-0.000 000 000 000 000 051 120 132 432 694 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.000 000 000 000 000 051 120 132 432 694(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
-0.000 000 000 000 000 051 120 132 432 694(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. Start with the positive version of the number:

|-0.000 000 000 000 000 051 120 132 432 694| = 0.000 000 000 000 000 051 120 132 432 694


2. 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;

3. 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)


4. Convert to binary (base 2) the fractional part: 0.000 000 000 000 000 051 120 132 432 694.

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) 0.000 000 000 000 000 051 120 132 432 694 × 2 = 0 + 0.000 000 000 000 000 102 240 264 865 388;
  • 2) 0.000 000 000 000 000 102 240 264 865 388 × 2 = 0 + 0.000 000 000 000 000 204 480 529 730 776;
  • 3) 0.000 000 000 000 000 204 480 529 730 776 × 2 = 0 + 0.000 000 000 000 000 408 961 059 461 552;
  • 4) 0.000 000 000 000 000 408 961 059 461 552 × 2 = 0 + 0.000 000 000 000 000 817 922 118 923 104;
  • 5) 0.000 000 000 000 000 817 922 118 923 104 × 2 = 0 + 0.000 000 000 000 001 635 844 237 846 208;
  • 6) 0.000 000 000 000 001 635 844 237 846 208 × 2 = 0 + 0.000 000 000 000 003 271 688 475 692 416;
  • 7) 0.000 000 000 000 003 271 688 475 692 416 × 2 = 0 + 0.000 000 000 000 006 543 376 951 384 832;
  • 8) 0.000 000 000 000 006 543 376 951 384 832 × 2 = 0 + 0.000 000 000 000 013 086 753 902 769 664;
  • 9) 0.000 000 000 000 013 086 753 902 769 664 × 2 = 0 + 0.000 000 000 000 026 173 507 805 539 328;
  • 10) 0.000 000 000 000 026 173 507 805 539 328 × 2 = 0 + 0.000 000 000 000 052 347 015 611 078 656;
  • 11) 0.000 000 000 000 052 347 015 611 078 656 × 2 = 0 + 0.000 000 000 000 104 694 031 222 157 312;
  • 12) 0.000 000 000 000 104 694 031 222 157 312 × 2 = 0 + 0.000 000 000 000 209 388 062 444 314 624;
  • 13) 0.000 000 000 000 209 388 062 444 314 624 × 2 = 0 + 0.000 000 000 000 418 776 124 888 629 248;
  • 14) 0.000 000 000 000 418 776 124 888 629 248 × 2 = 0 + 0.000 000 000 000 837 552 249 777 258 496;
  • 15) 0.000 000 000 000 837 552 249 777 258 496 × 2 = 0 + 0.000 000 000 001 675 104 499 554 516 992;
  • 16) 0.000 000 000 001 675 104 499 554 516 992 × 2 = 0 + 0.000 000 000 003 350 208 999 109 033 984;
  • 17) 0.000 000 000 003 350 208 999 109 033 984 × 2 = 0 + 0.000 000 000 006 700 417 998 218 067 968;
  • 18) 0.000 000 000 006 700 417 998 218 067 968 × 2 = 0 + 0.000 000 000 013 400 835 996 436 135 936;
  • 19) 0.000 000 000 013 400 835 996 436 135 936 × 2 = 0 + 0.000 000 000 026 801 671 992 872 271 872;
  • 20) 0.000 000 000 026 801 671 992 872 271 872 × 2 = 0 + 0.000 000 000 053 603 343 985 744 543 744;
  • 21) 0.000 000 000 053 603 343 985 744 543 744 × 2 = 0 + 0.000 000 000 107 206 687 971 489 087 488;
  • 22) 0.000 000 000 107 206 687 971 489 087 488 × 2 = 0 + 0.000 000 000 214 413 375 942 978 174 976;
  • 23) 0.000 000 000 214 413 375 942 978 174 976 × 2 = 0 + 0.000 000 000 428 826 751 885 956 349 952;
  • 24) 0.000 000 000 428 826 751 885 956 349 952 × 2 = 0 + 0.000 000 000 857 653 503 771 912 699 904;
  • 25) 0.000 000 000 857 653 503 771 912 699 904 × 2 = 0 + 0.000 000 001 715 307 007 543 825 399 808;
  • 26) 0.000 000 001 715 307 007 543 825 399 808 × 2 = 0 + 0.000 000 003 430 614 015 087 650 799 616;
  • 27) 0.000 000 003 430 614 015 087 650 799 616 × 2 = 0 + 0.000 000 006 861 228 030 175 301 599 232;
  • 28) 0.000 000 006 861 228 030 175 301 599 232 × 2 = 0 + 0.000 000 013 722 456 060 350 603 198 464;
  • 29) 0.000 000 013 722 456 060 350 603 198 464 × 2 = 0 + 0.000 000 027 444 912 120 701 206 396 928;
  • 30) 0.000 000 027 444 912 120 701 206 396 928 × 2 = 0 + 0.000 000 054 889 824 241 402 412 793 856;
  • 31) 0.000 000 054 889 824 241 402 412 793 856 × 2 = 0 + 0.000 000 109 779 648 482 804 825 587 712;
  • 32) 0.000 000 109 779 648 482 804 825 587 712 × 2 = 0 + 0.000 000 219 559 296 965 609 651 175 424;
  • 33) 0.000 000 219 559 296 965 609 651 175 424 × 2 = 0 + 0.000 000 439 118 593 931 219 302 350 848;
  • 34) 0.000 000 439 118 593 931 219 302 350 848 × 2 = 0 + 0.000 000 878 237 187 862 438 604 701 696;
  • 35) 0.000 000 878 237 187 862 438 604 701 696 × 2 = 0 + 0.000 001 756 474 375 724 877 209 403 392;
  • 36) 0.000 001 756 474 375 724 877 209 403 392 × 2 = 0 + 0.000 003 512 948 751 449 754 418 806 784;
  • 37) 0.000 003 512 948 751 449 754 418 806 784 × 2 = 0 + 0.000 007 025 897 502 899 508 837 613 568;
  • 38) 0.000 007 025 897 502 899 508 837 613 568 × 2 = 0 + 0.000 014 051 795 005 799 017 675 227 136;
  • 39) 0.000 014 051 795 005 799 017 675 227 136 × 2 = 0 + 0.000 028 103 590 011 598 035 350 454 272;
  • 40) 0.000 028 103 590 011 598 035 350 454 272 × 2 = 0 + 0.000 056 207 180 023 196 070 700 908 544;
  • 41) 0.000 056 207 180 023 196 070 700 908 544 × 2 = 0 + 0.000 112 414 360 046 392 141 401 817 088;
  • 42) 0.000 112 414 360 046 392 141 401 817 088 × 2 = 0 + 0.000 224 828 720 092 784 282 803 634 176;
  • 43) 0.000 224 828 720 092 784 282 803 634 176 × 2 = 0 + 0.000 449 657 440 185 568 565 607 268 352;
  • 44) 0.000 449 657 440 185 568 565 607 268 352 × 2 = 0 + 0.000 899 314 880 371 137 131 214 536 704;
  • 45) 0.000 899 314 880 371 137 131 214 536 704 × 2 = 0 + 0.001 798 629 760 742 274 262 429 073 408;
  • 46) 0.001 798 629 760 742 274 262 429 073 408 × 2 = 0 + 0.003 597 259 521 484 548 524 858 146 816;
  • 47) 0.003 597 259 521 484 548 524 858 146 816 × 2 = 0 + 0.007 194 519 042 969 097 049 716 293 632;
  • 48) 0.007 194 519 042 969 097 049 716 293 632 × 2 = 0 + 0.014 389 038 085 938 194 099 432 587 264;
  • 49) 0.014 389 038 085 938 194 099 432 587 264 × 2 = 0 + 0.028 778 076 171 876 388 198 865 174 528;
  • 50) 0.028 778 076 171 876 388 198 865 174 528 × 2 = 0 + 0.057 556 152 343 752 776 397 730 349 056;
  • 51) 0.057 556 152 343 752 776 397 730 349 056 × 2 = 0 + 0.115 112 304 687 505 552 795 460 698 112;
  • 52) 0.115 112 304 687 505 552 795 460 698 112 × 2 = 0 + 0.230 224 609 375 011 105 590 921 396 224;
  • 53) 0.230 224 609 375 011 105 590 921 396 224 × 2 = 0 + 0.460 449 218 750 022 211 181 842 792 448;
  • 54) 0.460 449 218 750 022 211 181 842 792 448 × 2 = 0 + 0.920 898 437 500 044 422 363 685 584 896;
  • 55) 0.920 898 437 500 044 422 363 685 584 896 × 2 = 1 + 0.841 796 875 000 088 844 727 371 169 792;
  • 56) 0.841 796 875 000 088 844 727 371 169 792 × 2 = 1 + 0.683 593 750 000 177 689 454 742 339 584;
  • 57) 0.683 593 750 000 177 689 454 742 339 584 × 2 = 1 + 0.367 187 500 000 355 378 909 484 679 168;
  • 58) 0.367 187 500 000 355 378 909 484 679 168 × 2 = 0 + 0.734 375 000 000 710 757 818 969 358 336;
  • 59) 0.734 375 000 000 710 757 818 969 358 336 × 2 = 1 + 0.468 750 000 001 421 515 637 938 716 672;
  • 60) 0.468 750 000 001 421 515 637 938 716 672 × 2 = 0 + 0.937 500 000 002 843 031 275 877 433 344;
  • 61) 0.937 500 000 002 843 031 275 877 433 344 × 2 = 1 + 0.875 000 000 005 686 062 551 754 866 688;
  • 62) 0.875 000 000 005 686 062 551 754 866 688 × 2 = 1 + 0.750 000 000 011 372 125 103 509 733 376;
  • 63) 0.750 000 000 011 372 125 103 509 733 376 × 2 = 1 + 0.500 000 000 022 744 250 207 019 466 752;
  • 64) 0.500 000 000 022 744 250 207 019 466 752 × 2 = 1 + 0.000 000 000 045 488 500 414 038 933 504;
  • 65) 0.000 000 000 045 488 500 414 038 933 504 × 2 = 0 + 0.000 000 000 090 977 000 828 077 867 008;
  • 66) 0.000 000 000 090 977 000 828 077 867 008 × 2 = 0 + 0.000 000 000 181 954 001 656 155 734 016;
  • 67) 0.000 000 000 181 954 001 656 155 734 016 × 2 = 0 + 0.000 000 000 363 908 003 312 311 468 032;
  • 68) 0.000 000 000 363 908 003 312 311 468 032 × 2 = 0 + 0.000 000 000 727 816 006 624 622 936 064;
  • 69) 0.000 000 000 727 816 006 624 622 936 064 × 2 = 0 + 0.000 000 001 455 632 013 249 245 872 128;
  • 70) 0.000 000 001 455 632 013 249 245 872 128 × 2 = 0 + 0.000 000 002 911 264 026 498 491 744 256;
  • 71) 0.000 000 002 911 264 026 498 491 744 256 × 2 = 0 + 0.000 000 005 822 528 052 996 983 488 512;
  • 72) 0.000 000 005 822 528 052 996 983 488 512 × 2 = 0 + 0.000 000 011 645 056 105 993 966 977 024;
  • 73) 0.000 000 011 645 056 105 993 966 977 024 × 2 = 0 + 0.000 000 023 290 112 211 987 933 954 048;
  • 74) 0.000 000 023 290 112 211 987 933 954 048 × 2 = 0 + 0.000 000 046 580 224 423 975 867 908 096;
  • 75) 0.000 000 046 580 224 423 975 867 908 096 × 2 = 0 + 0.000 000 093 160 448 847 951 735 816 192;
  • 76) 0.000 000 093 160 448 847 951 735 816 192 × 2 = 0 + 0.000 000 186 320 897 695 903 471 632 384;
  • 77) 0.000 000 186 320 897 695 903 471 632 384 × 2 = 0 + 0.000 000 372 641 795 391 806 943 264 768;
  • 78) 0.000 000 372 641 795 391 806 943 264 768 × 2 = 0 + 0.000 000 745 283 590 783 613 886 529 536;
  • 79) 0.000 000 745 283 590 783 613 886 529 536 × 2 = 0 + 0.000 001 490 567 181 567 227 773 059 072;
  • 80) 0.000 001 490 567 181 567 227 773 059 072 × 2 = 0 + 0.000 002 981 134 363 134 455 546 118 144;
  • 81) 0.000 002 981 134 363 134 455 546 118 144 × 2 = 0 + 0.000 005 962 268 726 268 911 092 236 288;
  • 82) 0.000 005 962 268 726 268 911 092 236 288 × 2 = 0 + 0.000 011 924 537 452 537 822 184 472 576;
  • 83) 0.000 011 924 537 452 537 822 184 472 576 × 2 = 0 + 0.000 023 849 074 905 075 644 368 945 152;
  • 84) 0.000 023 849 074 905 075 644 368 945 152 × 2 = 0 + 0.000 047 698 149 810 151 288 737 890 304;
  • 85) 0.000 047 698 149 810 151 288 737 890 304 × 2 = 0 + 0.000 095 396 299 620 302 577 475 780 608;
  • 86) 0.000 095 396 299 620 302 577 475 780 608 × 2 = 0 + 0.000 190 792 599 240 605 154 951 561 216;
  • 87) 0.000 190 792 599 240 605 154 951 561 216 × 2 = 0 + 0.000 381 585 198 481 210 309 903 122 432;
  • 88) 0.000 381 585 198 481 210 309 903 122 432 × 2 = 0 + 0.000 763 170 396 962 420 619 806 244 864;
  • 89) 0.000 763 170 396 962 420 619 806 244 864 × 2 = 0 + 0.001 526 340 793 924 841 239 612 489 728;
  • 90) 0.001 526 340 793 924 841 239 612 489 728 × 2 = 0 + 0.003 052 681 587 849 682 479 224 979 456;
  • 91) 0.003 052 681 587 849 682 479 224 979 456 × 2 = 0 + 0.006 105 363 175 699 364 958 449 958 912;
  • 92) 0.006 105 363 175 699 364 958 449 958 912 × 2 = 0 + 0.012 210 726 351 398 729 916 899 917 824;
  • 93) 0.012 210 726 351 398 729 916 899 917 824 × 2 = 0 + 0.024 421 452 702 797 459 833 799 835 648;
  • 94) 0.024 421 452 702 797 459 833 799 835 648 × 2 = 0 + 0.048 842 905 405 594 919 667 599 671 296;
  • 95) 0.048 842 905 405 594 919 667 599 671 296 × 2 = 0 + 0.097 685 810 811 189 839 335 199 342 592;
  • 96) 0.097 685 810 811 189 839 335 199 342 592 × 2 = 0 + 0.195 371 621 622 379 678 670 398 685 184;
  • 97) 0.195 371 621 622 379 678 670 398 685 184 × 2 = 0 + 0.390 743 243 244 759 357 340 797 370 368;
  • 98) 0.390 743 243 244 759 357 340 797 370 368 × 2 = 0 + 0.781 486 486 489 518 714 681 594 740 736;
  • 99) 0.781 486 486 489 518 714 681 594 740 736 × 2 = 1 + 0.562 972 972 979 037 429 363 189 481 472;
  • 100) 0.562 972 972 979 037 429 363 189 481 472 × 2 = 1 + 0.125 945 945 958 074 858 726 378 962 944;
  • 101) 0.125 945 945 958 074 858 726 378 962 944 × 2 = 0 + 0.251 891 891 916 149 717 452 757 925 888;
  • 102) 0.251 891 891 916 149 717 452 757 925 888 × 2 = 0 + 0.503 783 783 832 299 434 905 515 851 776;
  • 103) 0.503 783 783 832 299 434 905 515 851 776 × 2 = 1 + 0.007 567 567 664 598 869 811 031 703 552;
  • 104) 0.007 567 567 664 598 869 811 031 703 552 × 2 = 0 + 0.015 135 135 329 197 739 622 063 407 104;
  • 105) 0.015 135 135 329 197 739 622 063 407 104 × 2 = 0 + 0.030 270 270 658 395 479 244 126 814 208;
  • 106) 0.030 270 270 658 395 479 244 126 814 208 × 2 = 0 + 0.060 540 541 316 790 958 488 253 628 416;
  • 107) 0.060 540 541 316 790 958 488 253 628 416 × 2 = 0 + 0.121 081 082 633 581 916 976 507 256 832;

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).


5. 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:


0.000 000 000 000 000 051 120 132 432 694(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 1010 1111 0000 0000 0000 0000 0000 0000 0000 0000 0011 0010 000(2)

6. Positive number before normalization:

0.000 000 000 000 000 051 120 132 432 694(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 1010 1111 0000 0000 0000 0000 0000 0000 0000 0000 0011 0010 000(2)

7. Normalize the binary representation of the number.

Shift the decimal mark 55 positions to the right, so that only one non zero digit remains to the left of it:


0.000 000 000 000 000 051 120 132 432 694(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 1010 1111 0000 0000 0000 0000 0000 0000 0000 0000 0011 0010 000(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 1010 1111 0000 0000 0000 0000 0000 0000 0000 0000 0011 0010 000(2) × 20 =

1.1101 0111 1000 0000 0000 0000 0000 0000 0000 0000 0001 1001 0000(2) × 2-55


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): -55

Mantissa (not normalized):
1.1101 0111 1000 0000 0000 0000 0000 0000 0000 0000 0001 1001 0000


9. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

Exponent (unadjusted) + 2(11-1) - 1 =

-55 + 2(11-1) - 1 =

(-55 + 1 023)(10) =

968(10)


10. 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;
  • 968 ÷ 2 = 484 + 0;
  • 484 ÷ 2 = 242 + 0;
  • 242 ÷ 2 = 121 + 0;
  • 121 ÷ 2 = 60 + 1;
  • 60 ÷ 2 = 30 + 0;
  • 30 ÷ 2 = 15 + 0;
  • 15 ÷ 2 = 7 + 1;
  • 7 ÷ 2 = 3 + 1;
  • 3 ÷ 2 = 1 + 1;
  • 1 ÷ 2 = 0 + 1;

11. Construct the base 2 representation of the adjusted exponent.

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


Exponent (adjusted) =

968(10) =

011 1100 1000(2)


12. 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. 1101 0111 1000 0000 0000 0000 0000 0000 0000 0000 0001 1001 0000 =

1101 0111 1000 0000 0000 0000 0000 0000 0000 0000 0001 1001 0000


13. The three elements that make up the number's 64 bit double precision IEEE 754 binary floating point representation:

Sign (1 bit) =
1 (a negative number)

Exponent (11 bits) =
011 1100 1000

Mantissa (52 bits) =
1101 0111 1000 0000 0000 0000 0000 0000 0000 0000 0001 1001 0000


Decimal number -0.000 000 000 000 000 051 120 132 432 694 converted to 64 bit double precision IEEE 754 binary floating point representation:

1 - 011 1100 1000 - 1101 0111 1000 0000 0000 0000 0000 0000 0000 0000 0001 1001 0000

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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