0.000 000 000 000 000 000 008 535 37 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 000 000 008 535 37(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
0.000 000 000 000 000 000 008 535 37(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: 0.000 000 000 000 000 000 008 535 37.

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) 0.000 000 000 000 000 000 008 535 37 × 2 = 0 + 0.000 000 000 000 000 000 017 070 74;
  • 2) 0.000 000 000 000 000 000 017 070 74 × 2 = 0 + 0.000 000 000 000 000 000 034 141 48;
  • 3) 0.000 000 000 000 000 000 034 141 48 × 2 = 0 + 0.000 000 000 000 000 000 068 282 96;
  • 4) 0.000 000 000 000 000 000 068 282 96 × 2 = 0 + 0.000 000 000 000 000 000 136 565 92;
  • 5) 0.000 000 000 000 000 000 136 565 92 × 2 = 0 + 0.000 000 000 000 000 000 273 131 84;
  • 6) 0.000 000 000 000 000 000 273 131 84 × 2 = 0 + 0.000 000 000 000 000 000 546 263 68;
  • 7) 0.000 000 000 000 000 000 546 263 68 × 2 = 0 + 0.000 000 000 000 000 001 092 527 36;
  • 8) 0.000 000 000 000 000 001 092 527 36 × 2 = 0 + 0.000 000 000 000 000 002 185 054 72;
  • 9) 0.000 000 000 000 000 002 185 054 72 × 2 = 0 + 0.000 000 000 000 000 004 370 109 44;
  • 10) 0.000 000 000 000 000 004 370 109 44 × 2 = 0 + 0.000 000 000 000 000 008 740 218 88;
  • 11) 0.000 000 000 000 000 008 740 218 88 × 2 = 0 + 0.000 000 000 000 000 017 480 437 76;
  • 12) 0.000 000 000 000 000 017 480 437 76 × 2 = 0 + 0.000 000 000 000 000 034 960 875 52;
  • 13) 0.000 000 000 000 000 034 960 875 52 × 2 = 0 + 0.000 000 000 000 000 069 921 751 04;
  • 14) 0.000 000 000 000 000 069 921 751 04 × 2 = 0 + 0.000 000 000 000 000 139 843 502 08;
  • 15) 0.000 000 000 000 000 139 843 502 08 × 2 = 0 + 0.000 000 000 000 000 279 687 004 16;
  • 16) 0.000 000 000 000 000 279 687 004 16 × 2 = 0 + 0.000 000 000 000 000 559 374 008 32;
  • 17) 0.000 000 000 000 000 559 374 008 32 × 2 = 0 + 0.000 000 000 000 001 118 748 016 64;
  • 18) 0.000 000 000 000 001 118 748 016 64 × 2 = 0 + 0.000 000 000 000 002 237 496 033 28;
  • 19) 0.000 000 000 000 002 237 496 033 28 × 2 = 0 + 0.000 000 000 000 004 474 992 066 56;
  • 20) 0.000 000 000 000 004 474 992 066 56 × 2 = 0 + 0.000 000 000 000 008 949 984 133 12;
  • 21) 0.000 000 000 000 008 949 984 133 12 × 2 = 0 + 0.000 000 000 000 017 899 968 266 24;
  • 22) 0.000 000 000 000 017 899 968 266 24 × 2 = 0 + 0.000 000 000 000 035 799 936 532 48;
  • 23) 0.000 000 000 000 035 799 936 532 48 × 2 = 0 + 0.000 000 000 000 071 599 873 064 96;
  • 24) 0.000 000 000 000 071 599 873 064 96 × 2 = 0 + 0.000 000 000 000 143 199 746 129 92;
  • 25) 0.000 000 000 000 143 199 746 129 92 × 2 = 0 + 0.000 000 000 000 286 399 492 259 84;
  • 26) 0.000 000 000 000 286 399 492 259 84 × 2 = 0 + 0.000 000 000 000 572 798 984 519 68;
  • 27) 0.000 000 000 000 572 798 984 519 68 × 2 = 0 + 0.000 000 000 001 145 597 969 039 36;
  • 28) 0.000 000 000 001 145 597 969 039 36 × 2 = 0 + 0.000 000 000 002 291 195 938 078 72;
  • 29) 0.000 000 000 002 291 195 938 078 72 × 2 = 0 + 0.000 000 000 004 582 391 876 157 44;
  • 30) 0.000 000 000 004 582 391 876 157 44 × 2 = 0 + 0.000 000 000 009 164 783 752 314 88;
  • 31) 0.000 000 000 009 164 783 752 314 88 × 2 = 0 + 0.000 000 000 018 329 567 504 629 76;
  • 32) 0.000 000 000 018 329 567 504 629 76 × 2 = 0 + 0.000 000 000 036 659 135 009 259 52;
  • 33) 0.000 000 000 036 659 135 009 259 52 × 2 = 0 + 0.000 000 000 073 318 270 018 519 04;
  • 34) 0.000 000 000 073 318 270 018 519 04 × 2 = 0 + 0.000 000 000 146 636 540 037 038 08;
  • 35) 0.000 000 000 146 636 540 037 038 08 × 2 = 0 + 0.000 000 000 293 273 080 074 076 16;
  • 36) 0.000 000 000 293 273 080 074 076 16 × 2 = 0 + 0.000 000 000 586 546 160 148 152 32;
  • 37) 0.000 000 000 586 546 160 148 152 32 × 2 = 0 + 0.000 000 001 173 092 320 296 304 64;
  • 38) 0.000 000 001 173 092 320 296 304 64 × 2 = 0 + 0.000 000 002 346 184 640 592 609 28;
  • 39) 0.000 000 002 346 184 640 592 609 28 × 2 = 0 + 0.000 000 004 692 369 281 185 218 56;
  • 40) 0.000 000 004 692 369 281 185 218 56 × 2 = 0 + 0.000 000 009 384 738 562 370 437 12;
  • 41) 0.000 000 009 384 738 562 370 437 12 × 2 = 0 + 0.000 000 018 769 477 124 740 874 24;
  • 42) 0.000 000 018 769 477 124 740 874 24 × 2 = 0 + 0.000 000 037 538 954 249 481 748 48;
  • 43) 0.000 000 037 538 954 249 481 748 48 × 2 = 0 + 0.000 000 075 077 908 498 963 496 96;
  • 44) 0.000 000 075 077 908 498 963 496 96 × 2 = 0 + 0.000 000 150 155 816 997 926 993 92;
  • 45) 0.000 000 150 155 816 997 926 993 92 × 2 = 0 + 0.000 000 300 311 633 995 853 987 84;
  • 46) 0.000 000 300 311 633 995 853 987 84 × 2 = 0 + 0.000 000 600 623 267 991 707 975 68;
  • 47) 0.000 000 600 623 267 991 707 975 68 × 2 = 0 + 0.000 001 201 246 535 983 415 951 36;
  • 48) 0.000 001 201 246 535 983 415 951 36 × 2 = 0 + 0.000 002 402 493 071 966 831 902 72;
  • 49) 0.000 002 402 493 071 966 831 902 72 × 2 = 0 + 0.000 004 804 986 143 933 663 805 44;
  • 50) 0.000 004 804 986 143 933 663 805 44 × 2 = 0 + 0.000 009 609 972 287 867 327 610 88;
  • 51) 0.000 009 609 972 287 867 327 610 88 × 2 = 0 + 0.000 019 219 944 575 734 655 221 76;
  • 52) 0.000 019 219 944 575 734 655 221 76 × 2 = 0 + 0.000 038 439 889 151 469 310 443 52;
  • 53) 0.000 038 439 889 151 469 310 443 52 × 2 = 0 + 0.000 076 879 778 302 938 620 887 04;
  • 54) 0.000 076 879 778 302 938 620 887 04 × 2 = 0 + 0.000 153 759 556 605 877 241 774 08;
  • 55) 0.000 153 759 556 605 877 241 774 08 × 2 = 0 + 0.000 307 519 113 211 754 483 548 16;
  • 56) 0.000 307 519 113 211 754 483 548 16 × 2 = 0 + 0.000 615 038 226 423 508 967 096 32;
  • 57) 0.000 615 038 226 423 508 967 096 32 × 2 = 0 + 0.001 230 076 452 847 017 934 192 64;
  • 58) 0.001 230 076 452 847 017 934 192 64 × 2 = 0 + 0.002 460 152 905 694 035 868 385 28;
  • 59) 0.002 460 152 905 694 035 868 385 28 × 2 = 0 + 0.004 920 305 811 388 071 736 770 56;
  • 60) 0.004 920 305 811 388 071 736 770 56 × 2 = 0 + 0.009 840 611 622 776 143 473 541 12;
  • 61) 0.009 840 611 622 776 143 473 541 12 × 2 = 0 + 0.019 681 223 245 552 286 947 082 24;
  • 62) 0.019 681 223 245 552 286 947 082 24 × 2 = 0 + 0.039 362 446 491 104 573 894 164 48;
  • 63) 0.039 362 446 491 104 573 894 164 48 × 2 = 0 + 0.078 724 892 982 209 147 788 328 96;
  • 64) 0.078 724 892 982 209 147 788 328 96 × 2 = 0 + 0.157 449 785 964 418 295 576 657 92;
  • 65) 0.157 449 785 964 418 295 576 657 92 × 2 = 0 + 0.314 899 571 928 836 591 153 315 84;
  • 66) 0.314 899 571 928 836 591 153 315 84 × 2 = 0 + 0.629 799 143 857 673 182 306 631 68;
  • 67) 0.629 799 143 857 673 182 306 631 68 × 2 = 1 + 0.259 598 287 715 346 364 613 263 36;
  • 68) 0.259 598 287 715 346 364 613 263 36 × 2 = 0 + 0.519 196 575 430 692 729 226 526 72;
  • 69) 0.519 196 575 430 692 729 226 526 72 × 2 = 1 + 0.038 393 150 861 385 458 453 053 44;
  • 70) 0.038 393 150 861 385 458 453 053 44 × 2 = 0 + 0.076 786 301 722 770 916 906 106 88;
  • 71) 0.076 786 301 722 770 916 906 106 88 × 2 = 0 + 0.153 572 603 445 541 833 812 213 76;
  • 72) 0.153 572 603 445 541 833 812 213 76 × 2 = 0 + 0.307 145 206 891 083 667 624 427 52;
  • 73) 0.307 145 206 891 083 667 624 427 52 × 2 = 0 + 0.614 290 413 782 167 335 248 855 04;
  • 74) 0.614 290 413 782 167 335 248 855 04 × 2 = 1 + 0.228 580 827 564 334 670 497 710 08;
  • 75) 0.228 580 827 564 334 670 497 710 08 × 2 = 0 + 0.457 161 655 128 669 340 995 420 16;
  • 76) 0.457 161 655 128 669 340 995 420 16 × 2 = 0 + 0.914 323 310 257 338 681 990 840 32;
  • 77) 0.914 323 310 257 338 681 990 840 32 × 2 = 1 + 0.828 646 620 514 677 363 981 680 64;
  • 78) 0.828 646 620 514 677 363 981 680 64 × 2 = 1 + 0.657 293 241 029 354 727 963 361 28;
  • 79) 0.657 293 241 029 354 727 963 361 28 × 2 = 1 + 0.314 586 482 058 709 455 926 722 56;
  • 80) 0.314 586 482 058 709 455 926 722 56 × 2 = 0 + 0.629 172 964 117 418 911 853 445 12;
  • 81) 0.629 172 964 117 418 911 853 445 12 × 2 = 1 + 0.258 345 928 234 837 823 706 890 24;
  • 82) 0.258 345 928 234 837 823 706 890 24 × 2 = 0 + 0.516 691 856 469 675 647 413 780 48;
  • 83) 0.516 691 856 469 675 647 413 780 48 × 2 = 1 + 0.033 383 712 939 351 294 827 560 96;
  • 84) 0.033 383 712 939 351 294 827 560 96 × 2 = 0 + 0.066 767 425 878 702 589 655 121 92;
  • 85) 0.066 767 425 878 702 589 655 121 92 × 2 = 0 + 0.133 534 851 757 405 179 310 243 84;
  • 86) 0.133 534 851 757 405 179 310 243 84 × 2 = 0 + 0.267 069 703 514 810 358 620 487 68;
  • 87) 0.267 069 703 514 810 358 620 487 68 × 2 = 0 + 0.534 139 407 029 620 717 240 975 36;
  • 88) 0.534 139 407 029 620 717 240 975 36 × 2 = 1 + 0.068 278 814 059 241 434 481 950 72;
  • 89) 0.068 278 814 059 241 434 481 950 72 × 2 = 0 + 0.136 557 628 118 482 868 963 901 44;
  • 90) 0.136 557 628 118 482 868 963 901 44 × 2 = 0 + 0.273 115 256 236 965 737 927 802 88;
  • 91) 0.273 115 256 236 965 737 927 802 88 × 2 = 0 + 0.546 230 512 473 931 475 855 605 76;
  • 92) 0.546 230 512 473 931 475 855 605 76 × 2 = 1 + 0.092 461 024 947 862 951 711 211 52;
  • 93) 0.092 461 024 947 862 951 711 211 52 × 2 = 0 + 0.184 922 049 895 725 903 422 423 04;
  • 94) 0.184 922 049 895 725 903 422 423 04 × 2 = 0 + 0.369 844 099 791 451 806 844 846 08;
  • 95) 0.369 844 099 791 451 806 844 846 08 × 2 = 0 + 0.739 688 199 582 903 613 689 692 16;
  • 96) 0.739 688 199 582 903 613 689 692 16 × 2 = 1 + 0.479 376 399 165 807 227 379 384 32;
  • 97) 0.479 376 399 165 807 227 379 384 32 × 2 = 0 + 0.958 752 798 331 614 454 758 768 64;
  • 98) 0.958 752 798 331 614 454 758 768 64 × 2 = 1 + 0.917 505 596 663 228 909 517 537 28;
  • 99) 0.917 505 596 663 228 909 517 537 28 × 2 = 1 + 0.835 011 193 326 457 819 035 074 56;
  • 100) 0.835 011 193 326 457 819 035 074 56 × 2 = 1 + 0.670 022 386 652 915 638 070 149 12;
  • 101) 0.670 022 386 652 915 638 070 149 12 × 2 = 1 + 0.340 044 773 305 831 276 140 298 24;
  • 102) 0.340 044 773 305 831 276 140 298 24 × 2 = 0 + 0.680 089 546 611 662 552 280 596 48;
  • 103) 0.680 089 546 611 662 552 280 596 48 × 2 = 1 + 0.360 179 093 223 325 104 561 192 96;
  • 104) 0.360 179 093 223 325 104 561 192 96 × 2 = 0 + 0.720 358 186 446 650 209 122 385 92;
  • 105) 0.720 358 186 446 650 209 122 385 92 × 2 = 1 + 0.440 716 372 893 300 418 244 771 84;
  • 106) 0.440 716 372 893 300 418 244 771 84 × 2 = 0 + 0.881 432 745 786 600 836 489 543 68;
  • 107) 0.881 432 745 786 600 836 489 543 68 × 2 = 1 + 0.762 865 491 573 201 672 979 087 36;
  • 108) 0.762 865 491 573 201 672 979 087 36 × 2 = 1 + 0.525 730 983 146 403 345 958 174 72;
  • 109) 0.525 730 983 146 403 345 958 174 72 × 2 = 1 + 0.051 461 966 292 806 691 916 349 44;
  • 110) 0.051 461 966 292 806 691 916 349 44 × 2 = 0 + 0.102 923 932 585 613 383 832 698 88;
  • 111) 0.102 923 932 585 613 383 832 698 88 × 2 = 0 + 0.205 847 865 171 226 767 665 397 76;
  • 112) 0.205 847 865 171 226 767 665 397 76 × 2 = 0 + 0.411 695 730 342 453 535 330 795 52;
  • 113) 0.411 695 730 342 453 535 330 795 52 × 2 = 0 + 0.823 391 460 684 907 070 661 591 04;
  • 114) 0.823 391 460 684 907 070 661 591 04 × 2 = 1 + 0.646 782 921 369 814 141 323 182 08;
  • 115) 0.646 782 921 369 814 141 323 182 08 × 2 = 1 + 0.293 565 842 739 628 282 646 364 16;
  • 116) 0.293 565 842 739 628 282 646 364 16 × 2 = 0 + 0.587 131 685 479 256 565 292 728 32;
  • 117) 0.587 131 685 479 256 565 292 728 32 × 2 = 1 + 0.174 263 370 958 513 130 585 456 64;
  • 118) 0.174 263 370 958 513 130 585 456 64 × 2 = 0 + 0.348 526 741 917 026 261 170 913 28;
  • 119) 0.348 526 741 917 026 261 170 913 28 × 2 = 0 + 0.697 053 483 834 052 522 341 826 56;

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:


0.000 000 000 000 000 000 008 535 37(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1110 1010 0001 0001 0001 0111 1010 1011 1000 0110 100(2)

5. Positive number before normalization:

0.000 000 000 000 000 000 008 535 37(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1110 1010 0001 0001 0001 0111 1010 1011 1000 0110 100(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 67 positions to the right, so that only one non zero digit remains to the left of it:


0.000 000 000 000 000 000 008 535 37(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1110 1010 0001 0001 0001 0111 1010 1011 1000 0110 100(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1110 1010 0001 0001 0001 0111 1010 1011 1000 0110 100(2) × 20 =

1.0100 0010 0111 0101 0000 1000 1000 1011 1101 0101 1100 0011 0100(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 0111 0101 0000 1000 1000 1011 1101 0101 1100 0011 0100


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 0111 0101 0000 1000 1000 1011 1101 0101 1100 0011 0100 =

0100 0010 0111 0101 0000 1000 1000 1011 1101 0101 1100 0011 0100


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 0111 0101 0000 1000 1000 1011 1101 0101 1100 0011 0100


Decimal number 0.000 000 000 000 000 000 008 535 37 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1011 1100 - 0100 0010 0111 0101 0000 1000 1000 1011 1101 0101 1100 0011 0100

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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