0.000 000 000 000 000 000 008 533 4 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 000 000 008 533 4(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
0.000 000 000 000 000 000 008 533 4(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: 0.000 000 000 000 000 000 008 533 4.

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) 0.000 000 000 000 000 000 008 533 4 × 2 = 0 + 0.000 000 000 000 000 000 017 066 8;
  • 2) 0.000 000 000 000 000 000 017 066 8 × 2 = 0 + 0.000 000 000 000 000 000 034 133 6;
  • 3) 0.000 000 000 000 000 000 034 133 6 × 2 = 0 + 0.000 000 000 000 000 000 068 267 2;
  • 4) 0.000 000 000 000 000 000 068 267 2 × 2 = 0 + 0.000 000 000 000 000 000 136 534 4;
  • 5) 0.000 000 000 000 000 000 136 534 4 × 2 = 0 + 0.000 000 000 000 000 000 273 068 8;
  • 6) 0.000 000 000 000 000 000 273 068 8 × 2 = 0 + 0.000 000 000 000 000 000 546 137 6;
  • 7) 0.000 000 000 000 000 000 546 137 6 × 2 = 0 + 0.000 000 000 000 000 001 092 275 2;
  • 8) 0.000 000 000 000 000 001 092 275 2 × 2 = 0 + 0.000 000 000 000 000 002 184 550 4;
  • 9) 0.000 000 000 000 000 002 184 550 4 × 2 = 0 + 0.000 000 000 000 000 004 369 100 8;
  • 10) 0.000 000 000 000 000 004 369 100 8 × 2 = 0 + 0.000 000 000 000 000 008 738 201 6;
  • 11) 0.000 000 000 000 000 008 738 201 6 × 2 = 0 + 0.000 000 000 000 000 017 476 403 2;
  • 12) 0.000 000 000 000 000 017 476 403 2 × 2 = 0 + 0.000 000 000 000 000 034 952 806 4;
  • 13) 0.000 000 000 000 000 034 952 806 4 × 2 = 0 + 0.000 000 000 000 000 069 905 612 8;
  • 14) 0.000 000 000 000 000 069 905 612 8 × 2 = 0 + 0.000 000 000 000 000 139 811 225 6;
  • 15) 0.000 000 000 000 000 139 811 225 6 × 2 = 0 + 0.000 000 000 000 000 279 622 451 2;
  • 16) 0.000 000 000 000 000 279 622 451 2 × 2 = 0 + 0.000 000 000 000 000 559 244 902 4;
  • 17) 0.000 000 000 000 000 559 244 902 4 × 2 = 0 + 0.000 000 000 000 001 118 489 804 8;
  • 18) 0.000 000 000 000 001 118 489 804 8 × 2 = 0 + 0.000 000 000 000 002 236 979 609 6;
  • 19) 0.000 000 000 000 002 236 979 609 6 × 2 = 0 + 0.000 000 000 000 004 473 959 219 2;
  • 20) 0.000 000 000 000 004 473 959 219 2 × 2 = 0 + 0.000 000 000 000 008 947 918 438 4;
  • 21) 0.000 000 000 000 008 947 918 438 4 × 2 = 0 + 0.000 000 000 000 017 895 836 876 8;
  • 22) 0.000 000 000 000 017 895 836 876 8 × 2 = 0 + 0.000 000 000 000 035 791 673 753 6;
  • 23) 0.000 000 000 000 035 791 673 753 6 × 2 = 0 + 0.000 000 000 000 071 583 347 507 2;
  • 24) 0.000 000 000 000 071 583 347 507 2 × 2 = 0 + 0.000 000 000 000 143 166 695 014 4;
  • 25) 0.000 000 000 000 143 166 695 014 4 × 2 = 0 + 0.000 000 000 000 286 333 390 028 8;
  • 26) 0.000 000 000 000 286 333 390 028 8 × 2 = 0 + 0.000 000 000 000 572 666 780 057 6;
  • 27) 0.000 000 000 000 572 666 780 057 6 × 2 = 0 + 0.000 000 000 001 145 333 560 115 2;
  • 28) 0.000 000 000 001 145 333 560 115 2 × 2 = 0 + 0.000 000 000 002 290 667 120 230 4;
  • 29) 0.000 000 000 002 290 667 120 230 4 × 2 = 0 + 0.000 000 000 004 581 334 240 460 8;
  • 30) 0.000 000 000 004 581 334 240 460 8 × 2 = 0 + 0.000 000 000 009 162 668 480 921 6;
  • 31) 0.000 000 000 009 162 668 480 921 6 × 2 = 0 + 0.000 000 000 018 325 336 961 843 2;
  • 32) 0.000 000 000 018 325 336 961 843 2 × 2 = 0 + 0.000 000 000 036 650 673 923 686 4;
  • 33) 0.000 000 000 036 650 673 923 686 4 × 2 = 0 + 0.000 000 000 073 301 347 847 372 8;
  • 34) 0.000 000 000 073 301 347 847 372 8 × 2 = 0 + 0.000 000 000 146 602 695 694 745 6;
  • 35) 0.000 000 000 146 602 695 694 745 6 × 2 = 0 + 0.000 000 000 293 205 391 389 491 2;
  • 36) 0.000 000 000 293 205 391 389 491 2 × 2 = 0 + 0.000 000 000 586 410 782 778 982 4;
  • 37) 0.000 000 000 586 410 782 778 982 4 × 2 = 0 + 0.000 000 001 172 821 565 557 964 8;
  • 38) 0.000 000 001 172 821 565 557 964 8 × 2 = 0 + 0.000 000 002 345 643 131 115 929 6;
  • 39) 0.000 000 002 345 643 131 115 929 6 × 2 = 0 + 0.000 000 004 691 286 262 231 859 2;
  • 40) 0.000 000 004 691 286 262 231 859 2 × 2 = 0 + 0.000 000 009 382 572 524 463 718 4;
  • 41) 0.000 000 009 382 572 524 463 718 4 × 2 = 0 + 0.000 000 018 765 145 048 927 436 8;
  • 42) 0.000 000 018 765 145 048 927 436 8 × 2 = 0 + 0.000 000 037 530 290 097 854 873 6;
  • 43) 0.000 000 037 530 290 097 854 873 6 × 2 = 0 + 0.000 000 075 060 580 195 709 747 2;
  • 44) 0.000 000 075 060 580 195 709 747 2 × 2 = 0 + 0.000 000 150 121 160 391 419 494 4;
  • 45) 0.000 000 150 121 160 391 419 494 4 × 2 = 0 + 0.000 000 300 242 320 782 838 988 8;
  • 46) 0.000 000 300 242 320 782 838 988 8 × 2 = 0 + 0.000 000 600 484 641 565 677 977 6;
  • 47) 0.000 000 600 484 641 565 677 977 6 × 2 = 0 + 0.000 001 200 969 283 131 355 955 2;
  • 48) 0.000 001 200 969 283 131 355 955 2 × 2 = 0 + 0.000 002 401 938 566 262 711 910 4;
  • 49) 0.000 002 401 938 566 262 711 910 4 × 2 = 0 + 0.000 004 803 877 132 525 423 820 8;
  • 50) 0.000 004 803 877 132 525 423 820 8 × 2 = 0 + 0.000 009 607 754 265 050 847 641 6;
  • 51) 0.000 009 607 754 265 050 847 641 6 × 2 = 0 + 0.000 019 215 508 530 101 695 283 2;
  • 52) 0.000 019 215 508 530 101 695 283 2 × 2 = 0 + 0.000 038 431 017 060 203 390 566 4;
  • 53) 0.000 038 431 017 060 203 390 566 4 × 2 = 0 + 0.000 076 862 034 120 406 781 132 8;
  • 54) 0.000 076 862 034 120 406 781 132 8 × 2 = 0 + 0.000 153 724 068 240 813 562 265 6;
  • 55) 0.000 153 724 068 240 813 562 265 6 × 2 = 0 + 0.000 307 448 136 481 627 124 531 2;
  • 56) 0.000 307 448 136 481 627 124 531 2 × 2 = 0 + 0.000 614 896 272 963 254 249 062 4;
  • 57) 0.000 614 896 272 963 254 249 062 4 × 2 = 0 + 0.001 229 792 545 926 508 498 124 8;
  • 58) 0.001 229 792 545 926 508 498 124 8 × 2 = 0 + 0.002 459 585 091 853 016 996 249 6;
  • 59) 0.002 459 585 091 853 016 996 249 6 × 2 = 0 + 0.004 919 170 183 706 033 992 499 2;
  • 60) 0.004 919 170 183 706 033 992 499 2 × 2 = 0 + 0.009 838 340 367 412 067 984 998 4;
  • 61) 0.009 838 340 367 412 067 984 998 4 × 2 = 0 + 0.019 676 680 734 824 135 969 996 8;
  • 62) 0.019 676 680 734 824 135 969 996 8 × 2 = 0 + 0.039 353 361 469 648 271 939 993 6;
  • 63) 0.039 353 361 469 648 271 939 993 6 × 2 = 0 + 0.078 706 722 939 296 543 879 987 2;
  • 64) 0.078 706 722 939 296 543 879 987 2 × 2 = 0 + 0.157 413 445 878 593 087 759 974 4;
  • 65) 0.157 413 445 878 593 087 759 974 4 × 2 = 0 + 0.314 826 891 757 186 175 519 948 8;
  • 66) 0.314 826 891 757 186 175 519 948 8 × 2 = 0 + 0.629 653 783 514 372 351 039 897 6;
  • 67) 0.629 653 783 514 372 351 039 897 6 × 2 = 1 + 0.259 307 567 028 744 702 079 795 2;
  • 68) 0.259 307 567 028 744 702 079 795 2 × 2 = 0 + 0.518 615 134 057 489 404 159 590 4;
  • 69) 0.518 615 134 057 489 404 159 590 4 × 2 = 1 + 0.037 230 268 114 978 808 319 180 8;
  • 70) 0.037 230 268 114 978 808 319 180 8 × 2 = 0 + 0.074 460 536 229 957 616 638 361 6;
  • 71) 0.074 460 536 229 957 616 638 361 6 × 2 = 0 + 0.148 921 072 459 915 233 276 723 2;
  • 72) 0.148 921 072 459 915 233 276 723 2 × 2 = 0 + 0.297 842 144 919 830 466 553 446 4;
  • 73) 0.297 842 144 919 830 466 553 446 4 × 2 = 0 + 0.595 684 289 839 660 933 106 892 8;
  • 74) 0.595 684 289 839 660 933 106 892 8 × 2 = 1 + 0.191 368 579 679 321 866 213 785 6;
  • 75) 0.191 368 579 679 321 866 213 785 6 × 2 = 0 + 0.382 737 159 358 643 732 427 571 2;
  • 76) 0.382 737 159 358 643 732 427 571 2 × 2 = 0 + 0.765 474 318 717 287 464 855 142 4;
  • 77) 0.765 474 318 717 287 464 855 142 4 × 2 = 1 + 0.530 948 637 434 574 929 710 284 8;
  • 78) 0.530 948 637 434 574 929 710 284 8 × 2 = 1 + 0.061 897 274 869 149 859 420 569 6;
  • 79) 0.061 897 274 869 149 859 420 569 6 × 2 = 0 + 0.123 794 549 738 299 718 841 139 2;
  • 80) 0.123 794 549 738 299 718 841 139 2 × 2 = 0 + 0.247 589 099 476 599 437 682 278 4;
  • 81) 0.247 589 099 476 599 437 682 278 4 × 2 = 0 + 0.495 178 198 953 198 875 364 556 8;
  • 82) 0.495 178 198 953 198 875 364 556 8 × 2 = 0 + 0.990 356 397 906 397 750 729 113 6;
  • 83) 0.990 356 397 906 397 750 729 113 6 × 2 = 1 + 0.980 712 795 812 795 501 458 227 2;
  • 84) 0.980 712 795 812 795 501 458 227 2 × 2 = 1 + 0.961 425 591 625 591 002 916 454 4;
  • 85) 0.961 425 591 625 591 002 916 454 4 × 2 = 1 + 0.922 851 183 251 182 005 832 908 8;
  • 86) 0.922 851 183 251 182 005 832 908 8 × 2 = 1 + 0.845 702 366 502 364 011 665 817 6;
  • 87) 0.845 702 366 502 364 011 665 817 6 × 2 = 1 + 0.691 404 733 004 728 023 331 635 2;
  • 88) 0.691 404 733 004 728 023 331 635 2 × 2 = 1 + 0.382 809 466 009 456 046 663 270 4;
  • 89) 0.382 809 466 009 456 046 663 270 4 × 2 = 0 + 0.765 618 932 018 912 093 326 540 8;
  • 90) 0.765 618 932 018 912 093 326 540 8 × 2 = 1 + 0.531 237 864 037 824 186 653 081 6;
  • 91) 0.531 237 864 037 824 186 653 081 6 × 2 = 1 + 0.062 475 728 075 648 373 306 163 2;
  • 92) 0.062 475 728 075 648 373 306 163 2 × 2 = 0 + 0.124 951 456 151 296 746 612 326 4;
  • 93) 0.124 951 456 151 296 746 612 326 4 × 2 = 0 + 0.249 902 912 302 593 493 224 652 8;
  • 94) 0.249 902 912 302 593 493 224 652 8 × 2 = 0 + 0.499 805 824 605 186 986 449 305 6;
  • 95) 0.499 805 824 605 186 986 449 305 6 × 2 = 0 + 0.999 611 649 210 373 972 898 611 2;
  • 96) 0.999 611 649 210 373 972 898 611 2 × 2 = 1 + 0.999 223 298 420 747 945 797 222 4;
  • 97) 0.999 223 298 420 747 945 797 222 4 × 2 = 1 + 0.998 446 596 841 495 891 594 444 8;
  • 98) 0.998 446 596 841 495 891 594 444 8 × 2 = 1 + 0.996 893 193 682 991 783 188 889 6;
  • 99) 0.996 893 193 682 991 783 188 889 6 × 2 = 1 + 0.993 786 387 365 983 566 377 779 2;
  • 100) 0.993 786 387 365 983 566 377 779 2 × 2 = 1 + 0.987 572 774 731 967 132 755 558 4;
  • 101) 0.987 572 774 731 967 132 755 558 4 × 2 = 1 + 0.975 145 549 463 934 265 511 116 8;
  • 102) 0.975 145 549 463 934 265 511 116 8 × 2 = 1 + 0.950 291 098 927 868 531 022 233 6;
  • 103) 0.950 291 098 927 868 531 022 233 6 × 2 = 1 + 0.900 582 197 855 737 062 044 467 2;
  • 104) 0.900 582 197 855 737 062 044 467 2 × 2 = 1 + 0.801 164 395 711 474 124 088 934 4;
  • 105) 0.801 164 395 711 474 124 088 934 4 × 2 = 1 + 0.602 328 791 422 948 248 177 868 8;
  • 106) 0.602 328 791 422 948 248 177 868 8 × 2 = 1 + 0.204 657 582 845 896 496 355 737 6;
  • 107) 0.204 657 582 845 896 496 355 737 6 × 2 = 0 + 0.409 315 165 691 792 992 711 475 2;
  • 108) 0.409 315 165 691 792 992 711 475 2 × 2 = 0 + 0.818 630 331 383 585 985 422 950 4;
  • 109) 0.818 630 331 383 585 985 422 950 4 × 2 = 1 + 0.637 260 662 767 171 970 845 900 8;
  • 110) 0.637 260 662 767 171 970 845 900 8 × 2 = 1 + 0.274 521 325 534 343 941 691 801 6;
  • 111) 0.274 521 325 534 343 941 691 801 6 × 2 = 0 + 0.549 042 651 068 687 883 383 603 2;
  • 112) 0.549 042 651 068 687 883 383 603 2 × 2 = 1 + 0.098 085 302 137 375 766 767 206 4;
  • 113) 0.098 085 302 137 375 766 767 206 4 × 2 = 0 + 0.196 170 604 274 751 533 534 412 8;
  • 114) 0.196 170 604 274 751 533 534 412 8 × 2 = 0 + 0.392 341 208 549 503 067 068 825 6;
  • 115) 0.392 341 208 549 503 067 068 825 6 × 2 = 0 + 0.784 682 417 099 006 134 137 651 2;
  • 116) 0.784 682 417 099 006 134 137 651 2 × 2 = 1 + 0.569 364 834 198 012 268 275 302 4;
  • 117) 0.569 364 834 198 012 268 275 302 4 × 2 = 1 + 0.138 729 668 396 024 536 550 604 8;
  • 118) 0.138 729 668 396 024 536 550 604 8 × 2 = 0 + 0.277 459 336 792 049 073 101 209 6;
  • 119) 0.277 459 336 792 049 073 101 209 6 × 2 = 0 + 0.554 918 673 584 098 146 202 419 2;

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:


0.000 000 000 000 000 000 008 533 4(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1100 0011 1111 0110 0001 1111 1111 1100 1101 0001 100(2)

5. Positive number before normalization:

0.000 000 000 000 000 000 008 533 4(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1100 0011 1111 0110 0001 1111 1111 1100 1101 0001 100(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:


0.000 000 000 000 000 000 008 533 4(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1100 0011 1111 0110 0001 1111 1111 1100 1101 0001 100(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1100 0011 1111 0110 0001 1111 1111 1100 1101 0001 100(2) × 20 =

1.0100 0010 0110 0001 1111 1011 0000 1111 1111 1110 0110 1000 1100(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 0110 0001 1111 1011 0000 1111 1111 1110 0110 1000 1100


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 0110 0001 1111 1011 0000 1111 1111 1110 0110 1000 1100 =

0100 0010 0110 0001 1111 1011 0000 1111 1111 1110 0110 1000 1100


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 0110 0001 1111 1011 0000 1111 1111 1110 0110 1000 1100


Decimal number 0.000 000 000 000 000 000 008 533 4 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1011 1100 - 0100 0010 0110 0001 1111 1011 0000 1111 1111 1110 0110 1000 1100

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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