) 0.785 529 974 792 428 963 915 366 4 × 2 = 1 + 0.571 059 949 584 857 927 830 732 8;
  • 112) 0.571 059 949 584 857 927 830 732 8 × 2 = 1 + 0.142 119 899 169 715 855 661 465 6;
  • 113) 0.142 119 899 169 715 855 661 465 6 × 2 = 0 + 0.284 239 798 339 431 711 322 931 2;
  • 114) 0.284 239 798 339 431 711 322 931 2 × 2 = 0 + 0.568 479 596 678 863 422 645 862 4;
  • 115) 0.568 479 596 678 863 422 645 862 4 × 2 = 1 + 0.136 959 193 357 726 845 291 724 8;
  • 116) 0.136 959 193 357 726 845 291 724 8 × 2 = 0 + 0.273 918 386 715 453 690 583 449 6;
  • 117) 0.273 918 386 715 453 690 583 449 6 × 2 = 0 + 0.547 836 773 430 907 381 166 899 2;
  • 118) 0.547 836 773 430 907 381 166 899 2 × 2 = 1 + 0.095 673 546 861 814 762 333 798 4;
  • 119) 0.095 673 546 861 814 762 333 798 4 × 2 = 0 + 0.191 347 093 723 629 524 667 596 8;

We didn't get any fractional part that was equal to zero. But we had enough iterations (over Mantissa limit) and at least one integer that was different from zero => FULL STOP (Losing precision - the converted number we get in the end will be just a very good approximation of the initial one).


4. Construct the base 2 representation of the fractional part of the number.

Take all the integer parts of the multiplying operations, starting from the top of the constructed list above:


0.000 000 000 000 000 000 008 536 1(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1111 1000 0010 1111 1110 0000 1001 1011 1011 0010 010(2)

5. Positive number before normalization:

0.000 000 000 000 000 000 008 536 1(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1111 1000 0010 1111 1110 0000 1001 1011 1011 0010 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:


0.000 000 000 000 000 000 008 536 1(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1111 1000 0010 1111 1110 0000 1001 1011 1011 0010 010(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1111 1000 0010 1111 1110 0000 1001 1011 1011 0010 010(2) × 20 =

1.0100 0010 0111 1100 0001 0111 1111 0000 0100 1101 1101 1001 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 0111 1100 0001 0111 1111 0000 0100 1101 1101 1001 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 0111 1100 0001 0111 1111 0000 0100 1101 1101 1001 0010 =

0100 0010 0111 1100 0001 0111 1111 0000 0100 1101 1101 1001 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 0111 1100 0001 0111 1111 0000 0100 1101 1101 1001 0010


Decimal number 0.000 000 000 000 000 000 008 536 1 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1011 1100 - 0100 0010 0111 1100 0001 0111 1111 0000 0100 1101 1101 1001 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