0.000 000 000 000 000 000 008 536 6 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 000 000 008 536 6(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
0.000 000 000 000 000 000 008 536 6(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: 0.000 000 000 000 000 000 008 536 6.

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) 0.000 000 000 000 000 000 008 536 6 × 2 = 0 + 0.000 000 000 000 000 000 017 073 2;
  • 2) 0.000 000 000 000 000 000 017 073 2 × 2 = 0 + 0.000 000 000 000 000 000 034 146 4;
  • 3) 0.000 000 000 000 000 000 034 146 4 × 2 = 0 + 0.000 000 000 000 000 000 068 292 8;
  • 4) 0.000 000 000 000 000 000 068 292 8 × 2 = 0 + 0.000 000 000 000 000 000 136 585 6;
  • 5) 0.000 000 000 000 000 000 136 585 6 × 2 = 0 + 0.000 000 000 000 000 000 273 171 2;
  • 6) 0.000 000 000 000 000 000 273 171 2 × 2 = 0 + 0.000 000 000 000 000 000 546 342 4;
  • 7) 0.000 000 000 000 000 000 546 342 4 × 2 = 0 + 0.000 000 000 000 000 001 092 684 8;
  • 8) 0.000 000 000 000 000 001 092 684 8 × 2 = 0 + 0.000 000 000 000 000 002 185 369 6;
  • 9) 0.000 000 000 000 000 002 185 369 6 × 2 = 0 + 0.000 000 000 000 000 004 370 739 2;
  • 10) 0.000 000 000 000 000 004 370 739 2 × 2 = 0 + 0.000 000 000 000 000 008 741 478 4;
  • 11) 0.000 000 000 000 000 008 741 478 4 × 2 = 0 + 0.000 000 000 000 000 017 482 956 8;
  • 12) 0.000 000 000 000 000 017 482 956 8 × 2 = 0 + 0.000 000 000 000 000 034 965 913 6;
  • 13) 0.000 000 000 000 000 034 965 913 6 × 2 = 0 + 0.000 000 000 000 000 069 931 827 2;
  • 14) 0.000 000 000 000 000 069 931 827 2 × 2 = 0 + 0.000 000 000 000 000 139 863 654 4;
  • 15) 0.000 000 000 000 000 139 863 654 4 × 2 = 0 + 0.000 000 000 000 000 279 727 308 8;
  • 16) 0.000 000 000 000 000 279 727 308 8 × 2 = 0 + 0.000 000 000 000 000 559 454 617 6;
  • 17) 0.000 000 000 000 000 559 454 617 6 × 2 = 0 + 0.000 000 000 000 001 118 909 235 2;
  • 18) 0.000 000 000 000 001 118 909 235 2 × 2 = 0 + 0.000 000 000 000 002 237 818 470 4;
  • 19) 0.000 000 000 000 002 237 818 470 4 × 2 = 0 + 0.000 000 000 000 004 475 636 940 8;
  • 20) 0.000 000 000 000 004 475 636 940 8 × 2 = 0 + 0.000 000 000 000 008 951 273 881 6;
  • 21) 0.000 000 000 000 008 951 273 881 6 × 2 = 0 + 0.000 000 000 000 017 902 547 763 2;
  • 22) 0.000 000 000 000 017 902 547 763 2 × 2 = 0 + 0.000 000 000 000 035 805 095 526 4;
  • 23) 0.000 000 000 000 035 805 095 526 4 × 2 = 0 + 0.000 000 000 000 071 610 191 052 8;
  • 24) 0.000 000 000 000 071 610 191 052 8 × 2 = 0 + 0.000 000 000 000 143 220 382 105 6;
  • 25) 0.000 000 000 000 143 220 382 105 6 × 2 = 0 + 0.000 000 000 000 286 440 764 211 2;
  • 26) 0.000 000 000 000 286 440 764 211 2 × 2 = 0 + 0.000 000 000 000 572 881 528 422 4;
  • 27) 0.000 000 000 000 572 881 528 422 4 × 2 = 0 + 0.000 000 000 001 145 763 056 844 8;
  • 28) 0.000 000 000 001 145 763 056 844 8 × 2 = 0 + 0.000 000 000 002 291 526 113 689 6;
  • 29) 0.000 000 000 002 291 526 113 689 6 × 2 = 0 + 0.000 000 000 004 583 052 227 379 2;
  • 30) 0.000 000 000 004 583 052 227 379 2 × 2 = 0 + 0.000 000 000 009 166 104 454 758 4;
  • 31) 0.000 000 000 009 166 104 454 758 4 × 2 = 0 + 0.000 000 000 018 332 208 909 516 8;
  • 32) 0.000 000 000 018 332 208 909 516 8 × 2 = 0 + 0.000 000 000 036 664 417 819 033 6;
  • 33) 0.000 000 000 036 664 417 819 033 6 × 2 = 0 + 0.000 000 000 073 328 835 638 067 2;
  • 34) 0.000 000 000 073 328 835 638 067 2 × 2 = 0 + 0.000 000 000 146 657 671 276 134 4;
  • 35) 0.000 000 000 146 657 671 276 134 4 × 2 = 0 + 0.000 000 000 293 315 342 552 268 8;
  • 36) 0.000 000 000 293 315 342 552 268 8 × 2 = 0 + 0.000 000 000 586 630 685 104 537 6;
  • 37) 0.000 000 000 586 630 685 104 537 6 × 2 = 0 + 0.000 000 001 173 261 370 209 075 2;
  • 38) 0.000 000 001 173 261 370 209 075 2 × 2 = 0 + 0.000 000 002 346 522 740 418 150 4;
  • 39) 0.000 000 002 346 522 740 418 150 4 × 2 = 0 + 0.000 000 004 693 045 480 836 300 8;
  • 40) 0.000 000 004 693 045 480 836 300 8 × 2 = 0 + 0.000 000 009 386 090 961 672 601 6;
  • 41) 0.000 000 009 386 090 961 672 601 6 × 2 = 0 + 0.000 000 018 772 181 923 345 203 2;
  • 42) 0.000 000 018 772 181 923 345 203 2 × 2 = 0 + 0.000 000 037 544 363 846 690 406 4;
  • 43) 0.000 000 037 544 363 846 690 406 4 × 2 = 0 + 0.000 000 075 088 727 693 380 812 8;
  • 44) 0.000 000 075 088 727 693 380 812 8 × 2 = 0 + 0.000 000 150 177 455 386 761 625 6;
  • 45) 0.000 000 150 177 455 386 761 625 6 × 2 = 0 + 0.000 000 300 354 910 773 523 251 2;
  • 46) 0.000 000 300 354 910 773 523 251 2 × 2 = 0 + 0.000 000 600 709 821 547 046 502 4;
  • 47) 0.000 000 600 709 821 547 046 502 4 × 2 = 0 + 0.000 001 201 419 643 094 093 004 8;
  • 48) 0.000 001 201 419 643 094 093 004 8 × 2 = 0 + 0.000 002 402 839 286 188 186 009 6;
  • 49) 0.000 002 402 839 286 188 186 009 6 × 2 = 0 + 0.000 004 805 678 572 376 372 019 2;
  • 50) 0.000 004 805 678 572 376 372 019 2 × 2 = 0 + 0.000 009 611 357 144 752 744 038 4;
  • 51) 0.000 009 611 357 144 752 744 038 4 × 2 = 0 + 0.000 019 222 714 289 505 488 076 8;
  • 52) 0.000 019 222 714 289 505 488 076 8 × 2 = 0 + 0.000 038 445 428 579 010 976 153 6;
  • 53) 0.000 038 445 428 579 010 976 153 6 × 2 = 0 + 0.000 076 890 857 158 021 952 307 2;
  • 54) 0.000 076 890 857 158 021 952 307 2 × 2 = 0 + 0.000 153 781 714 316 043 904 614 4;
  • 55) 0.000 153 781 714 316 043 904 614 4 × 2 = 0 + 0.000 307 563 428 632 087 809 228 8;
  • 56) 0.000 307 563 428 632 087 809 228 8 × 2 = 0 + 0.000 615 126 857 264 175 618 457 6;
  • 57) 0.000 615 126 857 264 175 618 457 6 × 2 = 0 + 0.001 230 253 714 528 351 236 915 2;
  • 58) 0.001 230 253 714 528 351 236 915 2 × 2 = 0 + 0.002 460 507 429 056 702 473 830 4;
  • 59) 0.002 460 507 429 056 702 473 830 4 × 2 = 0 + 0.004 921 014 858 113 404 947 660 8;
  • 60) 0.004 921 014 858 113 404 947 660 8 × 2 = 0 + 0.009 842 029 716 226 809 895 321 6;
  • 61) 0.009 842 029 716 226 809 895 321 6 × 2 = 0 + 0.019 684 059 432 453 619 790 643 2;
  • 62) 0.019 684 059 432 453 619 790 643 2 × 2 = 0 + 0.039 368 118 864 907 239 581 286 4;
  • 63) 0.039 368 118 864 907 239 581 286 4 × 2 = 0 + 0.078 736 237 729 814 479 162 572 8;
  • 64) 0.078 736 237 729 814 479 162 572 8 × 2 = 0 + 0.157 472 475 459 628 958 325 145 6;
  • 65) 0.157 472 475 459 628 958 325 145 6 × 2 = 0 + 0.314 944 950 919 257 916 650 291 2;
  • 66) 0.314 944 950 919 257 916 650 291 2 × 2 = 0 + 0.629 889 901 838 515 833 300 582 4;
  • 67) 0.629 889 901 838 515 833 300 582 4 × 2 = 1 + 0.259 779 803 677 031 666 601 164 8;
  • 68) 0.259 779 803 677 031 666 601 164 8 × 2 = 0 + 0.519 559 607 354 063 333 202 329 6;
  • 69) 0.519 559 607 354 063 333 202 329 6 × 2 = 1 + 0.039 119 214 708 126 666 404 659 2;
  • 70) 0.039 119 214 708 126 666 404 659 2 × 2 = 0 + 0.078 238 429 416 253 332 809 318 4;
  • 71) 0.078 238 429 416 253 332 809 318 4 × 2 = 0 + 0.156 476 858 832 506 665 618 636 8;
  • 72) 0.156 476 858 832 506 665 618 636 8 × 2 = 0 + 0.312 953 717 665 013 331 237 273 6;
  • 73) 0.312 953 717 665 013 331 237 273 6 × 2 = 0 + 0.625 907 435 330 026 662 474 547 2;
  • 74) 0.625 907 435 330 026 662 474 547 2 × 2 = 1 + 0.251 814 870 660 053 324 949 094 4;
  • 75) 0.251 814 870 660 053 324 949 094 4 × 2 = 0 + 0.503 629 741 320 106 649 898 188 8;
  • 76) 0.503 629 741 320 106 649 898 188 8 × 2 = 1 + 0.007 259 482 640 213 299 796 377 6;
  • 77) 0.007 259 482 640 213 299 796 377 6 × 2 = 0 + 0.014 518 965 280 426 599 592 755 2;
  • 78) 0.014 518 965 280 426 599 592 755 2 × 2 = 0 + 0.029 037 930 560 853 199 185 510 4;
  • 79) 0.029 037 930 560 853 199 185 510 4 × 2 = 0 + 0.058 075 861 121 706 398 371 020 8;
  • 80) 0.058 075 861 121 706 398 371 020 8 × 2 = 0 + 0.116 151 722 243 412 796 742 041 6;
  • 81) 0.116 151 722 243 412 796 742 041 6 × 2 = 0 + 0.232 303 444 486 825 593 484 083 2;
  • 82) 0.232 303 444 486 825 593 484 083 2 × 2 = 0 + 0.464 606 888 973 651 186 968 166 4;
  • 83) 0.464 606 888 973 651 186 968 166 4 × 2 = 0 + 0.929 213 777 947 302 373 936 332 8;
  • 84) 0.929 213 777 947 302 373 936 332 8 × 2 = 1 + 0.858 427 555 894 604 747 872 665 6;
  • 85) 0.858 427 555 894 604 747 872 665 6 × 2 = 1 + 0.716 855 111 789 209 495 745 331 2;
  • 86) 0.716 855 111 789 209 495 745 331 2 × 2 = 1 + 0.433 710 223 578 418 991 490 662 4;
  • 87) 0.433 710 223 578 418 991 490 662 4 × 2 = 0 + 0.867 420 447 156 837 982 981 324 8;
  • 88) 0.867 420 447 156 837 982 981 324 8 × 2 = 1 + 0.734 840 894 313 675 965 962 649 6;
  • 89) 0.734 840 894 313 675 965 962 649 6 × 2 = 1 + 0.469 681 788 627 351 931 925 299 2;
  • 90) 0.469 681 788 627 351 931 925 299 2 × 2 = 0 + 0.939 363 577 254 703 863 850 598 4;
  • 91) 0.939 363 577 254 703 863 850 598 4 × 2 = 1 + 0.878 727 154 509 407 727 701 196 8;
  • 92) 0.878 727 154 509 407 727 701 196 8 × 2 = 1 + 0.757 454 309 018 815 455 402 393 6;
  • 93) 0.757 454 309 018 815 455 402 393 6 × 2 = 1 + 0.514 908 618 037 630 910 804 787 2;
  • 94) 0.514 908 618 037 630 910 804 787 2 × 2 = 1 + 0.029 817 236 075 261 821 609 574 4;
  • 95) 0.029 817 236 075 261 821 609 574 4 × 2 = 0 + 0.059 634 472 150 523 643 219 148 8;
  • 96) 0.059 634 472 150 523 643 219 148 8 × 2 = 0 + 0.119 268 944 301 047 286 438 297 6;
  • 97) 0.119 268 944 301 047 286 438 297 6 × 2 = 0 + 0.238 537 888 602 094 572 876 595 2;
  • 98) 0.238 537 888 602 094 572 876 595 2 × 2 = 0 + 0.477 075 777 204 189 145 753 190 4;
  • 99) 0.477 075 777 204 189 145 753 190 4 × 2 = 0 + 0.954 151 554 408 378 291 506 380 8;
  • 100) 0.954 151 554 408 378 291 506 380 8 × 2 = 1 + 0.908 303 108 816 756 583 012 761 6;
  • 101) 0.908 303 108 816 756 583 012 761 6 × 2 = 1 + 0.816 606 217 633 513 166 025 523 2;
  • 102) 0.816 606 217 633 513 166 025 523 2 × 2 = 1 + 0.633 212 435 267 026 332 051 046 4;
  • 103) 0.633 212 435 267 026 332 051 046 4 × 2 = 1 + 0.266 424 870 534 052 664 102 092 8;
  • 104) 0.266 424 870 534 052 664 102 092 8 × 2 = 0 + 0.532 849 741 068 105 328 204 185 6;
  • 105) 0.532 849 741 068 105 328 204 185 6 × 2 = 1 + 0.065 699 482 136 210 656 408 371 2;
  • 106) 0.065 699 482 136 210 656 408 371 2 × 2 = 0 + 0.131 398 964 272 421 312 816 742 4;
  • 107) 0.131 398 964 272 421 312 816 742 4 × 2 = 0 + 0.262 797 928 544 842 625 633 484 8;
  • 108) 0.262 797 928 544 842 625 633 484 8 × 2 = 0 + 0.525 595 857 089 685 251 266 969 6;
  • 109) 0.525 595 857 089 685 251 266 969 6 × 2 = 1 + 0.051 191 714 179 370 502 533 939 2;
  • 110) 0.051 191 714 179 370 502 533 939 2 × 2 = 0 + 0.102 383 428 358 741 005 067 878 4;
  • 111) 0.102 383 428 358 741 005 067 878 4 × 2 = 0 + 0.204 766 856 717 482 010 135 756 8;
  • 112) 0.204 766 856 717 482 010 135 756 8 × 2 = 0 + 0.409 533 713 434 964 020 271 513 6;
  • 113) 0.409 533 713 434 964 020 271 513 6 × 2 = 0 + 0.819 067 426 869 928 040 543 027 2;
  • 114) 0.819 067 426 869 928 040 543 027 2 × 2 = 1 + 0.638 134 853 739 856 081 086 054 4;
  • 115) 0.638 134 853 739 856 081 086 054 4 × 2 = 1 + 0.276 269 707 479 712 162 172 108 8;
  • 116) 0.276 269 707 479 712 162 172 108 8 × 2 = 0 + 0.552 539 414 959 424 324 344 217 6;
  • 117) 0.552 539 414 959 424 324 344 217 6 × 2 = 1 + 0.105 078 829 918 848 648 688 435 2;
  • 118) 0.105 078 829 918 848 648 688 435 2 × 2 = 0 + 0.210 157 659 837 697 297 376 870 4;
  • 119) 0.210 157 659 837 697 297 376 870 4 × 2 = 0 + 0.420 315 319 675 394 594 753 740 8;

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:


0.000 000 000 000 000 000 008 536 6(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0101 0000 0001 1101 1011 1100 0001 1110 1000 1000 0110 100(2)

5. Positive number before normalization:

0.000 000 000 000 000 000 008 536 6(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0101 0000 0001 1101 1011 1100 0001 1110 1000 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:


0.000 000 000 000 000 000 008 536 6(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0101 0000 0001 1101 1011 1100 0001 1110 1000 1000 0110 100(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0101 0000 0001 1101 1011 1100 0001 1110 1000 1000 0110 100(2) × 20 =

1.0100 0010 1000 0000 1110 1101 1110 0000 1111 0100 0100 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 1000 0000 1110 1101 1110 0000 1111 0100 0100 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 1000 0000 1110 1101 1110 0000 1111 0100 0100 0011 0100 =

0100 0010 1000 0000 1110 1101 1110 0000 1111 0100 0100 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 1000 0000 1110 1101 1110 0000 1111 0100 0100 0011 0100


Decimal number 0.000 000 000 000 000 000 008 536 6 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1011 1100 - 0100 0010 1000 0000 1110 1101 1110 0000 1111 0100 0100 0011 0100

Share

» Convert decimal 0.000 000 000 000 000 000 008 533 4 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: Pull vs. Push Leadership: The Invisible Power. NotebookLM YouTube Video of 11.31 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