0.000 000 000 000 000 000 008 537 181 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 000 000 008 537 181(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
0.000 000 000 000 000 000 008 537 181(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: 0.000 000 000 000 000 000 008 537 181.

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) 0.000 000 000 000 000 000 008 537 181 × 2 = 0 + 0.000 000 000 000 000 000 017 074 362;
  • 2) 0.000 000 000 000 000 000 017 074 362 × 2 = 0 + 0.000 000 000 000 000 000 034 148 724;
  • 3) 0.000 000 000 000 000 000 034 148 724 × 2 = 0 + 0.000 000 000 000 000 000 068 297 448;
  • 4) 0.000 000 000 000 000 000 068 297 448 × 2 = 0 + 0.000 000 000 000 000 000 136 594 896;
  • 5) 0.000 000 000 000 000 000 136 594 896 × 2 = 0 + 0.000 000 000 000 000 000 273 189 792;
  • 6) 0.000 000 000 000 000 000 273 189 792 × 2 = 0 + 0.000 000 000 000 000 000 546 379 584;
  • 7) 0.000 000 000 000 000 000 546 379 584 × 2 = 0 + 0.000 000 000 000 000 001 092 759 168;
  • 8) 0.000 000 000 000 000 001 092 759 168 × 2 = 0 + 0.000 000 000 000 000 002 185 518 336;
  • 9) 0.000 000 000 000 000 002 185 518 336 × 2 = 0 + 0.000 000 000 000 000 004 371 036 672;
  • 10) 0.000 000 000 000 000 004 371 036 672 × 2 = 0 + 0.000 000 000 000 000 008 742 073 344;
  • 11) 0.000 000 000 000 000 008 742 073 344 × 2 = 0 + 0.000 000 000 000 000 017 484 146 688;
  • 12) 0.000 000 000 000 000 017 484 146 688 × 2 = 0 + 0.000 000 000 000 000 034 968 293 376;
  • 13) 0.000 000 000 000 000 034 968 293 376 × 2 = 0 + 0.000 000 000 000 000 069 936 586 752;
  • 14) 0.000 000 000 000 000 069 936 586 752 × 2 = 0 + 0.000 000 000 000 000 139 873 173 504;
  • 15) 0.000 000 000 000 000 139 873 173 504 × 2 = 0 + 0.000 000 000 000 000 279 746 347 008;
  • 16) 0.000 000 000 000 000 279 746 347 008 × 2 = 0 + 0.000 000 000 000 000 559 492 694 016;
  • 17) 0.000 000 000 000 000 559 492 694 016 × 2 = 0 + 0.000 000 000 000 001 118 985 388 032;
  • 18) 0.000 000 000 000 001 118 985 388 032 × 2 = 0 + 0.000 000 000 000 002 237 970 776 064;
  • 19) 0.000 000 000 000 002 237 970 776 064 × 2 = 0 + 0.000 000 000 000 004 475 941 552 128;
  • 20) 0.000 000 000 000 004 475 941 552 128 × 2 = 0 + 0.000 000 000 000 008 951 883 104 256;
  • 21) 0.000 000 000 000 008 951 883 104 256 × 2 = 0 + 0.000 000 000 000 017 903 766 208 512;
  • 22) 0.000 000 000 000 017 903 766 208 512 × 2 = 0 + 0.000 000 000 000 035 807 532 417 024;
  • 23) 0.000 000 000 000 035 807 532 417 024 × 2 = 0 + 0.000 000 000 000 071 615 064 834 048;
  • 24) 0.000 000 000 000 071 615 064 834 048 × 2 = 0 + 0.000 000 000 000 143 230 129 668 096;
  • 25) 0.000 000 000 000 143 230 129 668 096 × 2 = 0 + 0.000 000 000 000 286 460 259 336 192;
  • 26) 0.000 000 000 000 286 460 259 336 192 × 2 = 0 + 0.000 000 000 000 572 920 518 672 384;
  • 27) 0.000 000 000 000 572 920 518 672 384 × 2 = 0 + 0.000 000 000 001 145 841 037 344 768;
  • 28) 0.000 000 000 001 145 841 037 344 768 × 2 = 0 + 0.000 000 000 002 291 682 074 689 536;
  • 29) 0.000 000 000 002 291 682 074 689 536 × 2 = 0 + 0.000 000 000 004 583 364 149 379 072;
  • 30) 0.000 000 000 004 583 364 149 379 072 × 2 = 0 + 0.000 000 000 009 166 728 298 758 144;
  • 31) 0.000 000 000 009 166 728 298 758 144 × 2 = 0 + 0.000 000 000 018 333 456 597 516 288;
  • 32) 0.000 000 000 018 333 456 597 516 288 × 2 = 0 + 0.000 000 000 036 666 913 195 032 576;
  • 33) 0.000 000 000 036 666 913 195 032 576 × 2 = 0 + 0.000 000 000 073 333 826 390 065 152;
  • 34) 0.000 000 000 073 333 826 390 065 152 × 2 = 0 + 0.000 000 000 146 667 652 780 130 304;
  • 35) 0.000 000 000 146 667 652 780 130 304 × 2 = 0 + 0.000 000 000 293 335 305 560 260 608;
  • 36) 0.000 000 000 293 335 305 560 260 608 × 2 = 0 + 0.000 000 000 586 670 611 120 521 216;
  • 37) 0.000 000 000 586 670 611 120 521 216 × 2 = 0 + 0.000 000 001 173 341 222 241 042 432;
  • 38) 0.000 000 001 173 341 222 241 042 432 × 2 = 0 + 0.000 000 002 346 682 444 482 084 864;
  • 39) 0.000 000 002 346 682 444 482 084 864 × 2 = 0 + 0.000 000 004 693 364 888 964 169 728;
  • 40) 0.000 000 004 693 364 888 964 169 728 × 2 = 0 + 0.000 000 009 386 729 777 928 339 456;
  • 41) 0.000 000 009 386 729 777 928 339 456 × 2 = 0 + 0.000 000 018 773 459 555 856 678 912;
  • 42) 0.000 000 018 773 459 555 856 678 912 × 2 = 0 + 0.000 000 037 546 919 111 713 357 824;
  • 43) 0.000 000 037 546 919 111 713 357 824 × 2 = 0 + 0.000 000 075 093 838 223 426 715 648;
  • 44) 0.000 000 075 093 838 223 426 715 648 × 2 = 0 + 0.000 000 150 187 676 446 853 431 296;
  • 45) 0.000 000 150 187 676 446 853 431 296 × 2 = 0 + 0.000 000 300 375 352 893 706 862 592;
  • 46) 0.000 000 300 375 352 893 706 862 592 × 2 = 0 + 0.000 000 600 750 705 787 413 725 184;
  • 47) 0.000 000 600 750 705 787 413 725 184 × 2 = 0 + 0.000 001 201 501 411 574 827 450 368;
  • 48) 0.000 001 201 501 411 574 827 450 368 × 2 = 0 + 0.000 002 403 002 823 149 654 900 736;
  • 49) 0.000 002 403 002 823 149 654 900 736 × 2 = 0 + 0.000 004 806 005 646 299 309 801 472;
  • 50) 0.000 004 806 005 646 299 309 801 472 × 2 = 0 + 0.000 009 612 011 292 598 619 602 944;
  • 51) 0.000 009 612 011 292 598 619 602 944 × 2 = 0 + 0.000 019 224 022 585 197 239 205 888;
  • 52) 0.000 019 224 022 585 197 239 205 888 × 2 = 0 + 0.000 038 448 045 170 394 478 411 776;
  • 53) 0.000 038 448 045 170 394 478 411 776 × 2 = 0 + 0.000 076 896 090 340 788 956 823 552;
  • 54) 0.000 076 896 090 340 788 956 823 552 × 2 = 0 + 0.000 153 792 180 681 577 913 647 104;
  • 55) 0.000 153 792 180 681 577 913 647 104 × 2 = 0 + 0.000 307 584 361 363 155 827 294 208;
  • 56) 0.000 307 584 361 363 155 827 294 208 × 2 = 0 + 0.000 615 168 722 726 311 654 588 416;
  • 57) 0.000 615 168 722 726 311 654 588 416 × 2 = 0 + 0.001 230 337 445 452 623 309 176 832;
  • 58) 0.001 230 337 445 452 623 309 176 832 × 2 = 0 + 0.002 460 674 890 905 246 618 353 664;
  • 59) 0.002 460 674 890 905 246 618 353 664 × 2 = 0 + 0.004 921 349 781 810 493 236 707 328;
  • 60) 0.004 921 349 781 810 493 236 707 328 × 2 = 0 + 0.009 842 699 563 620 986 473 414 656;
  • 61) 0.009 842 699 563 620 986 473 414 656 × 2 = 0 + 0.019 685 399 127 241 972 946 829 312;
  • 62) 0.019 685 399 127 241 972 946 829 312 × 2 = 0 + 0.039 370 798 254 483 945 893 658 624;
  • 63) 0.039 370 798 254 483 945 893 658 624 × 2 = 0 + 0.078 741 596 508 967 891 787 317 248;
  • 64) 0.078 741 596 508 967 891 787 317 248 × 2 = 0 + 0.157 483 193 017 935 783 574 634 496;
  • 65) 0.157 483 193 017 935 783 574 634 496 × 2 = 0 + 0.314 966 386 035 871 567 149 268 992;
  • 66) 0.314 966 386 035 871 567 149 268 992 × 2 = 0 + 0.629 932 772 071 743 134 298 537 984;
  • 67) 0.629 932 772 071 743 134 298 537 984 × 2 = 1 + 0.259 865 544 143 486 268 597 075 968;
  • 68) 0.259 865 544 143 486 268 597 075 968 × 2 = 0 + 0.519 731 088 286 972 537 194 151 936;
  • 69) 0.519 731 088 286 972 537 194 151 936 × 2 = 1 + 0.039 462 176 573 945 074 388 303 872;
  • 70) 0.039 462 176 573 945 074 388 303 872 × 2 = 0 + 0.078 924 353 147 890 148 776 607 744;
  • 71) 0.078 924 353 147 890 148 776 607 744 × 2 = 0 + 0.157 848 706 295 780 297 553 215 488;
  • 72) 0.157 848 706 295 780 297 553 215 488 × 2 = 0 + 0.315 697 412 591 560 595 106 430 976;
  • 73) 0.315 697 412 591 560 595 106 430 976 × 2 = 0 + 0.631 394 825 183 121 190 212 861 952;
  • 74) 0.631 394 825 183 121 190 212 861 952 × 2 = 1 + 0.262 789 650 366 242 380 425 723 904;
  • 75) 0.262 789 650 366 242 380 425 723 904 × 2 = 0 + 0.525 579 300 732 484 760 851 447 808;
  • 76) 0.525 579 300 732 484 760 851 447 808 × 2 = 1 + 0.051 158 601 464 969 521 702 895 616;
  • 77) 0.051 158 601 464 969 521 702 895 616 × 2 = 0 + 0.102 317 202 929 939 043 405 791 232;
  • 78) 0.102 317 202 929 939 043 405 791 232 × 2 = 0 + 0.204 634 405 859 878 086 811 582 464;
  • 79) 0.204 634 405 859 878 086 811 582 464 × 2 = 0 + 0.409 268 811 719 756 173 623 164 928;
  • 80) 0.409 268 811 719 756 173 623 164 928 × 2 = 0 + 0.818 537 623 439 512 347 246 329 856;
  • 81) 0.818 537 623 439 512 347 246 329 856 × 2 = 1 + 0.637 075 246 879 024 694 492 659 712;
  • 82) 0.637 075 246 879 024 694 492 659 712 × 2 = 1 + 0.274 150 493 758 049 388 985 319 424;
  • 83) 0.274 150 493 758 049 388 985 319 424 × 2 = 0 + 0.548 300 987 516 098 777 970 638 848;
  • 84) 0.548 300 987 516 098 777 970 638 848 × 2 = 1 + 0.096 601 975 032 197 555 941 277 696;
  • 85) 0.096 601 975 032 197 555 941 277 696 × 2 = 0 + 0.193 203 950 064 395 111 882 555 392;
  • 86) 0.193 203 950 064 395 111 882 555 392 × 2 = 0 + 0.386 407 900 128 790 223 765 110 784;
  • 87) 0.386 407 900 128 790 223 765 110 784 × 2 = 0 + 0.772 815 800 257 580 447 530 221 568;
  • 88) 0.772 815 800 257 580 447 530 221 568 × 2 = 1 + 0.545 631 600 515 160 895 060 443 136;
  • 89) 0.545 631 600 515 160 895 060 443 136 × 2 = 1 + 0.091 263 201 030 321 790 120 886 272;
  • 90) 0.091 263 201 030 321 790 120 886 272 × 2 = 0 + 0.182 526 402 060 643 580 241 772 544;
  • 91) 0.182 526 402 060 643 580 241 772 544 × 2 = 0 + 0.365 052 804 121 287 160 483 545 088;
  • 92) 0.365 052 804 121 287 160 483 545 088 × 2 = 0 + 0.730 105 608 242 574 320 967 090 176;
  • 93) 0.730 105 608 242 574 320 967 090 176 × 2 = 1 + 0.460 211 216 485 148 641 934 180 352;
  • 94) 0.460 211 216 485 148 641 934 180 352 × 2 = 0 + 0.920 422 432 970 297 283 868 360 704;
  • 95) 0.920 422 432 970 297 283 868 360 704 × 2 = 1 + 0.840 844 865 940 594 567 736 721 408;
  • 96) 0.840 844 865 940 594 567 736 721 408 × 2 = 1 + 0.681 689 731 881 189 135 473 442 816;
  • 97) 0.681 689 731 881 189 135 473 442 816 × 2 = 1 + 0.363 379 463 762 378 270 946 885 632;
  • 98) 0.363 379 463 762 378 270 946 885 632 × 2 = 0 + 0.726 758 927 524 756 541 893 771 264;
  • 99) 0.726 758 927 524 756 541 893 771 264 × 2 = 1 + 0.453 517 855 049 513 083 787 542 528;
  • 100) 0.453 517 855 049 513 083 787 542 528 × 2 = 0 + 0.907 035 710 099 026 167 575 085 056;
  • 101) 0.907 035 710 099 026 167 575 085 056 × 2 = 1 + 0.814 071 420 198 052 335 150 170 112;
  • 102) 0.814 071 420 198 052 335 150 170 112 × 2 = 1 + 0.628 142 840 396 104 670 300 340 224;
  • 103) 0.628 142 840 396 104 670 300 340 224 × 2 = 1 + 0.256 285 680 792 209 340 600 680 448;
  • 104) 0.256 285 680 792 209 340 600 680 448 × 2 = 0 + 0.512 571 361 584 418 681 201 360 896;
  • 105) 0.512 571 361 584 418 681 201 360 896 × 2 = 1 + 0.025 142 723 168 837 362 402 721 792;
  • 106) 0.025 142 723 168 837 362 402 721 792 × 2 = 0 + 0.050 285 446 337 674 724 805 443 584;
  • 107) 0.050 285 446 337 674 724 805 443 584 × 2 = 0 + 0.100 570 892 675 349 449 610 887 168;
  • 108) 0.100 570 892 675 349 449 610 887 168 × 2 = 0 + 0.201 141 785 350 698 899 221 774 336;
  • 109) 0.201 141 785 350 698 899 221 774 336 × 2 = 0 + 0.402 283 570 701 397 798 443 548 672;
  • 110) 0.402 283 570 701 397 798 443 548 672 × 2 = 0 + 0.804 567 141 402 795 596 887 097 344;
  • 111) 0.804 567 141 402 795 596 887 097 344 × 2 = 1 + 0.609 134 282 805 591 193 774 194 688;
  • 112) 0.609 134 282 805 591 193 774 194 688 × 2 = 1 + 0.218 268 565 611 182 387 548 389 376;
  • 113) 0.218 268 565 611 182 387 548 389 376 × 2 = 0 + 0.436 537 131 222 364 775 096 778 752;
  • 114) 0.436 537 131 222 364 775 096 778 752 × 2 = 0 + 0.873 074 262 444 729 550 193 557 504;
  • 115) 0.873 074 262 444 729 550 193 557 504 × 2 = 1 + 0.746 148 524 889 459 100 387 115 008;
  • 116) 0.746 148 524 889 459 100 387 115 008 × 2 = 1 + 0.492 297 049 778 918 200 774 230 016;
  • 117) 0.492 297 049 778 918 200 774 230 016 × 2 = 0 + 0.984 594 099 557 836 401 548 460 032;
  • 118) 0.984 594 099 557 836 401 548 460 032 × 2 = 1 + 0.969 188 199 115 672 803 096 920 064;
  • 119) 0.969 188 199 115 672 803 096 920 064 × 2 = 1 + 0.938 376 398 231 345 606 193 840 128;

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:


0.000 000 000 000 000 000 008 537 181(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0101 0000 1101 0001 1000 1011 1010 1110 1000 0011 0011 011(2)

5. Positive number before normalization:

0.000 000 000 000 000 000 008 537 181(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0101 0000 1101 0001 1000 1011 1010 1110 1000 0011 0011 011(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:


0.000 000 000 000 000 000 008 537 181(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0101 0000 1101 0001 1000 1011 1010 1110 1000 0011 0011 011(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0101 0000 1101 0001 1000 1011 1010 1110 1000 0011 0011 011(2) × 20 =

1.0100 0010 1000 0110 1000 1100 0101 1101 0111 0100 0001 1001 1011(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 0110 1000 1100 0101 1101 0111 0100 0001 1001 1011


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 0110 1000 1100 0101 1101 0111 0100 0001 1001 1011 =

0100 0010 1000 0110 1000 1100 0101 1101 0111 0100 0001 1001 1011


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 0110 1000 1100 0101 1101 0111 0100 0001 1001 1011


Decimal number 0.000 000 000 000 000 000 008 537 181 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1011 1100 - 0100 0010 1000 0110 1000 1100 0101 1101 0111 0100 0001 1001 1011

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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