0.000 000 000 000 000 000 008 534 48 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 000 000 008 534 48(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 48(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 48.

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 48 × 2 = 0 + 0.000 000 000 000 000 000 017 068 96;
  • 2) 0.000 000 000 000 000 000 017 068 96 × 2 = 0 + 0.000 000 000 000 000 000 034 137 92;
  • 3) 0.000 000 000 000 000 000 034 137 92 × 2 = 0 + 0.000 000 000 000 000 000 068 275 84;
  • 4) 0.000 000 000 000 000 000 068 275 84 × 2 = 0 + 0.000 000 000 000 000 000 136 551 68;
  • 5) 0.000 000 000 000 000 000 136 551 68 × 2 = 0 + 0.000 000 000 000 000 000 273 103 36;
  • 6) 0.000 000 000 000 000 000 273 103 36 × 2 = 0 + 0.000 000 000 000 000 000 546 206 72;
  • 7) 0.000 000 000 000 000 000 546 206 72 × 2 = 0 + 0.000 000 000 000 000 001 092 413 44;
  • 8) 0.000 000 000 000 000 001 092 413 44 × 2 = 0 + 0.000 000 000 000 000 002 184 826 88;
  • 9) 0.000 000 000 000 000 002 184 826 88 × 2 = 0 + 0.000 000 000 000 000 004 369 653 76;
  • 10) 0.000 000 000 000 000 004 369 653 76 × 2 = 0 + 0.000 000 000 000 000 008 739 307 52;
  • 11) 0.000 000 000 000 000 008 739 307 52 × 2 = 0 + 0.000 000 000 000 000 017 478 615 04;
  • 12) 0.000 000 000 000 000 017 478 615 04 × 2 = 0 + 0.000 000 000 000 000 034 957 230 08;
  • 13) 0.000 000 000 000 000 034 957 230 08 × 2 = 0 + 0.000 000 000 000 000 069 914 460 16;
  • 14) 0.000 000 000 000 000 069 914 460 16 × 2 = 0 + 0.000 000 000 000 000 139 828 920 32;
  • 15) 0.000 000 000 000 000 139 828 920 32 × 2 = 0 + 0.000 000 000 000 000 279 657 840 64;
  • 16) 0.000 000 000 000 000 279 657 840 64 × 2 = 0 + 0.000 000 000 000 000 559 315 681 28;
  • 17) 0.000 000 000 000 000 559 315 681 28 × 2 = 0 + 0.000 000 000 000 001 118 631 362 56;
  • 18) 0.000 000 000 000 001 118 631 362 56 × 2 = 0 + 0.000 000 000 000 002 237 262 725 12;
  • 19) 0.000 000 000 000 002 237 262 725 12 × 2 = 0 + 0.000 000 000 000 004 474 525 450 24;
  • 20) 0.000 000 000 000 004 474 525 450 24 × 2 = 0 + 0.000 000 000 000 008 949 050 900 48;
  • 21) 0.000 000 000 000 008 949 050 900 48 × 2 = 0 + 0.000 000 000 000 017 898 101 800 96;
  • 22) 0.000 000 000 000 017 898 101 800 96 × 2 = 0 + 0.000 000 000 000 035 796 203 601 92;
  • 23) 0.000 000 000 000 035 796 203 601 92 × 2 = 0 + 0.000 000 000 000 071 592 407 203 84;
  • 24) 0.000 000 000 000 071 592 407 203 84 × 2 = 0 + 0.000 000 000 000 143 184 814 407 68;
  • 25) 0.000 000 000 000 143 184 814 407 68 × 2 = 0 + 0.000 000 000 000 286 369 628 815 36;
  • 26) 0.000 000 000 000 286 369 628 815 36 × 2 = 0 + 0.000 000 000 000 572 739 257 630 72;
  • 27) 0.000 000 000 000 572 739 257 630 72 × 2 = 0 + 0.000 000 000 001 145 478 515 261 44;
  • 28) 0.000 000 000 001 145 478 515 261 44 × 2 = 0 + 0.000 000 000 002 290 957 030 522 88;
  • 29) 0.000 000 000 002 290 957 030 522 88 × 2 = 0 + 0.000 000 000 004 581 914 061 045 76;
  • 30) 0.000 000 000 004 581 914 061 045 76 × 2 = 0 + 0.000 000 000 009 163 828 122 091 52;
  • 31) 0.000 000 000 009 163 828 122 091 52 × 2 = 0 + 0.000 000 000 018 327 656 244 183 04;
  • 32) 0.000 000 000 018 327 656 244 183 04 × 2 = 0 + 0.000 000 000 036 655 312 488 366 08;
  • 33) 0.000 000 000 036 655 312 488 366 08 × 2 = 0 + 0.000 000 000 073 310 624 976 732 16;
  • 34) 0.000 000 000 073 310 624 976 732 16 × 2 = 0 + 0.000 000 000 146 621 249 953 464 32;
  • 35) 0.000 000 000 146 621 249 953 464 32 × 2 = 0 + 0.000 000 000 293 242 499 906 928 64;
  • 36) 0.000 000 000 293 242 499 906 928 64 × 2 = 0 + 0.000 000 000 586 484 999 813 857 28;
  • 37) 0.000 000 000 586 484 999 813 857 28 × 2 = 0 + 0.000 000 001 172 969 999 627 714 56;
  • 38) 0.000 000 001 172 969 999 627 714 56 × 2 = 0 + 0.000 000 002 345 939 999 255 429 12;
  • 39) 0.000 000 002 345 939 999 255 429 12 × 2 = 0 + 0.000 000 004 691 879 998 510 858 24;
  • 40) 0.000 000 004 691 879 998 510 858 24 × 2 = 0 + 0.000 000 009 383 759 997 021 716 48;
  • 41) 0.000 000 009 383 759 997 021 716 48 × 2 = 0 + 0.000 000 018 767 519 994 043 432 96;
  • 42) 0.000 000 018 767 519 994 043 432 96 × 2 = 0 + 0.000 000 037 535 039 988 086 865 92;
  • 43) 0.000 000 037 535 039 988 086 865 92 × 2 = 0 + 0.000 000 075 070 079 976 173 731 84;
  • 44) 0.000 000 075 070 079 976 173 731 84 × 2 = 0 + 0.000 000 150 140 159 952 347 463 68;
  • 45) 0.000 000 150 140 159 952 347 463 68 × 2 = 0 + 0.000 000 300 280 319 904 694 927 36;
  • 46) 0.000 000 300 280 319 904 694 927 36 × 2 = 0 + 0.000 000 600 560 639 809 389 854 72;
  • 47) 0.000 000 600 560 639 809 389 854 72 × 2 = 0 + 0.000 001 201 121 279 618 779 709 44;
  • 48) 0.000 001 201 121 279 618 779 709 44 × 2 = 0 + 0.000 002 402 242 559 237 559 418 88;
  • 49) 0.000 002 402 242 559 237 559 418 88 × 2 = 0 + 0.000 004 804 485 118 475 118 837 76;
  • 50) 0.000 004 804 485 118 475 118 837 76 × 2 = 0 + 0.000 009 608 970 236 950 237 675 52;
  • 51) 0.000 009 608 970 236 950 237 675 52 × 2 = 0 + 0.000 019 217 940 473 900 475 351 04;
  • 52) 0.000 019 217 940 473 900 475 351 04 × 2 = 0 + 0.000 038 435 880 947 800 950 702 08;
  • 53) 0.000 038 435 880 947 800 950 702 08 × 2 = 0 + 0.000 076 871 761 895 601 901 404 16;
  • 54) 0.000 076 871 761 895 601 901 404 16 × 2 = 0 + 0.000 153 743 523 791 203 802 808 32;
  • 55) 0.000 153 743 523 791 203 802 808 32 × 2 = 0 + 0.000 307 487 047 582 407 605 616 64;
  • 56) 0.000 307 487 047 582 407 605 616 64 × 2 = 0 + 0.000 614 974 095 164 815 211 233 28;
  • 57) 0.000 614 974 095 164 815 211 233 28 × 2 = 0 + 0.001 229 948 190 329 630 422 466 56;
  • 58) 0.001 229 948 190 329 630 422 466 56 × 2 = 0 + 0.002 459 896 380 659 260 844 933 12;
  • 59) 0.002 459 896 380 659 260 844 933 12 × 2 = 0 + 0.004 919 792 761 318 521 689 866 24;
  • 60) 0.004 919 792 761 318 521 689 866 24 × 2 = 0 + 0.009 839 585 522 637 043 379 732 48;
  • 61) 0.009 839 585 522 637 043 379 732 48 × 2 = 0 + 0.019 679 171 045 274 086 759 464 96;
  • 62) 0.019 679 171 045 274 086 759 464 96 × 2 = 0 + 0.039 358 342 090 548 173 518 929 92;
  • 63) 0.039 358 342 090 548 173 518 929 92 × 2 = 0 + 0.078 716 684 181 096 347 037 859 84;
  • 64) 0.078 716 684 181 096 347 037 859 84 × 2 = 0 + 0.157 433 368 362 192 694 075 719 68;
  • 65) 0.157 433 368 362 192 694 075 719 68 × 2 = 0 + 0.314 866 736 724 385 388 151 439 36;
  • 66) 0.314 866 736 724 385 388 151 439 36 × 2 = 0 + 0.629 733 473 448 770 776 302 878 72;
  • 67) 0.629 733 473 448 770 776 302 878 72 × 2 = 1 + 0.259 466 946 897 541 552 605 757 44;
  • 68) 0.259 466 946 897 541 552 605 757 44 × 2 = 0 + 0.518 933 893 795 083 105 211 514 88;
  • 69) 0.518 933 893 795 083 105 211 514 88 × 2 = 1 + 0.037 867 787 590 166 210 423 029 76;
  • 70) 0.037 867 787 590 166 210 423 029 76 × 2 = 0 + 0.075 735 575 180 332 420 846 059 52;
  • 71) 0.075 735 575 180 332 420 846 059 52 × 2 = 0 + 0.151 471 150 360 664 841 692 119 04;
  • 72) 0.151 471 150 360 664 841 692 119 04 × 2 = 0 + 0.302 942 300 721 329 683 384 238 08;
  • 73) 0.302 942 300 721 329 683 384 238 08 × 2 = 0 + 0.605 884 601 442 659 366 768 476 16;
  • 74) 0.605 884 601 442 659 366 768 476 16 × 2 = 1 + 0.211 769 202 885 318 733 536 952 32;
  • 75) 0.211 769 202 885 318 733 536 952 32 × 2 = 0 + 0.423 538 405 770 637 467 073 904 64;
  • 76) 0.423 538 405 770 637 467 073 904 64 × 2 = 0 + 0.847 076 811 541 274 934 147 809 28;
  • 77) 0.847 076 811 541 274 934 147 809 28 × 2 = 1 + 0.694 153 623 082 549 868 295 618 56;
  • 78) 0.694 153 623 082 549 868 295 618 56 × 2 = 1 + 0.388 307 246 165 099 736 591 237 12;
  • 79) 0.388 307 246 165 099 736 591 237 12 × 2 = 0 + 0.776 614 492 330 199 473 182 474 24;
  • 80) 0.776 614 492 330 199 473 182 474 24 × 2 = 1 + 0.553 228 984 660 398 946 364 948 48;
  • 81) 0.553 228 984 660 398 946 364 948 48 × 2 = 1 + 0.106 457 969 320 797 892 729 896 96;
  • 82) 0.106 457 969 320 797 892 729 896 96 × 2 = 0 + 0.212 915 938 641 595 785 459 793 92;
  • 83) 0.212 915 938 641 595 785 459 793 92 × 2 = 0 + 0.425 831 877 283 191 570 919 587 84;
  • 84) 0.425 831 877 283 191 570 919 587 84 × 2 = 0 + 0.851 663 754 566 383 141 839 175 68;
  • 85) 0.851 663 754 566 383 141 839 175 68 × 2 = 1 + 0.703 327 509 132 766 283 678 351 36;
  • 86) 0.703 327 509 132 766 283 678 351 36 × 2 = 1 + 0.406 655 018 265 532 567 356 702 72;
  • 87) 0.406 655 018 265 532 567 356 702 72 × 2 = 0 + 0.813 310 036 531 065 134 713 405 44;
  • 88) 0.813 310 036 531 065 134 713 405 44 × 2 = 1 + 0.626 620 073 062 130 269 426 810 88;
  • 89) 0.626 620 073 062 130 269 426 810 88 × 2 = 1 + 0.253 240 146 124 260 538 853 621 76;
  • 90) 0.253 240 146 124 260 538 853 621 76 × 2 = 0 + 0.506 480 292 248 521 077 707 243 52;
  • 91) 0.506 480 292 248 521 077 707 243 52 × 2 = 1 + 0.012 960 584 497 042 155 414 487 04;
  • 92) 0.012 960 584 497 042 155 414 487 04 × 2 = 0 + 0.025 921 168 994 084 310 828 974 08;
  • 93) 0.025 921 168 994 084 310 828 974 08 × 2 = 0 + 0.051 842 337 988 168 621 657 948 16;
  • 94) 0.051 842 337 988 168 621 657 948 16 × 2 = 0 + 0.103 684 675 976 337 243 315 896 32;
  • 95) 0.103 684 675 976 337 243 315 896 32 × 2 = 0 + 0.207 369 351 952 674 486 631 792 64;
  • 96) 0.207 369 351 952 674 486 631 792 64 × 2 = 0 + 0.414 738 703 905 348 973 263 585 28;
  • 97) 0.414 738 703 905 348 973 263 585 28 × 2 = 0 + 0.829 477 407 810 697 946 527 170 56;
  • 98) 0.829 477 407 810 697 946 527 170 56 × 2 = 1 + 0.658 954 815 621 395 893 054 341 12;
  • 99) 0.658 954 815 621 395 893 054 341 12 × 2 = 1 + 0.317 909 631 242 791 786 108 682 24;
  • 100) 0.317 909 631 242 791 786 108 682 24 × 2 = 0 + 0.635 819 262 485 583 572 217 364 48;
  • 101) 0.635 819 262 485 583 572 217 364 48 × 2 = 1 + 0.271 638 524 971 167 144 434 728 96;
  • 102) 0.271 638 524 971 167 144 434 728 96 × 2 = 0 + 0.543 277 049 942 334 288 869 457 92;
  • 103) 0.543 277 049 942 334 288 869 457 92 × 2 = 1 + 0.086 554 099 884 668 577 738 915 84;
  • 104) 0.086 554 099 884 668 577 738 915 84 × 2 = 0 + 0.173 108 199 769 337 155 477 831 68;
  • 105) 0.173 108 199 769 337 155 477 831 68 × 2 = 0 + 0.346 216 399 538 674 310 955 663 36;
  • 106) 0.346 216 399 538 674 310 955 663 36 × 2 = 0 + 0.692 432 799 077 348 621 911 326 72;
  • 107) 0.692 432 799 077 348 621 911 326 72 × 2 = 1 + 0.384 865 598 154 697 243 822 653 44;
  • 108) 0.384 865 598 154 697 243 822 653 44 × 2 = 0 + 0.769 731 196 309 394 487 645 306 88;
  • 109) 0.769 731 196 309 394 487 645 306 88 × 2 = 1 + 0.539 462 392 618 788 975 290 613 76;
  • 110) 0.539 462 392 618 788 975 290 613 76 × 2 = 1 + 0.078 924 785 237 577 950 581 227 52;
  • 111) 0.078 924 785 237 577 950 581 227 52 × 2 = 0 + 0.157 849 570 475 155 901 162 455 04;
  • 112) 0.157 849 570 475 155 901 162 455 04 × 2 = 0 + 0.315 699 140 950 311 802 324 910 08;
  • 113) 0.315 699 140 950 311 802 324 910 08 × 2 = 0 + 0.631 398 281 900 623 604 649 820 16;
  • 114) 0.631 398 281 900 623 604 649 820 16 × 2 = 1 + 0.262 796 563 801 247 209 299 640 32;
  • 115) 0.262 796 563 801 247 209 299 640 32 × 2 = 0 + 0.525 593 127 602 494 418 599 280 64;
  • 116) 0.525 593 127 602 494 418 599 280 64 × 2 = 1 + 0.051 186 255 204 988 837 198 561 28;
  • 117) 0.051 186 255 204 988 837 198 561 28 × 2 = 0 + 0.102 372 510 409 977 674 397 122 56;
  • 118) 0.102 372 510 409 977 674 397 122 56 × 2 = 0 + 0.204 745 020 819 955 348 794 245 12;
  • 119) 0.204 745 020 819 955 348 794 245 12 × 2 = 0 + 0.409 490 041 639 910 697 588 490 24;

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 48(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1101 1000 1101 1010 0000 0110 1010 0010 1100 0101 000(2)

5. Positive number before normalization:

0.000 000 000 000 000 000 008 534 48(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1101 1000 1101 1010 0000 0110 1010 0010 1100 0101 000(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 48(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1101 1000 1101 1010 0000 0110 1010 0010 1100 0101 000(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1101 1000 1101 1010 0000 0110 1010 0010 1100 0101 000(2) × 20 =

1.0100 0010 0110 1100 0110 1101 0000 0011 0101 0001 0110 0010 1000(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 1100 0110 1101 0000 0011 0101 0001 0110 0010 1000


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 1100 0110 1101 0000 0011 0101 0001 0110 0010 1000 =

0100 0010 0110 1100 0110 1101 0000 0011 0101 0001 0110 0010 1000


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 1100 0110 1101 0000 0011 0101 0001 0110 0010 1000


Decimal number 0.000 000 000 000 000 000 008 534 48 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1011 1100 - 0100 0010 0110 1100 0110 1101 0000 0011 0101 0001 0110 0010 1000

Share

» Convert decimal 0.000 000 000 000 000 000 008 534 77 to 64 bit double precision IEEE 754 binary floating point representation standard

» Calculations Performed by Our Visitors: Decimal Numbers Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard. Data organized on a Monthly Basis

» Month 06, 2026 [June]: Decimal Numbers Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard. Calculations performed during the month of: June - by our visitors

» Improve your social skills: The Hidden Requirement in Every Single Job Description. NotebookLM YouTube Video of 8.15 minutes

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