0.000 000 000 000 000 000 008 536 37 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 000 000 008 536 37(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 37(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 37.

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 37 × 2 = 0 + 0.000 000 000 000 000 000 017 072 74;
  • 2) 0.000 000 000 000 000 000 017 072 74 × 2 = 0 + 0.000 000 000 000 000 000 034 145 48;
  • 3) 0.000 000 000 000 000 000 034 145 48 × 2 = 0 + 0.000 000 000 000 000 000 068 290 96;
  • 4) 0.000 000 000 000 000 000 068 290 96 × 2 = 0 + 0.000 000 000 000 000 000 136 581 92;
  • 5) 0.000 000 000 000 000 000 136 581 92 × 2 = 0 + 0.000 000 000 000 000 000 273 163 84;
  • 6) 0.000 000 000 000 000 000 273 163 84 × 2 = 0 + 0.000 000 000 000 000 000 546 327 68;
  • 7) 0.000 000 000 000 000 000 546 327 68 × 2 = 0 + 0.000 000 000 000 000 001 092 655 36;
  • 8) 0.000 000 000 000 000 001 092 655 36 × 2 = 0 + 0.000 000 000 000 000 002 185 310 72;
  • 9) 0.000 000 000 000 000 002 185 310 72 × 2 = 0 + 0.000 000 000 000 000 004 370 621 44;
  • 10) 0.000 000 000 000 000 004 370 621 44 × 2 = 0 + 0.000 000 000 000 000 008 741 242 88;
  • 11) 0.000 000 000 000 000 008 741 242 88 × 2 = 0 + 0.000 000 000 000 000 017 482 485 76;
  • 12) 0.000 000 000 000 000 017 482 485 76 × 2 = 0 + 0.000 000 000 000 000 034 964 971 52;
  • 13) 0.000 000 000 000 000 034 964 971 52 × 2 = 0 + 0.000 000 000 000 000 069 929 943 04;
  • 14) 0.000 000 000 000 000 069 929 943 04 × 2 = 0 + 0.000 000 000 000 000 139 859 886 08;
  • 15) 0.000 000 000 000 000 139 859 886 08 × 2 = 0 + 0.000 000 000 000 000 279 719 772 16;
  • 16) 0.000 000 000 000 000 279 719 772 16 × 2 = 0 + 0.000 000 000 000 000 559 439 544 32;
  • 17) 0.000 000 000 000 000 559 439 544 32 × 2 = 0 + 0.000 000 000 000 001 118 879 088 64;
  • 18) 0.000 000 000 000 001 118 879 088 64 × 2 = 0 + 0.000 000 000 000 002 237 758 177 28;
  • 19) 0.000 000 000 000 002 237 758 177 28 × 2 = 0 + 0.000 000 000 000 004 475 516 354 56;
  • 20) 0.000 000 000 000 004 475 516 354 56 × 2 = 0 + 0.000 000 000 000 008 951 032 709 12;
  • 21) 0.000 000 000 000 008 951 032 709 12 × 2 = 0 + 0.000 000 000 000 017 902 065 418 24;
  • 22) 0.000 000 000 000 017 902 065 418 24 × 2 = 0 + 0.000 000 000 000 035 804 130 836 48;
  • 23) 0.000 000 000 000 035 804 130 836 48 × 2 = 0 + 0.000 000 000 000 071 608 261 672 96;
  • 24) 0.000 000 000 000 071 608 261 672 96 × 2 = 0 + 0.000 000 000 000 143 216 523 345 92;
  • 25) 0.000 000 000 000 143 216 523 345 92 × 2 = 0 + 0.000 000 000 000 286 433 046 691 84;
  • 26) 0.000 000 000 000 286 433 046 691 84 × 2 = 0 + 0.000 000 000 000 572 866 093 383 68;
  • 27) 0.000 000 000 000 572 866 093 383 68 × 2 = 0 + 0.000 000 000 001 145 732 186 767 36;
  • 28) 0.000 000 000 001 145 732 186 767 36 × 2 = 0 + 0.000 000 000 002 291 464 373 534 72;
  • 29) 0.000 000 000 002 291 464 373 534 72 × 2 = 0 + 0.000 000 000 004 582 928 747 069 44;
  • 30) 0.000 000 000 004 582 928 747 069 44 × 2 = 0 + 0.000 000 000 009 165 857 494 138 88;
  • 31) 0.000 000 000 009 165 857 494 138 88 × 2 = 0 + 0.000 000 000 018 331 714 988 277 76;
  • 32) 0.000 000 000 018 331 714 988 277 76 × 2 = 0 + 0.000 000 000 036 663 429 976 555 52;
  • 33) 0.000 000 000 036 663 429 976 555 52 × 2 = 0 + 0.000 000 000 073 326 859 953 111 04;
  • 34) 0.000 000 000 073 326 859 953 111 04 × 2 = 0 + 0.000 000 000 146 653 719 906 222 08;
  • 35) 0.000 000 000 146 653 719 906 222 08 × 2 = 0 + 0.000 000 000 293 307 439 812 444 16;
  • 36) 0.000 000 000 293 307 439 812 444 16 × 2 = 0 + 0.000 000 000 586 614 879 624 888 32;
  • 37) 0.000 000 000 586 614 879 624 888 32 × 2 = 0 + 0.000 000 001 173 229 759 249 776 64;
  • 38) 0.000 000 001 173 229 759 249 776 64 × 2 = 0 + 0.000 000 002 346 459 518 499 553 28;
  • 39) 0.000 000 002 346 459 518 499 553 28 × 2 = 0 + 0.000 000 004 692 919 036 999 106 56;
  • 40) 0.000 000 004 692 919 036 999 106 56 × 2 = 0 + 0.000 000 009 385 838 073 998 213 12;
  • 41) 0.000 000 009 385 838 073 998 213 12 × 2 = 0 + 0.000 000 018 771 676 147 996 426 24;
  • 42) 0.000 000 018 771 676 147 996 426 24 × 2 = 0 + 0.000 000 037 543 352 295 992 852 48;
  • 43) 0.000 000 037 543 352 295 992 852 48 × 2 = 0 + 0.000 000 075 086 704 591 985 704 96;
  • 44) 0.000 000 075 086 704 591 985 704 96 × 2 = 0 + 0.000 000 150 173 409 183 971 409 92;
  • 45) 0.000 000 150 173 409 183 971 409 92 × 2 = 0 + 0.000 000 300 346 818 367 942 819 84;
  • 46) 0.000 000 300 346 818 367 942 819 84 × 2 = 0 + 0.000 000 600 693 636 735 885 639 68;
  • 47) 0.000 000 600 693 636 735 885 639 68 × 2 = 0 + 0.000 001 201 387 273 471 771 279 36;
  • 48) 0.000 001 201 387 273 471 771 279 36 × 2 = 0 + 0.000 002 402 774 546 943 542 558 72;
  • 49) 0.000 002 402 774 546 943 542 558 72 × 2 = 0 + 0.000 004 805 549 093 887 085 117 44;
  • 50) 0.000 004 805 549 093 887 085 117 44 × 2 = 0 + 0.000 009 611 098 187 774 170 234 88;
  • 51) 0.000 009 611 098 187 774 170 234 88 × 2 = 0 + 0.000 019 222 196 375 548 340 469 76;
  • 52) 0.000 019 222 196 375 548 340 469 76 × 2 = 0 + 0.000 038 444 392 751 096 680 939 52;
  • 53) 0.000 038 444 392 751 096 680 939 52 × 2 = 0 + 0.000 076 888 785 502 193 361 879 04;
  • 54) 0.000 076 888 785 502 193 361 879 04 × 2 = 0 + 0.000 153 777 571 004 386 723 758 08;
  • 55) 0.000 153 777 571 004 386 723 758 08 × 2 = 0 + 0.000 307 555 142 008 773 447 516 16;
  • 56) 0.000 307 555 142 008 773 447 516 16 × 2 = 0 + 0.000 615 110 284 017 546 895 032 32;
  • 57) 0.000 615 110 284 017 546 895 032 32 × 2 = 0 + 0.001 230 220 568 035 093 790 064 64;
  • 58) 0.001 230 220 568 035 093 790 064 64 × 2 = 0 + 0.002 460 441 136 070 187 580 129 28;
  • 59) 0.002 460 441 136 070 187 580 129 28 × 2 = 0 + 0.004 920 882 272 140 375 160 258 56;
  • 60) 0.004 920 882 272 140 375 160 258 56 × 2 = 0 + 0.009 841 764 544 280 750 320 517 12;
  • 61) 0.009 841 764 544 280 750 320 517 12 × 2 = 0 + 0.019 683 529 088 561 500 641 034 24;
  • 62) 0.019 683 529 088 561 500 641 034 24 × 2 = 0 + 0.039 367 058 177 123 001 282 068 48;
  • 63) 0.039 367 058 177 123 001 282 068 48 × 2 = 0 + 0.078 734 116 354 246 002 564 136 96;
  • 64) 0.078 734 116 354 246 002 564 136 96 × 2 = 0 + 0.157 468 232 708 492 005 128 273 92;
  • 65) 0.157 468 232 708 492 005 128 273 92 × 2 = 0 + 0.314 936 465 416 984 010 256 547 84;
  • 66) 0.314 936 465 416 984 010 256 547 84 × 2 = 0 + 0.629 872 930 833 968 020 513 095 68;
  • 67) 0.629 872 930 833 968 020 513 095 68 × 2 = 1 + 0.259 745 861 667 936 041 026 191 36;
  • 68) 0.259 745 861 667 936 041 026 191 36 × 2 = 0 + 0.519 491 723 335 872 082 052 382 72;
  • 69) 0.519 491 723 335 872 082 052 382 72 × 2 = 1 + 0.038 983 446 671 744 164 104 765 44;
  • 70) 0.038 983 446 671 744 164 104 765 44 × 2 = 0 + 0.077 966 893 343 488 328 209 530 88;
  • 71) 0.077 966 893 343 488 328 209 530 88 × 2 = 0 + 0.155 933 786 686 976 656 419 061 76;
  • 72) 0.155 933 786 686 976 656 419 061 76 × 2 = 0 + 0.311 867 573 373 953 312 838 123 52;
  • 73) 0.311 867 573 373 953 312 838 123 52 × 2 = 0 + 0.623 735 146 747 906 625 676 247 04;
  • 74) 0.623 735 146 747 906 625 676 247 04 × 2 = 1 + 0.247 470 293 495 813 251 352 494 08;
  • 75) 0.247 470 293 495 813 251 352 494 08 × 2 = 0 + 0.494 940 586 991 626 502 704 988 16;
  • 76) 0.494 940 586 991 626 502 704 988 16 × 2 = 0 + 0.989 881 173 983 253 005 409 976 32;
  • 77) 0.989 881 173 983 253 005 409 976 32 × 2 = 1 + 0.979 762 347 966 506 010 819 952 64;
  • 78) 0.979 762 347 966 506 010 819 952 64 × 2 = 1 + 0.959 524 695 933 012 021 639 905 28;
  • 79) 0.959 524 695 933 012 021 639 905 28 × 2 = 1 + 0.919 049 391 866 024 043 279 810 56;
  • 80) 0.919 049 391 866 024 043 279 810 56 × 2 = 1 + 0.838 098 783 732 048 086 559 621 12;
  • 81) 0.838 098 783 732 048 086 559 621 12 × 2 = 1 + 0.676 197 567 464 096 173 119 242 24;
  • 82) 0.676 197 567 464 096 173 119 242 24 × 2 = 1 + 0.352 395 134 928 192 346 238 484 48;
  • 83) 0.352 395 134 928 192 346 238 484 48 × 2 = 0 + 0.704 790 269 856 384 692 476 968 96;
  • 84) 0.704 790 269 856 384 692 476 968 96 × 2 = 1 + 0.409 580 539 712 769 384 953 937 92;
  • 85) 0.409 580 539 712 769 384 953 937 92 × 2 = 0 + 0.819 161 079 425 538 769 907 875 84;
  • 86) 0.819 161 079 425 538 769 907 875 84 × 2 = 1 + 0.638 322 158 851 077 539 815 751 68;
  • 87) 0.638 322 158 851 077 539 815 751 68 × 2 = 1 + 0.276 644 317 702 155 079 631 503 36;
  • 88) 0.276 644 317 702 155 079 631 503 36 × 2 = 0 + 0.553 288 635 404 310 159 263 006 72;
  • 89) 0.553 288 635 404 310 159 263 006 72 × 2 = 1 + 0.106 577 270 808 620 318 526 013 44;
  • 90) 0.106 577 270 808 620 318 526 013 44 × 2 = 0 + 0.213 154 541 617 240 637 052 026 88;
  • 91) 0.213 154 541 617 240 637 052 026 88 × 2 = 0 + 0.426 309 083 234 481 274 104 053 76;
  • 92) 0.426 309 083 234 481 274 104 053 76 × 2 = 0 + 0.852 618 166 468 962 548 208 107 52;
  • 93) 0.852 618 166 468 962 548 208 107 52 × 2 = 1 + 0.705 236 332 937 925 096 416 215 04;
  • 94) 0.705 236 332 937 925 096 416 215 04 × 2 = 1 + 0.410 472 665 875 850 192 832 430 08;
  • 95) 0.410 472 665 875 850 192 832 430 08 × 2 = 0 + 0.820 945 331 751 700 385 664 860 16;
  • 96) 0.820 945 331 751 700 385 664 860 16 × 2 = 1 + 0.641 890 663 503 400 771 329 720 32;
  • 97) 0.641 890 663 503 400 771 329 720 32 × 2 = 1 + 0.283 781 327 006 801 542 659 440 64;
  • 98) 0.283 781 327 006 801 542 659 440 64 × 2 = 0 + 0.567 562 654 013 603 085 318 881 28;
  • 99) 0.567 562 654 013 603 085 318 881 28 × 2 = 1 + 0.135 125 308 027 206 170 637 762 56;
  • 100) 0.135 125 308 027 206 170 637 762 56 × 2 = 0 + 0.270 250 616 054 412 341 275 525 12;
  • 101) 0.270 250 616 054 412 341 275 525 12 × 2 = 0 + 0.540 501 232 108 824 682 551 050 24;
  • 102) 0.540 501 232 108 824 682 551 050 24 × 2 = 1 + 0.081 002 464 217 649 365 102 100 48;
  • 103) 0.081 002 464 217 649 365 102 100 48 × 2 = 0 + 0.162 004 928 435 298 730 204 200 96;
  • 104) 0.162 004 928 435 298 730 204 200 96 × 2 = 0 + 0.324 009 856 870 597 460 408 401 92;
  • 105) 0.324 009 856 870 597 460 408 401 92 × 2 = 0 + 0.648 019 713 741 194 920 816 803 84;
  • 106) 0.648 019 713 741 194 920 816 803 84 × 2 = 1 + 0.296 039 427 482 389 841 633 607 68;
  • 107) 0.296 039 427 482 389 841 633 607 68 × 2 = 0 + 0.592 078 854 964 779 683 267 215 36;
  • 108) 0.592 078 854 964 779 683 267 215 36 × 2 = 1 + 0.184 157 709 929 559 366 534 430 72;
  • 109) 0.184 157 709 929 559 366 534 430 72 × 2 = 0 + 0.368 315 419 859 118 733 068 861 44;
  • 110) 0.368 315 419 859 118 733 068 861 44 × 2 = 0 + 0.736 630 839 718 237 466 137 722 88;
  • 111) 0.736 630 839 718 237 466 137 722 88 × 2 = 1 + 0.473 261 679 436 474 932 275 445 76;
  • 112) 0.473 261 679 436 474 932 275 445 76 × 2 = 0 + 0.946 523 358 872 949 864 550 891 52;
  • 113) 0.946 523 358 872 949 864 550 891 52 × 2 = 1 + 0.893 046 717 745 899 729 101 783 04;
  • 114) 0.893 046 717 745 899 729 101 783 04 × 2 = 1 + 0.786 093 435 491 799 458 203 566 08;
  • 115) 0.786 093 435 491 799 458 203 566 08 × 2 = 1 + 0.572 186 870 983 598 916 407 132 16;
  • 116) 0.572 186 870 983 598 916 407 132 16 × 2 = 1 + 0.144 373 741 967 197 832 814 264 32;
  • 117) 0.144 373 741 967 197 832 814 264 32 × 2 = 0 + 0.288 747 483 934 395 665 628 528 64;
  • 118) 0.288 747 483 934 395 665 628 528 64 × 2 = 0 + 0.577 494 967 868 791 331 257 057 28;
  • 119) 0.577 494 967 868 791 331 257 057 28 × 2 = 1 + 0.154 989 935 737 582 662 514 114 56;

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 37(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1111 1101 0110 1000 1101 1010 0100 0101 0010 1111 001(2)

5. Positive number before normalization:

0.000 000 000 000 000 000 008 536 37(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1111 1101 0110 1000 1101 1010 0100 0101 0010 1111 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:


0.000 000 000 000 000 000 008 536 37(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1111 1101 0110 1000 1101 1010 0100 0101 0010 1111 001(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1111 1101 0110 1000 1101 1010 0100 0101 0010 1111 001(2) × 20 =

1.0100 0010 0111 1110 1011 0100 0110 1101 0010 0010 1001 0111 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 1110 1011 0100 0110 1101 0010 0010 1001 0111 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 1110 1011 0100 0110 1101 0010 0010 1001 0111 1001 =

0100 0010 0111 1110 1011 0100 0110 1101 0010 0010 1001 0111 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 1110 1011 0100 0110 1101 0010 0010 1001 0111 1001


Decimal number 0.000 000 000 000 000 000 008 536 37 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1011 1100 - 0100 0010 0111 1110 1011 0100 0110 1101 0010 0010 1001 0111 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