0.000 000 000 000 000 000 008 495 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 000 000 008 495(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
0.000 000 000 000 000 000 008 495(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: 0.000 000 000 000 000 000 008 495.

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) 0.000 000 000 000 000 000 008 495 × 2 = 0 + 0.000 000 000 000 000 000 016 99;
  • 2) 0.000 000 000 000 000 000 016 99 × 2 = 0 + 0.000 000 000 000 000 000 033 98;
  • 3) 0.000 000 000 000 000 000 033 98 × 2 = 0 + 0.000 000 000 000 000 000 067 96;
  • 4) 0.000 000 000 000 000 000 067 96 × 2 = 0 + 0.000 000 000 000 000 000 135 92;
  • 5) 0.000 000 000 000 000 000 135 92 × 2 = 0 + 0.000 000 000 000 000 000 271 84;
  • 6) 0.000 000 000 000 000 000 271 84 × 2 = 0 + 0.000 000 000 000 000 000 543 68;
  • 7) 0.000 000 000 000 000 000 543 68 × 2 = 0 + 0.000 000 000 000 000 001 087 36;
  • 8) 0.000 000 000 000 000 001 087 36 × 2 = 0 + 0.000 000 000 000 000 002 174 72;
  • 9) 0.000 000 000 000 000 002 174 72 × 2 = 0 + 0.000 000 000 000 000 004 349 44;
  • 10) 0.000 000 000 000 000 004 349 44 × 2 = 0 + 0.000 000 000 000 000 008 698 88;
  • 11) 0.000 000 000 000 000 008 698 88 × 2 = 0 + 0.000 000 000 000 000 017 397 76;
  • 12) 0.000 000 000 000 000 017 397 76 × 2 = 0 + 0.000 000 000 000 000 034 795 52;
  • 13) 0.000 000 000 000 000 034 795 52 × 2 = 0 + 0.000 000 000 000 000 069 591 04;
  • 14) 0.000 000 000 000 000 069 591 04 × 2 = 0 + 0.000 000 000 000 000 139 182 08;
  • 15) 0.000 000 000 000 000 139 182 08 × 2 = 0 + 0.000 000 000 000 000 278 364 16;
  • 16) 0.000 000 000 000 000 278 364 16 × 2 = 0 + 0.000 000 000 000 000 556 728 32;
  • 17) 0.000 000 000 000 000 556 728 32 × 2 = 0 + 0.000 000 000 000 001 113 456 64;
  • 18) 0.000 000 000 000 001 113 456 64 × 2 = 0 + 0.000 000 000 000 002 226 913 28;
  • 19) 0.000 000 000 000 002 226 913 28 × 2 = 0 + 0.000 000 000 000 004 453 826 56;
  • 20) 0.000 000 000 000 004 453 826 56 × 2 = 0 + 0.000 000 000 000 008 907 653 12;
  • 21) 0.000 000 000 000 008 907 653 12 × 2 = 0 + 0.000 000 000 000 017 815 306 24;
  • 22) 0.000 000 000 000 017 815 306 24 × 2 = 0 + 0.000 000 000 000 035 630 612 48;
  • 23) 0.000 000 000 000 035 630 612 48 × 2 = 0 + 0.000 000 000 000 071 261 224 96;
  • 24) 0.000 000 000 000 071 261 224 96 × 2 = 0 + 0.000 000 000 000 142 522 449 92;
  • 25) 0.000 000 000 000 142 522 449 92 × 2 = 0 + 0.000 000 000 000 285 044 899 84;
  • 26) 0.000 000 000 000 285 044 899 84 × 2 = 0 + 0.000 000 000 000 570 089 799 68;
  • 27) 0.000 000 000 000 570 089 799 68 × 2 = 0 + 0.000 000 000 001 140 179 599 36;
  • 28) 0.000 000 000 001 140 179 599 36 × 2 = 0 + 0.000 000 000 002 280 359 198 72;
  • 29) 0.000 000 000 002 280 359 198 72 × 2 = 0 + 0.000 000 000 004 560 718 397 44;
  • 30) 0.000 000 000 004 560 718 397 44 × 2 = 0 + 0.000 000 000 009 121 436 794 88;
  • 31) 0.000 000 000 009 121 436 794 88 × 2 = 0 + 0.000 000 000 018 242 873 589 76;
  • 32) 0.000 000 000 018 242 873 589 76 × 2 = 0 + 0.000 000 000 036 485 747 179 52;
  • 33) 0.000 000 000 036 485 747 179 52 × 2 = 0 + 0.000 000 000 072 971 494 359 04;
  • 34) 0.000 000 000 072 971 494 359 04 × 2 = 0 + 0.000 000 000 145 942 988 718 08;
  • 35) 0.000 000 000 145 942 988 718 08 × 2 = 0 + 0.000 000 000 291 885 977 436 16;
  • 36) 0.000 000 000 291 885 977 436 16 × 2 = 0 + 0.000 000 000 583 771 954 872 32;
  • 37) 0.000 000 000 583 771 954 872 32 × 2 = 0 + 0.000 000 001 167 543 909 744 64;
  • 38) 0.000 000 001 167 543 909 744 64 × 2 = 0 + 0.000 000 002 335 087 819 489 28;
  • 39) 0.000 000 002 335 087 819 489 28 × 2 = 0 + 0.000 000 004 670 175 638 978 56;
  • 40) 0.000 000 004 670 175 638 978 56 × 2 = 0 + 0.000 000 009 340 351 277 957 12;
  • 41) 0.000 000 009 340 351 277 957 12 × 2 = 0 + 0.000 000 018 680 702 555 914 24;
  • 42) 0.000 000 018 680 702 555 914 24 × 2 = 0 + 0.000 000 037 361 405 111 828 48;
  • 43) 0.000 000 037 361 405 111 828 48 × 2 = 0 + 0.000 000 074 722 810 223 656 96;
  • 44) 0.000 000 074 722 810 223 656 96 × 2 = 0 + 0.000 000 149 445 620 447 313 92;
  • 45) 0.000 000 149 445 620 447 313 92 × 2 = 0 + 0.000 000 298 891 240 894 627 84;
  • 46) 0.000 000 298 891 240 894 627 84 × 2 = 0 + 0.000 000 597 782 481 789 255 68;
  • 47) 0.000 000 597 782 481 789 255 68 × 2 = 0 + 0.000 001 195 564 963 578 511 36;
  • 48) 0.000 001 195 564 963 578 511 36 × 2 = 0 + 0.000 002 391 129 927 157 022 72;
  • 49) 0.000 002 391 129 927 157 022 72 × 2 = 0 + 0.000 004 782 259 854 314 045 44;
  • 50) 0.000 004 782 259 854 314 045 44 × 2 = 0 + 0.000 009 564 519 708 628 090 88;
  • 51) 0.000 009 564 519 708 628 090 88 × 2 = 0 + 0.000 019 129 039 417 256 181 76;
  • 52) 0.000 019 129 039 417 256 181 76 × 2 = 0 + 0.000 038 258 078 834 512 363 52;
  • 53) 0.000 038 258 078 834 512 363 52 × 2 = 0 + 0.000 076 516 157 669 024 727 04;
  • 54) 0.000 076 516 157 669 024 727 04 × 2 = 0 + 0.000 153 032 315 338 049 454 08;
  • 55) 0.000 153 032 315 338 049 454 08 × 2 = 0 + 0.000 306 064 630 676 098 908 16;
  • 56) 0.000 306 064 630 676 098 908 16 × 2 = 0 + 0.000 612 129 261 352 197 816 32;
  • 57) 0.000 612 129 261 352 197 816 32 × 2 = 0 + 0.001 224 258 522 704 395 632 64;
  • 58) 0.001 224 258 522 704 395 632 64 × 2 = 0 + 0.002 448 517 045 408 791 265 28;
  • 59) 0.002 448 517 045 408 791 265 28 × 2 = 0 + 0.004 897 034 090 817 582 530 56;
  • 60) 0.004 897 034 090 817 582 530 56 × 2 = 0 + 0.009 794 068 181 635 165 061 12;
  • 61) 0.009 794 068 181 635 165 061 12 × 2 = 0 + 0.019 588 136 363 270 330 122 24;
  • 62) 0.019 588 136 363 270 330 122 24 × 2 = 0 + 0.039 176 272 726 540 660 244 48;
  • 63) 0.039 176 272 726 540 660 244 48 × 2 = 0 + 0.078 352 545 453 081 320 488 96;
  • 64) 0.078 352 545 453 081 320 488 96 × 2 = 0 + 0.156 705 090 906 162 640 977 92;
  • 65) 0.156 705 090 906 162 640 977 92 × 2 = 0 + 0.313 410 181 812 325 281 955 84;
  • 66) 0.313 410 181 812 325 281 955 84 × 2 = 0 + 0.626 820 363 624 650 563 911 68;
  • 67) 0.626 820 363 624 650 563 911 68 × 2 = 1 + 0.253 640 727 249 301 127 823 36;
  • 68) 0.253 640 727 249 301 127 823 36 × 2 = 0 + 0.507 281 454 498 602 255 646 72;
  • 69) 0.507 281 454 498 602 255 646 72 × 2 = 1 + 0.014 562 908 997 204 511 293 44;
  • 70) 0.014 562 908 997 204 511 293 44 × 2 = 0 + 0.029 125 817 994 409 022 586 88;
  • 71) 0.029 125 817 994 409 022 586 88 × 2 = 0 + 0.058 251 635 988 818 045 173 76;
  • 72) 0.058 251 635 988 818 045 173 76 × 2 = 0 + 0.116 503 271 977 636 090 347 52;
  • 73) 0.116 503 271 977 636 090 347 52 × 2 = 0 + 0.233 006 543 955 272 180 695 04;
  • 74) 0.233 006 543 955 272 180 695 04 × 2 = 0 + 0.466 013 087 910 544 361 390 08;
  • 75) 0.466 013 087 910 544 361 390 08 × 2 = 0 + 0.932 026 175 821 088 722 780 16;
  • 76) 0.932 026 175 821 088 722 780 16 × 2 = 1 + 0.864 052 351 642 177 445 560 32;
  • 77) 0.864 052 351 642 177 445 560 32 × 2 = 1 + 0.728 104 703 284 354 891 120 64;
  • 78) 0.728 104 703 284 354 891 120 64 × 2 = 1 + 0.456 209 406 568 709 782 241 28;
  • 79) 0.456 209 406 568 709 782 241 28 × 2 = 0 + 0.912 418 813 137 419 564 482 56;
  • 80) 0.912 418 813 137 419 564 482 56 × 2 = 1 + 0.824 837 626 274 839 128 965 12;
  • 81) 0.824 837 626 274 839 128 965 12 × 2 = 1 + 0.649 675 252 549 678 257 930 24;
  • 82) 0.649 675 252 549 678 257 930 24 × 2 = 1 + 0.299 350 505 099 356 515 860 48;
  • 83) 0.299 350 505 099 356 515 860 48 × 2 = 0 + 0.598 701 010 198 713 031 720 96;
  • 84) 0.598 701 010 198 713 031 720 96 × 2 = 1 + 0.197 402 020 397 426 063 441 92;
  • 85) 0.197 402 020 397 426 063 441 92 × 2 = 0 + 0.394 804 040 794 852 126 883 84;
  • 86) 0.394 804 040 794 852 126 883 84 × 2 = 0 + 0.789 608 081 589 704 253 767 68;
  • 87) 0.789 608 081 589 704 253 767 68 × 2 = 1 + 0.579 216 163 179 408 507 535 36;
  • 88) 0.579 216 163 179 408 507 535 36 × 2 = 1 + 0.158 432 326 358 817 015 070 72;
  • 89) 0.158 432 326 358 817 015 070 72 × 2 = 0 + 0.316 864 652 717 634 030 141 44;
  • 90) 0.316 864 652 717 634 030 141 44 × 2 = 0 + 0.633 729 305 435 268 060 282 88;
  • 91) 0.633 729 305 435 268 060 282 88 × 2 = 1 + 0.267 458 610 870 536 120 565 76;
  • 92) 0.267 458 610 870 536 120 565 76 × 2 = 0 + 0.534 917 221 741 072 241 131 52;
  • 93) 0.534 917 221 741 072 241 131 52 × 2 = 1 + 0.069 834 443 482 144 482 263 04;
  • 94) 0.069 834 443 482 144 482 263 04 × 2 = 0 + 0.139 668 886 964 288 964 526 08;
  • 95) 0.139 668 886 964 288 964 526 08 × 2 = 0 + 0.279 337 773 928 577 929 052 16;
  • 96) 0.279 337 773 928 577 929 052 16 × 2 = 0 + 0.558 675 547 857 155 858 104 32;
  • 97) 0.558 675 547 857 155 858 104 32 × 2 = 1 + 0.117 351 095 714 311 716 208 64;
  • 98) 0.117 351 095 714 311 716 208 64 × 2 = 0 + 0.234 702 191 428 623 432 417 28;
  • 99) 0.234 702 191 428 623 432 417 28 × 2 = 0 + 0.469 404 382 857 246 864 834 56;
  • 100) 0.469 404 382 857 246 864 834 56 × 2 = 0 + 0.938 808 765 714 493 729 669 12;
  • 101) 0.938 808 765 714 493 729 669 12 × 2 = 1 + 0.877 617 531 428 987 459 338 24;
  • 102) 0.877 617 531 428 987 459 338 24 × 2 = 1 + 0.755 235 062 857 974 918 676 48;
  • 103) 0.755 235 062 857 974 918 676 48 × 2 = 1 + 0.510 470 125 715 949 837 352 96;
  • 104) 0.510 470 125 715 949 837 352 96 × 2 = 1 + 0.020 940 251 431 899 674 705 92;
  • 105) 0.020 940 251 431 899 674 705 92 × 2 = 0 + 0.041 880 502 863 799 349 411 84;
  • 106) 0.041 880 502 863 799 349 411 84 × 2 = 0 + 0.083 761 005 727 598 698 823 68;
  • 107) 0.083 761 005 727 598 698 823 68 × 2 = 0 + 0.167 522 011 455 197 397 647 36;
  • 108) 0.167 522 011 455 197 397 647 36 × 2 = 0 + 0.335 044 022 910 394 795 294 72;
  • 109) 0.335 044 022 910 394 795 294 72 × 2 = 0 + 0.670 088 045 820 789 590 589 44;
  • 110) 0.670 088 045 820 789 590 589 44 × 2 = 1 + 0.340 176 091 641 579 181 178 88;
  • 111) 0.340 176 091 641 579 181 178 88 × 2 = 0 + 0.680 352 183 283 158 362 357 76;
  • 112) 0.680 352 183 283 158 362 357 76 × 2 = 1 + 0.360 704 366 566 316 724 715 52;
  • 113) 0.360 704 366 566 316 724 715 52 × 2 = 0 + 0.721 408 733 132 633 449 431 04;
  • 114) 0.721 408 733 132 633 449 431 04 × 2 = 1 + 0.442 817 466 265 266 898 862 08;
  • 115) 0.442 817 466 265 266 898 862 08 × 2 = 0 + 0.885 634 932 530 533 797 724 16;
  • 116) 0.885 634 932 530 533 797 724 16 × 2 = 1 + 0.771 269 865 061 067 595 448 32;
  • 117) 0.771 269 865 061 067 595 448 32 × 2 = 1 + 0.542 539 730 122 135 190 896 64;
  • 118) 0.542 539 730 122 135 190 896 64 × 2 = 1 + 0.085 079 460 244 270 381 793 28;
  • 119) 0.085 079 460 244 270 381 793 28 × 2 = 0 + 0.170 158 920 488 540 763 586 56;

We didn't get any fractional part that was equal to zero. But we had enough iterations (over Mantissa limit) and at least one integer that was different from zero => FULL STOP (Losing precision - the converted number we get in the end will be just a very good approximation of the initial one).


4. Construct the base 2 representation of the fractional part of the number.

Take all the integer parts of the multiplying operations, starting from the top of the constructed list above:


0.000 000 000 000 000 000 008 495(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0001 1101 1101 0011 0010 1000 1000 1111 0000 0101 0101 110(2)

5. Positive number before normalization:

0.000 000 000 000 000 000 008 495(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0001 1101 1101 0011 0010 1000 1000 1111 0000 0101 0101 110(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:


0.000 000 000 000 000 000 008 495(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0001 1101 1101 0011 0010 1000 1000 1111 0000 0101 0101 110(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0001 1101 1101 0011 0010 1000 1000 1111 0000 0101 0101 110(2) × 20 =

1.0100 0000 1110 1110 1001 1001 0100 0100 0111 1000 0010 1010 1110(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 0000 1110 1110 1001 1001 0100 0100 0111 1000 0010 1010 1110


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 0000 1110 1110 1001 1001 0100 0100 0111 1000 0010 1010 1110 =

0100 0000 1110 1110 1001 1001 0100 0100 0111 1000 0010 1010 1110


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 0000 1110 1110 1001 1001 0100 0100 0111 1000 0010 1010 1110


Decimal number 0.000 000 000 000 000 000 008 495 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1011 1100 - 0100 0000 1110 1110 1001 1001 0100 0100 0111 1000 0010 1010 1110

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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