0.000 000 000 000 000 000 008 534 77 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 000 000 008 534 77(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
0.000 000 000 000 000 000 008 534 77(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: 0.000 000 000 000 000 000 008 534 77.

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) 0.000 000 000 000 000 000 008 534 77 × 2 = 0 + 0.000 000 000 000 000 000 017 069 54;
  • 2) 0.000 000 000 000 000 000 017 069 54 × 2 = 0 + 0.000 000 000 000 000 000 034 139 08;
  • 3) 0.000 000 000 000 000 000 034 139 08 × 2 = 0 + 0.000 000 000 000 000 000 068 278 16;
  • 4) 0.000 000 000 000 000 000 068 278 16 × 2 = 0 + 0.000 000 000 000 000 000 136 556 32;
  • 5) 0.000 000 000 000 000 000 136 556 32 × 2 = 0 + 0.000 000 000 000 000 000 273 112 64;
  • 6) 0.000 000 000 000 000 000 273 112 64 × 2 = 0 + 0.000 000 000 000 000 000 546 225 28;
  • 7) 0.000 000 000 000 000 000 546 225 28 × 2 = 0 + 0.000 000 000 000 000 001 092 450 56;
  • 8) 0.000 000 000 000 000 001 092 450 56 × 2 = 0 + 0.000 000 000 000 000 002 184 901 12;
  • 9) 0.000 000 000 000 000 002 184 901 12 × 2 = 0 + 0.000 000 000 000 000 004 369 802 24;
  • 10) 0.000 000 000 000 000 004 369 802 24 × 2 = 0 + 0.000 000 000 000 000 008 739 604 48;
  • 11) 0.000 000 000 000 000 008 739 604 48 × 2 = 0 + 0.000 000 000 000 000 017 479 208 96;
  • 12) 0.000 000 000 000 000 017 479 208 96 × 2 = 0 + 0.000 000 000 000 000 034 958 417 92;
  • 13) 0.000 000 000 000 000 034 958 417 92 × 2 = 0 + 0.000 000 000 000 000 069 916 835 84;
  • 14) 0.000 000 000 000 000 069 916 835 84 × 2 = 0 + 0.000 000 000 000 000 139 833 671 68;
  • 15) 0.000 000 000 000 000 139 833 671 68 × 2 = 0 + 0.000 000 000 000 000 279 667 343 36;
  • 16) 0.000 000 000 000 000 279 667 343 36 × 2 = 0 + 0.000 000 000 000 000 559 334 686 72;
  • 17) 0.000 000 000 000 000 559 334 686 72 × 2 = 0 + 0.000 000 000 000 001 118 669 373 44;
  • 18) 0.000 000 000 000 001 118 669 373 44 × 2 = 0 + 0.000 000 000 000 002 237 338 746 88;
  • 19) 0.000 000 000 000 002 237 338 746 88 × 2 = 0 + 0.000 000 000 000 004 474 677 493 76;
  • 20) 0.000 000 000 000 004 474 677 493 76 × 2 = 0 + 0.000 000 000 000 008 949 354 987 52;
  • 21) 0.000 000 000 000 008 949 354 987 52 × 2 = 0 + 0.000 000 000 000 017 898 709 975 04;
  • 22) 0.000 000 000 000 017 898 709 975 04 × 2 = 0 + 0.000 000 000 000 035 797 419 950 08;
  • 23) 0.000 000 000 000 035 797 419 950 08 × 2 = 0 + 0.000 000 000 000 071 594 839 900 16;
  • 24) 0.000 000 000 000 071 594 839 900 16 × 2 = 0 + 0.000 000 000 000 143 189 679 800 32;
  • 25) 0.000 000 000 000 143 189 679 800 32 × 2 = 0 + 0.000 000 000 000 286 379 359 600 64;
  • 26) 0.000 000 000 000 286 379 359 600 64 × 2 = 0 + 0.000 000 000 000 572 758 719 201 28;
  • 27) 0.000 000 000 000 572 758 719 201 28 × 2 = 0 + 0.000 000 000 001 145 517 438 402 56;
  • 28) 0.000 000 000 001 145 517 438 402 56 × 2 = 0 + 0.000 000 000 002 291 034 876 805 12;
  • 29) 0.000 000 000 002 291 034 876 805 12 × 2 = 0 + 0.000 000 000 004 582 069 753 610 24;
  • 30) 0.000 000 000 004 582 069 753 610 24 × 2 = 0 + 0.000 000 000 009 164 139 507 220 48;
  • 31) 0.000 000 000 009 164 139 507 220 48 × 2 = 0 + 0.000 000 000 018 328 279 014 440 96;
  • 32) 0.000 000 000 018 328 279 014 440 96 × 2 = 0 + 0.000 000 000 036 656 558 028 881 92;
  • 33) 0.000 000 000 036 656 558 028 881 92 × 2 = 0 + 0.000 000 000 073 313 116 057 763 84;
  • 34) 0.000 000 000 073 313 116 057 763 84 × 2 = 0 + 0.000 000 000 146 626 232 115 527 68;
  • 35) 0.000 000 000 146 626 232 115 527 68 × 2 = 0 + 0.000 000 000 293 252 464 231 055 36;
  • 36) 0.000 000 000 293 252 464 231 055 36 × 2 = 0 + 0.000 000 000 586 504 928 462 110 72;
  • 37) 0.000 000 000 586 504 928 462 110 72 × 2 = 0 + 0.000 000 001 173 009 856 924 221 44;
  • 38) 0.000 000 001 173 009 856 924 221 44 × 2 = 0 + 0.000 000 002 346 019 713 848 442 88;
  • 39) 0.000 000 002 346 019 713 848 442 88 × 2 = 0 + 0.000 000 004 692 039 427 696 885 76;
  • 40) 0.000 000 004 692 039 427 696 885 76 × 2 = 0 + 0.000 000 009 384 078 855 393 771 52;
  • 41) 0.000 000 009 384 078 855 393 771 52 × 2 = 0 + 0.000 000 018 768 157 710 787 543 04;
  • 42) 0.000 000 018 768 157 710 787 543 04 × 2 = 0 + 0.000 000 037 536 315 421 575 086 08;
  • 43) 0.000 000 037 536 315 421 575 086 08 × 2 = 0 + 0.000 000 075 072 630 843 150 172 16;
  • 44) 0.000 000 075 072 630 843 150 172 16 × 2 = 0 + 0.000 000 150 145 261 686 300 344 32;
  • 45) 0.000 000 150 145 261 686 300 344 32 × 2 = 0 + 0.000 000 300 290 523 372 600 688 64;
  • 46) 0.000 000 300 290 523 372 600 688 64 × 2 = 0 + 0.000 000 600 581 046 745 201 377 28;
  • 47) 0.000 000 600 581 046 745 201 377 28 × 2 = 0 + 0.000 001 201 162 093 490 402 754 56;
  • 48) 0.000 001 201 162 093 490 402 754 56 × 2 = 0 + 0.000 002 402 324 186 980 805 509 12;
  • 49) 0.000 002 402 324 186 980 805 509 12 × 2 = 0 + 0.000 004 804 648 373 961 611 018 24;
  • 50) 0.000 004 804 648 373 961 611 018 24 × 2 = 0 + 0.000 009 609 296 747 923 222 036 48;
  • 51) 0.000 009 609 296 747 923 222 036 48 × 2 = 0 + 0.000 019 218 593 495 846 444 072 96;
  • 52) 0.000 019 218 593 495 846 444 072 96 × 2 = 0 + 0.000 038 437 186 991 692 888 145 92;
  • 53) 0.000 038 437 186 991 692 888 145 92 × 2 = 0 + 0.000 076 874 373 983 385 776 291 84;
  • 54) 0.000 076 874 373 983 385 776 291 84 × 2 = 0 + 0.000 153 748 747 966 771 552 583 68;
  • 55) 0.000 153 748 747 966 771 552 583 68 × 2 = 0 + 0.000 307 497 495 933 543 105 167 36;
  • 56) 0.000 307 497 495 933 543 105 167 36 × 2 = 0 + 0.000 614 994 991 867 086 210 334 72;
  • 57) 0.000 614 994 991 867 086 210 334 72 × 2 = 0 + 0.001 229 989 983 734 172 420 669 44;
  • 58) 0.001 229 989 983 734 172 420 669 44 × 2 = 0 + 0.002 459 979 967 468 344 841 338 88;
  • 59) 0.002 459 979 967 468 344 841 338 88 × 2 = 0 + 0.004 919 959 934 936 689 682 677 76;
  • 60) 0.004 919 959 934 936 689 682 677 76 × 2 = 0 + 0.009 839 919 869 873 379 365 355 52;
  • 61) 0.009 839 919 869 873 379 365 355 52 × 2 = 0 + 0.019 679 839 739 746 758 730 711 04;
  • 62) 0.019 679 839 739 746 758 730 711 04 × 2 = 0 + 0.039 359 679 479 493 517 461 422 08;
  • 63) 0.039 359 679 479 493 517 461 422 08 × 2 = 0 + 0.078 719 358 958 987 034 922 844 16;
  • 64) 0.078 719 358 958 987 034 922 844 16 × 2 = 0 + 0.157 438 717 917 974 069 845 688 32;
  • 65) 0.157 438 717 917 974 069 845 688 32 × 2 = 0 + 0.314 877 435 835 948 139 691 376 64;
  • 66) 0.314 877 435 835 948 139 691 376 64 × 2 = 0 + 0.629 754 871 671 896 279 382 753 28;
  • 67) 0.629 754 871 671 896 279 382 753 28 × 2 = 1 + 0.259 509 743 343 792 558 765 506 56;
  • 68) 0.259 509 743 343 792 558 765 506 56 × 2 = 0 + 0.519 019 486 687 585 117 531 013 12;
  • 69) 0.519 019 486 687 585 117 531 013 12 × 2 = 1 + 0.038 038 973 375 170 235 062 026 24;
  • 70) 0.038 038 973 375 170 235 062 026 24 × 2 = 0 + 0.076 077 946 750 340 470 124 052 48;
  • 71) 0.076 077 946 750 340 470 124 052 48 × 2 = 0 + 0.152 155 893 500 680 940 248 104 96;
  • 72) 0.152 155 893 500 680 940 248 104 96 × 2 = 0 + 0.304 311 787 001 361 880 496 209 92;
  • 73) 0.304 311 787 001 361 880 496 209 92 × 2 = 0 + 0.608 623 574 002 723 760 992 419 84;
  • 74) 0.608 623 574 002 723 760 992 419 84 × 2 = 1 + 0.217 247 148 005 447 521 984 839 68;
  • 75) 0.217 247 148 005 447 521 984 839 68 × 2 = 0 + 0.434 494 296 010 895 043 969 679 36;
  • 76) 0.434 494 296 010 895 043 969 679 36 × 2 = 0 + 0.868 988 592 021 790 087 939 358 72;
  • 77) 0.868 988 592 021 790 087 939 358 72 × 2 = 1 + 0.737 977 184 043 580 175 878 717 44;
  • 78) 0.737 977 184 043 580 175 878 717 44 × 2 = 1 + 0.475 954 368 087 160 351 757 434 88;
  • 79) 0.475 954 368 087 160 351 757 434 88 × 2 = 0 + 0.951 908 736 174 320 703 514 869 76;
  • 80) 0.951 908 736 174 320 703 514 869 76 × 2 = 1 + 0.903 817 472 348 641 407 029 739 52;
  • 81) 0.903 817 472 348 641 407 029 739 52 × 2 = 1 + 0.807 634 944 697 282 814 059 479 04;
  • 82) 0.807 634 944 697 282 814 059 479 04 × 2 = 1 + 0.615 269 889 394 565 628 118 958 08;
  • 83) 0.615 269 889 394 565 628 118 958 08 × 2 = 1 + 0.230 539 778 789 131 256 237 916 16;
  • 84) 0.230 539 778 789 131 256 237 916 16 × 2 = 0 + 0.461 079 557 578 262 512 475 832 32;
  • 85) 0.461 079 557 578 262 512 475 832 32 × 2 = 0 + 0.922 159 115 156 525 024 951 664 64;
  • 86) 0.922 159 115 156 525 024 951 664 64 × 2 = 1 + 0.844 318 230 313 050 049 903 329 28;
  • 87) 0.844 318 230 313 050 049 903 329 28 × 2 = 1 + 0.688 636 460 626 100 099 806 658 56;
  • 88) 0.688 636 460 626 100 099 806 658 56 × 2 = 1 + 0.377 272 921 252 200 199 613 317 12;
  • 89) 0.377 272 921 252 200 199 613 317 12 × 2 = 0 + 0.754 545 842 504 400 399 226 634 24;
  • 90) 0.754 545 842 504 400 399 226 634 24 × 2 = 1 + 0.509 091 685 008 800 798 453 268 48;
  • 91) 0.509 091 685 008 800 798 453 268 48 × 2 = 1 + 0.018 183 370 017 601 596 906 536 96;
  • 92) 0.018 183 370 017 601 596 906 536 96 × 2 = 0 + 0.036 366 740 035 203 193 813 073 92;
  • 93) 0.036 366 740 035 203 193 813 073 92 × 2 = 0 + 0.072 733 480 070 406 387 626 147 84;
  • 94) 0.072 733 480 070 406 387 626 147 84 × 2 = 0 + 0.145 466 960 140 812 775 252 295 68;
  • 95) 0.145 466 960 140 812 775 252 295 68 × 2 = 0 + 0.290 933 920 281 625 550 504 591 36;
  • 96) 0.290 933 920 281 625 550 504 591 36 × 2 = 0 + 0.581 867 840 563 251 101 009 182 72;
  • 97) 0.581 867 840 563 251 101 009 182 72 × 2 = 1 + 0.163 735 681 126 502 202 018 365 44;
  • 98) 0.163 735 681 126 502 202 018 365 44 × 2 = 0 + 0.327 471 362 253 004 404 036 730 88;
  • 99) 0.327 471 362 253 004 404 036 730 88 × 2 = 0 + 0.654 942 724 506 008 808 073 461 76;
  • 100) 0.654 942 724 506 008 808 073 461 76 × 2 = 1 + 0.309 885 449 012 017 616 146 923 52;
  • 101) 0.309 885 449 012 017 616 146 923 52 × 2 = 0 + 0.619 770 898 024 035 232 293 847 04;
  • 102) 0.619 770 898 024 035 232 293 847 04 × 2 = 1 + 0.239 541 796 048 070 464 587 694 08;
  • 103) 0.239 541 796 048 070 464 587 694 08 × 2 = 0 + 0.479 083 592 096 140 929 175 388 16;
  • 104) 0.479 083 592 096 140 929 175 388 16 × 2 = 0 + 0.958 167 184 192 281 858 350 776 32;
  • 105) 0.958 167 184 192 281 858 350 776 32 × 2 = 1 + 0.916 334 368 384 563 716 701 552 64;
  • 106) 0.916 334 368 384 563 716 701 552 64 × 2 = 1 + 0.832 668 736 769 127 433 403 105 28;
  • 107) 0.832 668 736 769 127 433 403 105 28 × 2 = 1 + 0.665 337 473 538 254 866 806 210 56;
  • 108) 0.665 337 473 538 254 866 806 210 56 × 2 = 1 + 0.330 674 947 076 509 733 612 421 12;
  • 109) 0.330 674 947 076 509 733 612 421 12 × 2 = 0 + 0.661 349 894 153 019 467 224 842 24;
  • 110) 0.661 349 894 153 019 467 224 842 24 × 2 = 1 + 0.322 699 788 306 038 934 449 684 48;
  • 111) 0.322 699 788 306 038 934 449 684 48 × 2 = 0 + 0.645 399 576 612 077 868 899 368 96;
  • 112) 0.645 399 576 612 077 868 899 368 96 × 2 = 1 + 0.290 799 153 224 155 737 798 737 92;
  • 113) 0.290 799 153 224 155 737 798 737 92 × 2 = 0 + 0.581 598 306 448 311 475 597 475 84;
  • 114) 0.581 598 306 448 311 475 597 475 84 × 2 = 1 + 0.163 196 612 896 622 951 194 951 68;
  • 115) 0.163 196 612 896 622 951 194 951 68 × 2 = 0 + 0.326 393 225 793 245 902 389 903 36;
  • 116) 0.326 393 225 793 245 902 389 903 36 × 2 = 0 + 0.652 786 451 586 491 804 779 806 72;
  • 117) 0.652 786 451 586 491 804 779 806 72 × 2 = 1 + 0.305 572 903 172 983 609 559 613 44;
  • 118) 0.305 572 903 172 983 609 559 613 44 × 2 = 0 + 0.611 145 806 345 967 219 119 226 88;
  • 119) 0.611 145 806 345 967 219 119 226 88 × 2 = 1 + 0.222 291 612 691 934 438 238 453 76;

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:


0.000 000 000 000 000 000 008 534 77(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1101 1110 0111 0110 0000 1001 0100 1111 0101 0100 101(2)

5. Positive number before normalization:

0.000 000 000 000 000 000 008 534 77(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1101 1110 0111 0110 0000 1001 0100 1111 0101 0100 101(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:


0.000 000 000 000 000 000 008 534 77(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1101 1110 0111 0110 0000 1001 0100 1111 0101 0100 101(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1101 1110 0111 0110 0000 1001 0100 1111 0101 0100 101(2) × 20 =

1.0100 0010 0110 1111 0011 1011 0000 0100 1010 0111 1010 1010 0101(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 0110 1111 0011 1011 0000 0100 1010 0111 1010 1010 0101


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 0110 1111 0011 1011 0000 0100 1010 0111 1010 1010 0101 =

0100 0010 0110 1111 0011 1011 0000 0100 1010 0111 1010 1010 0101


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 0110 1111 0011 1011 0000 0100 1010 0111 1010 1010 0101


Decimal number 0.000 000 000 000 000 000 008 534 77 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1011 1100 - 0100 0010 0110 1111 0011 1011 0000 0100 1010 0111 1010 1010 0101

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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