-0.000 000 000 000 000 222 044 604 925 003 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.000 000 000 000 000 222 044 604 925 003(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
-0.000 000 000 000 000 222 044 604 925 003(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:

|-0.000 000 000 000 000 222 044 604 925 003| = 0.000 000 000 000 000 222 044 604 925 003


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: 0.000 000 000 000 000 222 044 604 925 003.

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) 0.000 000 000 000 000 222 044 604 925 003 × 2 = 0 + 0.000 000 000 000 000 444 089 209 850 006;
  • 2) 0.000 000 000 000 000 444 089 209 850 006 × 2 = 0 + 0.000 000 000 000 000 888 178 419 700 012;
  • 3) 0.000 000 000 000 000 888 178 419 700 012 × 2 = 0 + 0.000 000 000 000 001 776 356 839 400 024;
  • 4) 0.000 000 000 000 001 776 356 839 400 024 × 2 = 0 + 0.000 000 000 000 003 552 713 678 800 048;
  • 5) 0.000 000 000 000 003 552 713 678 800 048 × 2 = 0 + 0.000 000 000 000 007 105 427 357 600 096;
  • 6) 0.000 000 000 000 007 105 427 357 600 096 × 2 = 0 + 0.000 000 000 000 014 210 854 715 200 192;
  • 7) 0.000 000 000 000 014 210 854 715 200 192 × 2 = 0 + 0.000 000 000 000 028 421 709 430 400 384;
  • 8) 0.000 000 000 000 028 421 709 430 400 384 × 2 = 0 + 0.000 000 000 000 056 843 418 860 800 768;
  • 9) 0.000 000 000 000 056 843 418 860 800 768 × 2 = 0 + 0.000 000 000 000 113 686 837 721 601 536;
  • 10) 0.000 000 000 000 113 686 837 721 601 536 × 2 = 0 + 0.000 000 000 000 227 373 675 443 203 072;
  • 11) 0.000 000 000 000 227 373 675 443 203 072 × 2 = 0 + 0.000 000 000 000 454 747 350 886 406 144;
  • 12) 0.000 000 000 000 454 747 350 886 406 144 × 2 = 0 + 0.000 000 000 000 909 494 701 772 812 288;
  • 13) 0.000 000 000 000 909 494 701 772 812 288 × 2 = 0 + 0.000 000 000 001 818 989 403 545 624 576;
  • 14) 0.000 000 000 001 818 989 403 545 624 576 × 2 = 0 + 0.000 000 000 003 637 978 807 091 249 152;
  • 15) 0.000 000 000 003 637 978 807 091 249 152 × 2 = 0 + 0.000 000 000 007 275 957 614 182 498 304;
  • 16) 0.000 000 000 007 275 957 614 182 498 304 × 2 = 0 + 0.000 000 000 014 551 915 228 364 996 608;
  • 17) 0.000 000 000 014 551 915 228 364 996 608 × 2 = 0 + 0.000 000 000 029 103 830 456 729 993 216;
  • 18) 0.000 000 000 029 103 830 456 729 993 216 × 2 = 0 + 0.000 000 000 058 207 660 913 459 986 432;
  • 19) 0.000 000 000 058 207 660 913 459 986 432 × 2 = 0 + 0.000 000 000 116 415 321 826 919 972 864;
  • 20) 0.000 000 000 116 415 321 826 919 972 864 × 2 = 0 + 0.000 000 000 232 830 643 653 839 945 728;
  • 21) 0.000 000 000 232 830 643 653 839 945 728 × 2 = 0 + 0.000 000 000 465 661 287 307 679 891 456;
  • 22) 0.000 000 000 465 661 287 307 679 891 456 × 2 = 0 + 0.000 000 000 931 322 574 615 359 782 912;
  • 23) 0.000 000 000 931 322 574 615 359 782 912 × 2 = 0 + 0.000 000 001 862 645 149 230 719 565 824;
  • 24) 0.000 000 001 862 645 149 230 719 565 824 × 2 = 0 + 0.000 000 003 725 290 298 461 439 131 648;
  • 25) 0.000 000 003 725 290 298 461 439 131 648 × 2 = 0 + 0.000 000 007 450 580 596 922 878 263 296;
  • 26) 0.000 000 007 450 580 596 922 878 263 296 × 2 = 0 + 0.000 000 014 901 161 193 845 756 526 592;
  • 27) 0.000 000 014 901 161 193 845 756 526 592 × 2 = 0 + 0.000 000 029 802 322 387 691 513 053 184;
  • 28) 0.000 000 029 802 322 387 691 513 053 184 × 2 = 0 + 0.000 000 059 604 644 775 383 026 106 368;
  • 29) 0.000 000 059 604 644 775 383 026 106 368 × 2 = 0 + 0.000 000 119 209 289 550 766 052 212 736;
  • 30) 0.000 000 119 209 289 550 766 052 212 736 × 2 = 0 + 0.000 000 238 418 579 101 532 104 425 472;
  • 31) 0.000 000 238 418 579 101 532 104 425 472 × 2 = 0 + 0.000 000 476 837 158 203 064 208 850 944;
  • 32) 0.000 000 476 837 158 203 064 208 850 944 × 2 = 0 + 0.000 000 953 674 316 406 128 417 701 888;
  • 33) 0.000 000 953 674 316 406 128 417 701 888 × 2 = 0 + 0.000 001 907 348 632 812 256 835 403 776;
  • 34) 0.000 001 907 348 632 812 256 835 403 776 × 2 = 0 + 0.000 003 814 697 265 624 513 670 807 552;
  • 35) 0.000 003 814 697 265 624 513 670 807 552 × 2 = 0 + 0.000 007 629 394 531 249 027 341 615 104;
  • 36) 0.000 007 629 394 531 249 027 341 615 104 × 2 = 0 + 0.000 015 258 789 062 498 054 683 230 208;
  • 37) 0.000 015 258 789 062 498 054 683 230 208 × 2 = 0 + 0.000 030 517 578 124 996 109 366 460 416;
  • 38) 0.000 030 517 578 124 996 109 366 460 416 × 2 = 0 + 0.000 061 035 156 249 992 218 732 920 832;
  • 39) 0.000 061 035 156 249 992 218 732 920 832 × 2 = 0 + 0.000 122 070 312 499 984 437 465 841 664;
  • 40) 0.000 122 070 312 499 984 437 465 841 664 × 2 = 0 + 0.000 244 140 624 999 968 874 931 683 328;
  • 41) 0.000 244 140 624 999 968 874 931 683 328 × 2 = 0 + 0.000 488 281 249 999 937 749 863 366 656;
  • 42) 0.000 488 281 249 999 937 749 863 366 656 × 2 = 0 + 0.000 976 562 499 999 875 499 726 733 312;
  • 43) 0.000 976 562 499 999 875 499 726 733 312 × 2 = 0 + 0.001 953 124 999 999 750 999 453 466 624;
  • 44) 0.001 953 124 999 999 750 999 453 466 624 × 2 = 0 + 0.003 906 249 999 999 501 998 906 933 248;
  • 45) 0.003 906 249 999 999 501 998 906 933 248 × 2 = 0 + 0.007 812 499 999 999 003 997 813 866 496;
  • 46) 0.007 812 499 999 999 003 997 813 866 496 × 2 = 0 + 0.015 624 999 999 998 007 995 627 732 992;
  • 47) 0.015 624 999 999 998 007 995 627 732 992 × 2 = 0 + 0.031 249 999 999 996 015 991 255 465 984;
  • 48) 0.031 249 999 999 996 015 991 255 465 984 × 2 = 0 + 0.062 499 999 999 992 031 982 510 931 968;
  • 49) 0.062 499 999 999 992 031 982 510 931 968 × 2 = 0 + 0.124 999 999 999 984 063 965 021 863 936;
  • 50) 0.124 999 999 999 984 063 965 021 863 936 × 2 = 0 + 0.249 999 999 999 968 127 930 043 727 872;
  • 51) 0.249 999 999 999 968 127 930 043 727 872 × 2 = 0 + 0.499 999 999 999 936 255 860 087 455 744;
  • 52) 0.499 999 999 999 936 255 860 087 455 744 × 2 = 0 + 0.999 999 999 999 872 511 720 174 911 488;
  • 53) 0.999 999 999 999 872 511 720 174 911 488 × 2 = 1 + 0.999 999 999 999 745 023 440 349 822 976;
  • 54) 0.999 999 999 999 745 023 440 349 822 976 × 2 = 1 + 0.999 999 999 999 490 046 880 699 645 952;
  • 55) 0.999 999 999 999 490 046 880 699 645 952 × 2 = 1 + 0.999 999 999 998 980 093 761 399 291 904;
  • 56) 0.999 999 999 998 980 093 761 399 291 904 × 2 = 1 + 0.999 999 999 997 960 187 522 798 583 808;
  • 57) 0.999 999 999 997 960 187 522 798 583 808 × 2 = 1 + 0.999 999 999 995 920 375 045 597 167 616;
  • 58) 0.999 999 999 995 920 375 045 597 167 616 × 2 = 1 + 0.999 999 999 991 840 750 091 194 335 232;
  • 59) 0.999 999 999 991 840 750 091 194 335 232 × 2 = 1 + 0.999 999 999 983 681 500 182 388 670 464;
  • 60) 0.999 999 999 983 681 500 182 388 670 464 × 2 = 1 + 0.999 999 999 967 363 000 364 777 340 928;
  • 61) 0.999 999 999 967 363 000 364 777 340 928 × 2 = 1 + 0.999 999 999 934 726 000 729 554 681 856;
  • 62) 0.999 999 999 934 726 000 729 554 681 856 × 2 = 1 + 0.999 999 999 869 452 001 459 109 363 712;
  • 63) 0.999 999 999 869 452 001 459 109 363 712 × 2 = 1 + 0.999 999 999 738 904 002 918 218 727 424;
  • 64) 0.999 999 999 738 904 002 918 218 727 424 × 2 = 1 + 0.999 999 999 477 808 005 836 437 454 848;
  • 65) 0.999 999 999 477 808 005 836 437 454 848 × 2 = 1 + 0.999 999 998 955 616 011 672 874 909 696;
  • 66) 0.999 999 998 955 616 011 672 874 909 696 × 2 = 1 + 0.999 999 997 911 232 023 345 749 819 392;
  • 67) 0.999 999 997 911 232 023 345 749 819 392 × 2 = 1 + 0.999 999 995 822 464 046 691 499 638 784;
  • 68) 0.999 999 995 822 464 046 691 499 638 784 × 2 = 1 + 0.999 999 991 644 928 093 382 999 277 568;
  • 69) 0.999 999 991 644 928 093 382 999 277 568 × 2 = 1 + 0.999 999 983 289 856 186 765 998 555 136;
  • 70) 0.999 999 983 289 856 186 765 998 555 136 × 2 = 1 + 0.999 999 966 579 712 373 531 997 110 272;
  • 71) 0.999 999 966 579 712 373 531 997 110 272 × 2 = 1 + 0.999 999 933 159 424 747 063 994 220 544;
  • 72) 0.999 999 933 159 424 747 063 994 220 544 × 2 = 1 + 0.999 999 866 318 849 494 127 988 441 088;
  • 73) 0.999 999 866 318 849 494 127 988 441 088 × 2 = 1 + 0.999 999 732 637 698 988 255 976 882 176;
  • 74) 0.999 999 732 637 698 988 255 976 882 176 × 2 = 1 + 0.999 999 465 275 397 976 511 953 764 352;
  • 75) 0.999 999 465 275 397 976 511 953 764 352 × 2 = 1 + 0.999 998 930 550 795 953 023 907 528 704;
  • 76) 0.999 998 930 550 795 953 023 907 528 704 × 2 = 1 + 0.999 997 861 101 591 906 047 815 057 408;
  • 77) 0.999 997 861 101 591 906 047 815 057 408 × 2 = 1 + 0.999 995 722 203 183 812 095 630 114 816;
  • 78) 0.999 995 722 203 183 812 095 630 114 816 × 2 = 1 + 0.999 991 444 406 367 624 191 260 229 632;
  • 79) 0.999 991 444 406 367 624 191 260 229 632 × 2 = 1 + 0.999 982 888 812 735 248 382 520 459 264;
  • 80) 0.999 982 888 812 735 248 382 520 459 264 × 2 = 1 + 0.999 965 777 625 470 496 765 040 918 528;
  • 81) 0.999 965 777 625 470 496 765 040 918 528 × 2 = 1 + 0.999 931 555 250 940 993 530 081 837 056;
  • 82) 0.999 931 555 250 940 993 530 081 837 056 × 2 = 1 + 0.999 863 110 501 881 987 060 163 674 112;
  • 83) 0.999 863 110 501 881 987 060 163 674 112 × 2 = 1 + 0.999 726 221 003 763 974 120 327 348 224;
  • 84) 0.999 726 221 003 763 974 120 327 348 224 × 2 = 1 + 0.999 452 442 007 527 948 240 654 696 448;
  • 85) 0.999 452 442 007 527 948 240 654 696 448 × 2 = 1 + 0.998 904 884 015 055 896 481 309 392 896;
  • 86) 0.998 904 884 015 055 896 481 309 392 896 × 2 = 1 + 0.997 809 768 030 111 792 962 618 785 792;
  • 87) 0.997 809 768 030 111 792 962 618 785 792 × 2 = 1 + 0.995 619 536 060 223 585 925 237 571 584;
  • 88) 0.995 619 536 060 223 585 925 237 571 584 × 2 = 1 + 0.991 239 072 120 447 171 850 475 143 168;
  • 89) 0.991 239 072 120 447 171 850 475 143 168 × 2 = 1 + 0.982 478 144 240 894 343 700 950 286 336;
  • 90) 0.982 478 144 240 894 343 700 950 286 336 × 2 = 1 + 0.964 956 288 481 788 687 401 900 572 672;
  • 91) 0.964 956 288 481 788 687 401 900 572 672 × 2 = 1 + 0.929 912 576 963 577 374 803 801 145 344;
  • 92) 0.929 912 576 963 577 374 803 801 145 344 × 2 = 1 + 0.859 825 153 927 154 749 607 602 290 688;
  • 93) 0.859 825 153 927 154 749 607 602 290 688 × 2 = 1 + 0.719 650 307 854 309 499 215 204 581 376;
  • 94) 0.719 650 307 854 309 499 215 204 581 376 × 2 = 1 + 0.439 300 615 708 618 998 430 409 162 752;
  • 95) 0.439 300 615 708 618 998 430 409 162 752 × 2 = 0 + 0.878 601 231 417 237 996 860 818 325 504;
  • 96) 0.878 601 231 417 237 996 860 818 325 504 × 2 = 1 + 0.757 202 462 834 475 993 721 636 651 008;
  • 97) 0.757 202 462 834 475 993 721 636 651 008 × 2 = 1 + 0.514 404 925 668 951 987 443 273 302 016;
  • 98) 0.514 404 925 668 951 987 443 273 302 016 × 2 = 1 + 0.028 809 851 337 903 974 886 546 604 032;
  • 99) 0.028 809 851 337 903 974 886 546 604 032 × 2 = 0 + 0.057 619 702 675 807 949 773 093 208 064;
  • 100) 0.057 619 702 675 807 949 773 093 208 064 × 2 = 0 + 0.115 239 405 351 615 899 546 186 416 128;
  • 101) 0.115 239 405 351 615 899 546 186 416 128 × 2 = 0 + 0.230 478 810 703 231 799 092 372 832 256;
  • 102) 0.230 478 810 703 231 799 092 372 832 256 × 2 = 0 + 0.460 957 621 406 463 598 184 745 664 512;
  • 103) 0.460 957 621 406 463 598 184 745 664 512 × 2 = 0 + 0.921 915 242 812 927 196 369 491 329 024;
  • 104) 0.921 915 242 812 927 196 369 491 329 024 × 2 = 1 + 0.843 830 485 625 854 392 738 982 658 048;
  • 105) 0.843 830 485 625 854 392 738 982 658 048 × 2 = 1 + 0.687 660 971 251 708 785 477 965 316 096;

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:


0.000 000 000 000 000 222 044 604 925 003(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1101 1100 0001 1(2)

6. Positive number before normalization:

0.000 000 000 000 000 222 044 604 925 003(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1101 1100 0001 1(2)

7. Normalize the binary representation of the number.

Shift the decimal mark 53 positions to the right, so that only one non zero digit remains to the left of it:


0.000 000 000 000 000 222 044 604 925 003(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1101 1100 0001 1(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1101 1100 0001 1(2) × 20 =

1.1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1011 1000 0011(2) × 2-53


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): -53

Mantissa (not normalized):
1.1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1011 1000 0011


9. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

-53 + 2(11-1) - 1 =

(-53 + 1 023)(10) =

970(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;
  • 970 ÷ 2 = 485 + 0;
  • 485 ÷ 2 = 242 + 1;
  • 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) =

970(10) =

011 1100 1010(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. 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1011 1000 0011 =

1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1011 1000 0011


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 1010

Mantissa (52 bits) =
1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1011 1000 0011


Decimal number -0.000 000 000 000 000 222 044 604 925 003 converted to 64 bit double precision IEEE 754 binary floating point representation:

1 - 011 1100 1010 - 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1011 1000 0011

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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