0.000 000 000 000 000 000 008 556 2 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 000 000 008 556 2(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
0.000 000 000 000 000 000 008 556 2(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: 0.000 000 000 000 000 000 008 556 2.

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) 0.000 000 000 000 000 000 008 556 2 × 2 = 0 + 0.000 000 000 000 000 000 017 112 4;
  • 2) 0.000 000 000 000 000 000 017 112 4 × 2 = 0 + 0.000 000 000 000 000 000 034 224 8;
  • 3) 0.000 000 000 000 000 000 034 224 8 × 2 = 0 + 0.000 000 000 000 000 000 068 449 6;
  • 4) 0.000 000 000 000 000 000 068 449 6 × 2 = 0 + 0.000 000 000 000 000 000 136 899 2;
  • 5) 0.000 000 000 000 000 000 136 899 2 × 2 = 0 + 0.000 000 000 000 000 000 273 798 4;
  • 6) 0.000 000 000 000 000 000 273 798 4 × 2 = 0 + 0.000 000 000 000 000 000 547 596 8;
  • 7) 0.000 000 000 000 000 000 547 596 8 × 2 = 0 + 0.000 000 000 000 000 001 095 193 6;
  • 8) 0.000 000 000 000 000 001 095 193 6 × 2 = 0 + 0.000 000 000 000 000 002 190 387 2;
  • 9) 0.000 000 000 000 000 002 190 387 2 × 2 = 0 + 0.000 000 000 000 000 004 380 774 4;
  • 10) 0.000 000 000 000 000 004 380 774 4 × 2 = 0 + 0.000 000 000 000 000 008 761 548 8;
  • 11) 0.000 000 000 000 000 008 761 548 8 × 2 = 0 + 0.000 000 000 000 000 017 523 097 6;
  • 12) 0.000 000 000 000 000 017 523 097 6 × 2 = 0 + 0.000 000 000 000 000 035 046 195 2;
  • 13) 0.000 000 000 000 000 035 046 195 2 × 2 = 0 + 0.000 000 000 000 000 070 092 390 4;
  • 14) 0.000 000 000 000 000 070 092 390 4 × 2 = 0 + 0.000 000 000 000 000 140 184 780 8;
  • 15) 0.000 000 000 000 000 140 184 780 8 × 2 = 0 + 0.000 000 000 000 000 280 369 561 6;
  • 16) 0.000 000 000 000 000 280 369 561 6 × 2 = 0 + 0.000 000 000 000 000 560 739 123 2;
  • 17) 0.000 000 000 000 000 560 739 123 2 × 2 = 0 + 0.000 000 000 000 001 121 478 246 4;
  • 18) 0.000 000 000 000 001 121 478 246 4 × 2 = 0 + 0.000 000 000 000 002 242 956 492 8;
  • 19) 0.000 000 000 000 002 242 956 492 8 × 2 = 0 + 0.000 000 000 000 004 485 912 985 6;
  • 20) 0.000 000 000 000 004 485 912 985 6 × 2 = 0 + 0.000 000 000 000 008 971 825 971 2;
  • 21) 0.000 000 000 000 008 971 825 971 2 × 2 = 0 + 0.000 000 000 000 017 943 651 942 4;
  • 22) 0.000 000 000 000 017 943 651 942 4 × 2 = 0 + 0.000 000 000 000 035 887 303 884 8;
  • 23) 0.000 000 000 000 035 887 303 884 8 × 2 = 0 + 0.000 000 000 000 071 774 607 769 6;
  • 24) 0.000 000 000 000 071 774 607 769 6 × 2 = 0 + 0.000 000 000 000 143 549 215 539 2;
  • 25) 0.000 000 000 000 143 549 215 539 2 × 2 = 0 + 0.000 000 000 000 287 098 431 078 4;
  • 26) 0.000 000 000 000 287 098 431 078 4 × 2 = 0 + 0.000 000 000 000 574 196 862 156 8;
  • 27) 0.000 000 000 000 574 196 862 156 8 × 2 = 0 + 0.000 000 000 001 148 393 724 313 6;
  • 28) 0.000 000 000 001 148 393 724 313 6 × 2 = 0 + 0.000 000 000 002 296 787 448 627 2;
  • 29) 0.000 000 000 002 296 787 448 627 2 × 2 = 0 + 0.000 000 000 004 593 574 897 254 4;
  • 30) 0.000 000 000 004 593 574 897 254 4 × 2 = 0 + 0.000 000 000 009 187 149 794 508 8;
  • 31) 0.000 000 000 009 187 149 794 508 8 × 2 = 0 + 0.000 000 000 018 374 299 589 017 6;
  • 32) 0.000 000 000 018 374 299 589 017 6 × 2 = 0 + 0.000 000 000 036 748 599 178 035 2;
  • 33) 0.000 000 000 036 748 599 178 035 2 × 2 = 0 + 0.000 000 000 073 497 198 356 070 4;
  • 34) 0.000 000 000 073 497 198 356 070 4 × 2 = 0 + 0.000 000 000 146 994 396 712 140 8;
  • 35) 0.000 000 000 146 994 396 712 140 8 × 2 = 0 + 0.000 000 000 293 988 793 424 281 6;
  • 36) 0.000 000 000 293 988 793 424 281 6 × 2 = 0 + 0.000 000 000 587 977 586 848 563 2;
  • 37) 0.000 000 000 587 977 586 848 563 2 × 2 = 0 + 0.000 000 001 175 955 173 697 126 4;
  • 38) 0.000 000 001 175 955 173 697 126 4 × 2 = 0 + 0.000 000 002 351 910 347 394 252 8;
  • 39) 0.000 000 002 351 910 347 394 252 8 × 2 = 0 + 0.000 000 004 703 820 694 788 505 6;
  • 40) 0.000 000 004 703 820 694 788 505 6 × 2 = 0 + 0.000 000 009 407 641 389 577 011 2;
  • 41) 0.000 000 009 407 641 389 577 011 2 × 2 = 0 + 0.000 000 018 815 282 779 154 022 4;
  • 42) 0.000 000 018 815 282 779 154 022 4 × 2 = 0 + 0.000 000 037 630 565 558 308 044 8;
  • 43) 0.000 000 037 630 565 558 308 044 8 × 2 = 0 + 0.000 000 075 261 131 116 616 089 6;
  • 44) 0.000 000 075 261 131 116 616 089 6 × 2 = 0 + 0.000 000 150 522 262 233 232 179 2;
  • 45) 0.000 000 150 522 262 233 232 179 2 × 2 = 0 + 0.000 000 301 044 524 466 464 358 4;
  • 46) 0.000 000 301 044 524 466 464 358 4 × 2 = 0 + 0.000 000 602 089 048 932 928 716 8;
  • 47) 0.000 000 602 089 048 932 928 716 8 × 2 = 0 + 0.000 001 204 178 097 865 857 433 6;
  • 48) 0.000 001 204 178 097 865 857 433 6 × 2 = 0 + 0.000 002 408 356 195 731 714 867 2;
  • 49) 0.000 002 408 356 195 731 714 867 2 × 2 = 0 + 0.000 004 816 712 391 463 429 734 4;
  • 50) 0.000 004 816 712 391 463 429 734 4 × 2 = 0 + 0.000 009 633 424 782 926 859 468 8;
  • 51) 0.000 009 633 424 782 926 859 468 8 × 2 = 0 + 0.000 019 266 849 565 853 718 937 6;
  • 52) 0.000 019 266 849 565 853 718 937 6 × 2 = 0 + 0.000 038 533 699 131 707 437 875 2;
  • 53) 0.000 038 533 699 131 707 437 875 2 × 2 = 0 + 0.000 077 067 398 263 414 875 750 4;
  • 54) 0.000 077 067 398 263 414 875 750 4 × 2 = 0 + 0.000 154 134 796 526 829 751 500 8;
  • 55) 0.000 154 134 796 526 829 751 500 8 × 2 = 0 + 0.000 308 269 593 053 659 503 001 6;
  • 56) 0.000 308 269 593 053 659 503 001 6 × 2 = 0 + 0.000 616 539 186 107 319 006 003 2;
  • 57) 0.000 616 539 186 107 319 006 003 2 × 2 = 0 + 0.001 233 078 372 214 638 012 006 4;
  • 58) 0.001 233 078 372 214 638 012 006 4 × 2 = 0 + 0.002 466 156 744 429 276 024 012 8;
  • 59) 0.002 466 156 744 429 276 024 012 8 × 2 = 0 + 0.004 932 313 488 858 552 048 025 6;
  • 60) 0.004 932 313 488 858 552 048 025 6 × 2 = 0 + 0.009 864 626 977 717 104 096 051 2;
  • 61) 0.009 864 626 977 717 104 096 051 2 × 2 = 0 + 0.019 729 253 955 434 208 192 102 4;
  • 62) 0.019 729 253 955 434 208 192 102 4 × 2 = 0 + 0.039 458 507 910 868 416 384 204 8;
  • 63) 0.039 458 507 910 868 416 384 204 8 × 2 = 0 + 0.078 917 015 821 736 832 768 409 6;
  • 64) 0.078 917 015 821 736 832 768 409 6 × 2 = 0 + 0.157 834 031 643 473 665 536 819 2;
  • 65) 0.157 834 031 643 473 665 536 819 2 × 2 = 0 + 0.315 668 063 286 947 331 073 638 4;
  • 66) 0.315 668 063 286 947 331 073 638 4 × 2 = 0 + 0.631 336 126 573 894 662 147 276 8;
  • 67) 0.631 336 126 573 894 662 147 276 8 × 2 = 1 + 0.262 672 253 147 789 324 294 553 6;
  • 68) 0.262 672 253 147 789 324 294 553 6 × 2 = 0 + 0.525 344 506 295 578 648 589 107 2;
  • 69) 0.525 344 506 295 578 648 589 107 2 × 2 = 1 + 0.050 689 012 591 157 297 178 214 4;
  • 70) 0.050 689 012 591 157 297 178 214 4 × 2 = 0 + 0.101 378 025 182 314 594 356 428 8;
  • 71) 0.101 378 025 182 314 594 356 428 8 × 2 = 0 + 0.202 756 050 364 629 188 712 857 6;
  • 72) 0.202 756 050 364 629 188 712 857 6 × 2 = 0 + 0.405 512 100 729 258 377 425 715 2;
  • 73) 0.405 512 100 729 258 377 425 715 2 × 2 = 0 + 0.811 024 201 458 516 754 851 430 4;
  • 74) 0.811 024 201 458 516 754 851 430 4 × 2 = 1 + 0.622 048 402 917 033 509 702 860 8;
  • 75) 0.622 048 402 917 033 509 702 860 8 × 2 = 1 + 0.244 096 805 834 067 019 405 721 6;
  • 76) 0.244 096 805 834 067 019 405 721 6 × 2 = 0 + 0.488 193 611 668 134 038 811 443 2;
  • 77) 0.488 193 611 668 134 038 811 443 2 × 2 = 0 + 0.976 387 223 336 268 077 622 886 4;
  • 78) 0.976 387 223 336 268 077 622 886 4 × 2 = 1 + 0.952 774 446 672 536 155 245 772 8;
  • 79) 0.952 774 446 672 536 155 245 772 8 × 2 = 1 + 0.905 548 893 345 072 310 491 545 6;
  • 80) 0.905 548 893 345 072 310 491 545 6 × 2 = 1 + 0.811 097 786 690 144 620 983 091 2;
  • 81) 0.811 097 786 690 144 620 983 091 2 × 2 = 1 + 0.622 195 573 380 289 241 966 182 4;
  • 82) 0.622 195 573 380 289 241 966 182 4 × 2 = 1 + 0.244 391 146 760 578 483 932 364 8;
  • 83) 0.244 391 146 760 578 483 932 364 8 × 2 = 0 + 0.488 782 293 521 156 967 864 729 6;
  • 84) 0.488 782 293 521 156 967 864 729 6 × 2 = 0 + 0.977 564 587 042 313 935 729 459 2;
  • 85) 0.977 564 587 042 313 935 729 459 2 × 2 = 1 + 0.955 129 174 084 627 871 458 918 4;
  • 86) 0.955 129 174 084 627 871 458 918 4 × 2 = 1 + 0.910 258 348 169 255 742 917 836 8;
  • 87) 0.910 258 348 169 255 742 917 836 8 × 2 = 1 + 0.820 516 696 338 511 485 835 673 6;
  • 88) 0.820 516 696 338 511 485 835 673 6 × 2 = 1 + 0.641 033 392 677 022 971 671 347 2;
  • 89) 0.641 033 392 677 022 971 671 347 2 × 2 = 1 + 0.282 066 785 354 045 943 342 694 4;
  • 90) 0.282 066 785 354 045 943 342 694 4 × 2 = 0 + 0.564 133 570 708 091 886 685 388 8;
  • 91) 0.564 133 570 708 091 886 685 388 8 × 2 = 1 + 0.128 267 141 416 183 773 370 777 6;
  • 92) 0.128 267 141 416 183 773 370 777 6 × 2 = 0 + 0.256 534 282 832 367 546 741 555 2;
  • 93) 0.256 534 282 832 367 546 741 555 2 × 2 = 0 + 0.513 068 565 664 735 093 483 110 4;
  • 94) 0.513 068 565 664 735 093 483 110 4 × 2 = 1 + 0.026 137 131 329 470 186 966 220 8;
  • 95) 0.026 137 131 329 470 186 966 220 8 × 2 = 0 + 0.052 274 262 658 940 373 932 441 6;
  • 96) 0.052 274 262 658 940 373 932 441 6 × 2 = 0 + 0.104 548 525 317 880 747 864 883 2;
  • 97) 0.104 548 525 317 880 747 864 883 2 × 2 = 0 + 0.209 097 050 635 761 495 729 766 4;
  • 98) 0.209 097 050 635 761 495 729 766 4 × 2 = 0 + 0.418 194 101 271 522 991 459 532 8;
  • 99) 0.418 194 101 271 522 991 459 532 8 × 2 = 0 + 0.836 388 202 543 045 982 919 065 6;
  • 100) 0.836 388 202 543 045 982 919 065 6 × 2 = 1 + 0.672 776 405 086 091 965 838 131 2;
  • 101) 0.672 776 405 086 091 965 838 131 2 × 2 = 1 + 0.345 552 810 172 183 931 676 262 4;
  • 102) 0.345 552 810 172 183 931 676 262 4 × 2 = 0 + 0.691 105 620 344 367 863 352 524 8;
  • 103) 0.691 105 620 344 367 863 352 524 8 × 2 = 1 + 0.382 211 240 688 735 726 705 049 6;
  • 104) 0.382 211 240 688 735 726 705 049 6 × 2 = 0 + 0.764 422 481 377 471 453 410 099 2;
  • 105) 0.764 422 481 377 471 453 410 099 2 × 2 = 1 + 0.528 844 962 754 942 906 820 198 4;
  • 106) 0.528 844 962 754 942 906 820 198 4 × 2 = 1 + 0.057 689 925 509 885 813 640 396 8;
  • 107) 0.057 689 925 509 885 813 640 396 8 × 2 = 0 + 0.115 379 851 019 771 627 280 793 6;
  • 108) 0.115 379 851 019 771 627 280 793 6 × 2 = 0 + 0.230 759 702 039 543 254 561 587 2;
  • 109) 0.230 759 702 039 543 254 561 587 2 × 2 = 0 + 0.461 519 404 079 086 509 123 174 4;
  • 110) 0.461 519 404 079 086 509 123 174 4 × 2 = 0 + 0.923 038 808 158 173 018 246 348 8;
  • 111) 0.923 038 808 158 173 018 246 348 8 × 2 = 1 + 0.846 077 616 316 346 036 492 697 6;
  • 112) 0.846 077 616 316 346 036 492 697 6 × 2 = 1 + 0.692 155 232 632 692 072 985 395 2;
  • 113) 0.692 155 232 632 692 072 985 395 2 × 2 = 1 + 0.384 310 465 265 384 145 970 790 4;
  • 114) 0.384 310 465 265 384 145 970 790 4 × 2 = 0 + 0.768 620 930 530 768 291 941 580 8;
  • 115) 0.768 620 930 530 768 291 941 580 8 × 2 = 1 + 0.537 241 861 061 536 583 883 161 6;
  • 116) 0.537 241 861 061 536 583 883 161 6 × 2 = 1 + 0.074 483 722 123 073 167 766 323 2;
  • 117) 0.074 483 722 123 073 167 766 323 2 × 2 = 0 + 0.148 967 444 246 146 335 532 646 4;
  • 118) 0.148 967 444 246 146 335 532 646 4 × 2 = 0 + 0.297 934 888 492 292 671 065 292 8;
  • 119) 0.297 934 888 492 292 671 065 292 8 × 2 = 0 + 0.595 869 776 984 585 342 130 585 6;

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:


0.000 000 000 000 000 000 008 556 2(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0110 0111 1100 1111 1010 0100 0001 1010 1100 0011 1011 000(2)

5. Positive number before normalization:

0.000 000 000 000 000 000 008 556 2(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0110 0111 1100 1111 1010 0100 0001 1010 1100 0011 1011 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:


0.000 000 000 000 000 000 008 556 2(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0110 0111 1100 1111 1010 0100 0001 1010 1100 0011 1011 000(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0110 0111 1100 1111 1010 0100 0001 1010 1100 0011 1011 000(2) × 20 =

1.0100 0011 0011 1110 0111 1101 0010 0000 1101 0110 0001 1101 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 0011 0011 1110 0111 1101 0010 0000 1101 0110 0001 1101 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 0011 0011 1110 0111 1101 0010 0000 1101 0110 0001 1101 1000 =

0100 0011 0011 1110 0111 1101 0010 0000 1101 0110 0001 1101 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 0011 0011 1110 0111 1101 0010 0000 1101 0110 0001 1101 1000


Decimal number 0.000 000 000 000 000 000 008 556 2 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1011 1100 - 0100 0011 0011 1110 0111 1101 0010 0000 1101 0110 0001 1101 1000

Share

» Convert decimal 0.000 000 000 000 000 000 008 565 7 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: Reputation: Bridge Between Company's Actions and Market Value. NotebookLM YouTube Video of 6.55 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