0.000 000 000 000 000 000 008 537 38 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 000 000 008 537 38(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
0.000 000 000 000 000 000 008 537 38(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: 0.000 000 000 000 000 000 008 537 38.

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) 0.000 000 000 000 000 000 008 537 38 × 2 = 0 + 0.000 000 000 000 000 000 017 074 76;
  • 2) 0.000 000 000 000 000 000 017 074 76 × 2 = 0 + 0.000 000 000 000 000 000 034 149 52;
  • 3) 0.000 000 000 000 000 000 034 149 52 × 2 = 0 + 0.000 000 000 000 000 000 068 299 04;
  • 4) 0.000 000 000 000 000 000 068 299 04 × 2 = 0 + 0.000 000 000 000 000 000 136 598 08;
  • 5) 0.000 000 000 000 000 000 136 598 08 × 2 = 0 + 0.000 000 000 000 000 000 273 196 16;
  • 6) 0.000 000 000 000 000 000 273 196 16 × 2 = 0 + 0.000 000 000 000 000 000 546 392 32;
  • 7) 0.000 000 000 000 000 000 546 392 32 × 2 = 0 + 0.000 000 000 000 000 001 092 784 64;
  • 8) 0.000 000 000 000 000 001 092 784 64 × 2 = 0 + 0.000 000 000 000 000 002 185 569 28;
  • 9) 0.000 000 000 000 000 002 185 569 28 × 2 = 0 + 0.000 000 000 000 000 004 371 138 56;
  • 10) 0.000 000 000 000 000 004 371 138 56 × 2 = 0 + 0.000 000 000 000 000 008 742 277 12;
  • 11) 0.000 000 000 000 000 008 742 277 12 × 2 = 0 + 0.000 000 000 000 000 017 484 554 24;
  • 12) 0.000 000 000 000 000 017 484 554 24 × 2 = 0 + 0.000 000 000 000 000 034 969 108 48;
  • 13) 0.000 000 000 000 000 034 969 108 48 × 2 = 0 + 0.000 000 000 000 000 069 938 216 96;
  • 14) 0.000 000 000 000 000 069 938 216 96 × 2 = 0 + 0.000 000 000 000 000 139 876 433 92;
  • 15) 0.000 000 000 000 000 139 876 433 92 × 2 = 0 + 0.000 000 000 000 000 279 752 867 84;
  • 16) 0.000 000 000 000 000 279 752 867 84 × 2 = 0 + 0.000 000 000 000 000 559 505 735 68;
  • 17) 0.000 000 000 000 000 559 505 735 68 × 2 = 0 + 0.000 000 000 000 001 119 011 471 36;
  • 18) 0.000 000 000 000 001 119 011 471 36 × 2 = 0 + 0.000 000 000 000 002 238 022 942 72;
  • 19) 0.000 000 000 000 002 238 022 942 72 × 2 = 0 + 0.000 000 000 000 004 476 045 885 44;
  • 20) 0.000 000 000 000 004 476 045 885 44 × 2 = 0 + 0.000 000 000 000 008 952 091 770 88;
  • 21) 0.000 000 000 000 008 952 091 770 88 × 2 = 0 + 0.000 000 000 000 017 904 183 541 76;
  • 22) 0.000 000 000 000 017 904 183 541 76 × 2 = 0 + 0.000 000 000 000 035 808 367 083 52;
  • 23) 0.000 000 000 000 035 808 367 083 52 × 2 = 0 + 0.000 000 000 000 071 616 734 167 04;
  • 24) 0.000 000 000 000 071 616 734 167 04 × 2 = 0 + 0.000 000 000 000 143 233 468 334 08;
  • 25) 0.000 000 000 000 143 233 468 334 08 × 2 = 0 + 0.000 000 000 000 286 466 936 668 16;
  • 26) 0.000 000 000 000 286 466 936 668 16 × 2 = 0 + 0.000 000 000 000 572 933 873 336 32;
  • 27) 0.000 000 000 000 572 933 873 336 32 × 2 = 0 + 0.000 000 000 001 145 867 746 672 64;
  • 28) 0.000 000 000 001 145 867 746 672 64 × 2 = 0 + 0.000 000 000 002 291 735 493 345 28;
  • 29) 0.000 000 000 002 291 735 493 345 28 × 2 = 0 + 0.000 000 000 004 583 470 986 690 56;
  • 30) 0.000 000 000 004 583 470 986 690 56 × 2 = 0 + 0.000 000 000 009 166 941 973 381 12;
  • 31) 0.000 000 000 009 166 941 973 381 12 × 2 = 0 + 0.000 000 000 018 333 883 946 762 24;
  • 32) 0.000 000 000 018 333 883 946 762 24 × 2 = 0 + 0.000 000 000 036 667 767 893 524 48;
  • 33) 0.000 000 000 036 667 767 893 524 48 × 2 = 0 + 0.000 000 000 073 335 535 787 048 96;
  • 34) 0.000 000 000 073 335 535 787 048 96 × 2 = 0 + 0.000 000 000 146 671 071 574 097 92;
  • 35) 0.000 000 000 146 671 071 574 097 92 × 2 = 0 + 0.000 000 000 293 342 143 148 195 84;
  • 36) 0.000 000 000 293 342 143 148 195 84 × 2 = 0 + 0.000 000 000 586 684 286 296 391 68;
  • 37) 0.000 000 000 586 684 286 296 391 68 × 2 = 0 + 0.000 000 001 173 368 572 592 783 36;
  • 38) 0.000 000 001 173 368 572 592 783 36 × 2 = 0 + 0.000 000 002 346 737 145 185 566 72;
  • 39) 0.000 000 002 346 737 145 185 566 72 × 2 = 0 + 0.000 000 004 693 474 290 371 133 44;
  • 40) 0.000 000 004 693 474 290 371 133 44 × 2 = 0 + 0.000 000 009 386 948 580 742 266 88;
  • 41) 0.000 000 009 386 948 580 742 266 88 × 2 = 0 + 0.000 000 018 773 897 161 484 533 76;
  • 42) 0.000 000 018 773 897 161 484 533 76 × 2 = 0 + 0.000 000 037 547 794 322 969 067 52;
  • 43) 0.000 000 037 547 794 322 969 067 52 × 2 = 0 + 0.000 000 075 095 588 645 938 135 04;
  • 44) 0.000 000 075 095 588 645 938 135 04 × 2 = 0 + 0.000 000 150 191 177 291 876 270 08;
  • 45) 0.000 000 150 191 177 291 876 270 08 × 2 = 0 + 0.000 000 300 382 354 583 752 540 16;
  • 46) 0.000 000 300 382 354 583 752 540 16 × 2 = 0 + 0.000 000 600 764 709 167 505 080 32;
  • 47) 0.000 000 600 764 709 167 505 080 32 × 2 = 0 + 0.000 001 201 529 418 335 010 160 64;
  • 48) 0.000 001 201 529 418 335 010 160 64 × 2 = 0 + 0.000 002 403 058 836 670 020 321 28;
  • 49) 0.000 002 403 058 836 670 020 321 28 × 2 = 0 + 0.000 004 806 117 673 340 040 642 56;
  • 50) 0.000 004 806 117 673 340 040 642 56 × 2 = 0 + 0.000 009 612 235 346 680 081 285 12;
  • 51) 0.000 009 612 235 346 680 081 285 12 × 2 = 0 + 0.000 019 224 470 693 360 162 570 24;
  • 52) 0.000 019 224 470 693 360 162 570 24 × 2 = 0 + 0.000 038 448 941 386 720 325 140 48;
  • 53) 0.000 038 448 941 386 720 325 140 48 × 2 = 0 + 0.000 076 897 882 773 440 650 280 96;
  • 54) 0.000 076 897 882 773 440 650 280 96 × 2 = 0 + 0.000 153 795 765 546 881 300 561 92;
  • 55) 0.000 153 795 765 546 881 300 561 92 × 2 = 0 + 0.000 307 591 531 093 762 601 123 84;
  • 56) 0.000 307 591 531 093 762 601 123 84 × 2 = 0 + 0.000 615 183 062 187 525 202 247 68;
  • 57) 0.000 615 183 062 187 525 202 247 68 × 2 = 0 + 0.001 230 366 124 375 050 404 495 36;
  • 58) 0.001 230 366 124 375 050 404 495 36 × 2 = 0 + 0.002 460 732 248 750 100 808 990 72;
  • 59) 0.002 460 732 248 750 100 808 990 72 × 2 = 0 + 0.004 921 464 497 500 201 617 981 44;
  • 60) 0.004 921 464 497 500 201 617 981 44 × 2 = 0 + 0.009 842 928 995 000 403 235 962 88;
  • 61) 0.009 842 928 995 000 403 235 962 88 × 2 = 0 + 0.019 685 857 990 000 806 471 925 76;
  • 62) 0.019 685 857 990 000 806 471 925 76 × 2 = 0 + 0.039 371 715 980 001 612 943 851 52;
  • 63) 0.039 371 715 980 001 612 943 851 52 × 2 = 0 + 0.078 743 431 960 003 225 887 703 04;
  • 64) 0.078 743 431 960 003 225 887 703 04 × 2 = 0 + 0.157 486 863 920 006 451 775 406 08;
  • 65) 0.157 486 863 920 006 451 775 406 08 × 2 = 0 + 0.314 973 727 840 012 903 550 812 16;
  • 66) 0.314 973 727 840 012 903 550 812 16 × 2 = 0 + 0.629 947 455 680 025 807 101 624 32;
  • 67) 0.629 947 455 680 025 807 101 624 32 × 2 = 1 + 0.259 894 911 360 051 614 203 248 64;
  • 68) 0.259 894 911 360 051 614 203 248 64 × 2 = 0 + 0.519 789 822 720 103 228 406 497 28;
  • 69) 0.519 789 822 720 103 228 406 497 28 × 2 = 1 + 0.039 579 645 440 206 456 812 994 56;
  • 70) 0.039 579 645 440 206 456 812 994 56 × 2 = 0 + 0.079 159 290 880 412 913 625 989 12;
  • 71) 0.079 159 290 880 412 913 625 989 12 × 2 = 0 + 0.158 318 581 760 825 827 251 978 24;
  • 72) 0.158 318 581 760 825 827 251 978 24 × 2 = 0 + 0.316 637 163 521 651 654 503 956 48;
  • 73) 0.316 637 163 521 651 654 503 956 48 × 2 = 0 + 0.633 274 327 043 303 309 007 912 96;
  • 74) 0.633 274 327 043 303 309 007 912 96 × 2 = 1 + 0.266 548 654 086 606 618 015 825 92;
  • 75) 0.266 548 654 086 606 618 015 825 92 × 2 = 0 + 0.533 097 308 173 213 236 031 651 84;
  • 76) 0.533 097 308 173 213 236 031 651 84 × 2 = 1 + 0.066 194 616 346 426 472 063 303 68;
  • 77) 0.066 194 616 346 426 472 063 303 68 × 2 = 0 + 0.132 389 232 692 852 944 126 607 36;
  • 78) 0.132 389 232 692 852 944 126 607 36 × 2 = 0 + 0.264 778 465 385 705 888 253 214 72;
  • 79) 0.264 778 465 385 705 888 253 214 72 × 2 = 0 + 0.529 556 930 771 411 776 506 429 44;
  • 80) 0.529 556 930 771 411 776 506 429 44 × 2 = 1 + 0.059 113 861 542 823 553 012 858 88;
  • 81) 0.059 113 861 542 823 553 012 858 88 × 2 = 0 + 0.118 227 723 085 647 106 025 717 76;
  • 82) 0.118 227 723 085 647 106 025 717 76 × 2 = 0 + 0.236 455 446 171 294 212 051 435 52;
  • 83) 0.236 455 446 171 294 212 051 435 52 × 2 = 0 + 0.472 910 892 342 588 424 102 871 04;
  • 84) 0.472 910 892 342 588 424 102 871 04 × 2 = 0 + 0.945 821 784 685 176 848 205 742 08;
  • 85) 0.945 821 784 685 176 848 205 742 08 × 2 = 1 + 0.891 643 569 370 353 696 411 484 16;
  • 86) 0.891 643 569 370 353 696 411 484 16 × 2 = 1 + 0.783 287 138 740 707 392 822 968 32;
  • 87) 0.783 287 138 740 707 392 822 968 32 × 2 = 1 + 0.566 574 277 481 414 785 645 936 64;
  • 88) 0.566 574 277 481 414 785 645 936 64 × 2 = 1 + 0.133 148 554 962 829 571 291 873 28;
  • 89) 0.133 148 554 962 829 571 291 873 28 × 2 = 0 + 0.266 297 109 925 659 142 583 746 56;
  • 90) 0.266 297 109 925 659 142 583 746 56 × 2 = 0 + 0.532 594 219 851 318 285 167 493 12;
  • 91) 0.532 594 219 851 318 285 167 493 12 × 2 = 1 + 0.065 188 439 702 636 570 334 986 24;
  • 92) 0.065 188 439 702 636 570 334 986 24 × 2 = 0 + 0.130 376 879 405 273 140 669 972 48;
  • 93) 0.130 376 879 405 273 140 669 972 48 × 2 = 0 + 0.260 753 758 810 546 281 339 944 96;
  • 94) 0.260 753 758 810 546 281 339 944 96 × 2 = 0 + 0.521 507 517 621 092 562 679 889 92;
  • 95) 0.521 507 517 621 092 562 679 889 92 × 2 = 1 + 0.043 015 035 242 185 125 359 779 84;
  • 96) 0.043 015 035 242 185 125 359 779 84 × 2 = 0 + 0.086 030 070 484 370 250 719 559 68;
  • 97) 0.086 030 070 484 370 250 719 559 68 × 2 = 0 + 0.172 060 140 968 740 501 439 119 36;
  • 98) 0.172 060 140 968 740 501 439 119 36 × 2 = 0 + 0.344 120 281 937 481 002 878 238 72;
  • 99) 0.344 120 281 937 481 002 878 238 72 × 2 = 0 + 0.688 240 563 874 962 005 756 477 44;
  • 100) 0.688 240 563 874 962 005 756 477 44 × 2 = 1 + 0.376 481 127 749 924 011 512 954 88;
  • 101) 0.376 481 127 749 924 011 512 954 88 × 2 = 0 + 0.752 962 255 499 848 023 025 909 76;
  • 102) 0.752 962 255 499 848 023 025 909 76 × 2 = 1 + 0.505 924 510 999 696 046 051 819 52;
  • 103) 0.505 924 510 999 696 046 051 819 52 × 2 = 1 + 0.011 849 021 999 392 092 103 639 04;
  • 104) 0.011 849 021 999 392 092 103 639 04 × 2 = 0 + 0.023 698 043 998 784 184 207 278 08;
  • 105) 0.023 698 043 998 784 184 207 278 08 × 2 = 0 + 0.047 396 087 997 568 368 414 556 16;
  • 106) 0.047 396 087 997 568 368 414 556 16 × 2 = 0 + 0.094 792 175 995 136 736 829 112 32;
  • 107) 0.094 792 175 995 136 736 829 112 32 × 2 = 0 + 0.189 584 351 990 273 473 658 224 64;
  • 108) 0.189 584 351 990 273 473 658 224 64 × 2 = 0 + 0.379 168 703 980 546 947 316 449 28;
  • 109) 0.379 168 703 980 546 947 316 449 28 × 2 = 0 + 0.758 337 407 961 093 894 632 898 56;
  • 110) 0.758 337 407 961 093 894 632 898 56 × 2 = 1 + 0.516 674 815 922 187 789 265 797 12;
  • 111) 0.516 674 815 922 187 789 265 797 12 × 2 = 1 + 0.033 349 631 844 375 578 531 594 24;
  • 112) 0.033 349 631 844 375 578 531 594 24 × 2 = 0 + 0.066 699 263 688 751 157 063 188 48;
  • 113) 0.066 699 263 688 751 157 063 188 48 × 2 = 0 + 0.133 398 527 377 502 314 126 376 96;
  • 114) 0.133 398 527 377 502 314 126 376 96 × 2 = 0 + 0.266 797 054 755 004 628 252 753 92;
  • 115) 0.266 797 054 755 004 628 252 753 92 × 2 = 0 + 0.533 594 109 510 009 256 505 507 84;
  • 116) 0.533 594 109 510 009 256 505 507 84 × 2 = 1 + 0.067 188 219 020 018 513 011 015 68;
  • 117) 0.067 188 219 020 018 513 011 015 68 × 2 = 0 + 0.134 376 438 040 037 026 022 031 36;
  • 118) 0.134 376 438 040 037 026 022 031 36 × 2 = 0 + 0.268 752 876 080 074 052 044 062 72;
  • 119) 0.268 752 876 080 074 052 044 062 72 × 2 = 0 + 0.537 505 752 160 148 104 088 125 44;

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:


0.000 000 000 000 000 000 008 537 38(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0101 0001 0000 1111 0010 0010 0001 0110 0000 0110 0001 000(2)

5. Positive number before normalization:

0.000 000 000 000 000 000 008 537 38(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0101 0001 0000 1111 0010 0010 0001 0110 0000 0110 0001 000(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:


0.000 000 000 000 000 000 008 537 38(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0101 0001 0000 1111 0010 0010 0001 0110 0000 0110 0001 000(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0101 0001 0000 1111 0010 0010 0001 0110 0000 0110 0001 000(2) × 20 =

1.0100 0010 1000 1000 0111 1001 0001 0000 1011 0000 0011 0000 1000(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 1000 1000 0111 1001 0001 0000 1011 0000 0011 0000 1000


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 1000 1000 0111 1001 0001 0000 1011 0000 0011 0000 1000 =

0100 0010 1000 1000 0111 1001 0001 0000 1011 0000 0011 0000 1000


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 1000 1000 0111 1001 0001 0000 1011 0000 0011 0000 1000


Decimal number 0.000 000 000 000 000 000 008 537 38 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1011 1100 - 0100 0010 1000 1000 0111 1001 0001 0000 1011 0000 0011 0000 1000

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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