0.000 000 000 000 000 000 008 467 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 000 000 008 467(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
0.000 000 000 000 000 000 008 467(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: 0.000 000 000 000 000 000 008 467.

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) 0.000 000 000 000 000 000 008 467 × 2 = 0 + 0.000 000 000 000 000 000 016 934;
  • 2) 0.000 000 000 000 000 000 016 934 × 2 = 0 + 0.000 000 000 000 000 000 033 868;
  • 3) 0.000 000 000 000 000 000 033 868 × 2 = 0 + 0.000 000 000 000 000 000 067 736;
  • 4) 0.000 000 000 000 000 000 067 736 × 2 = 0 + 0.000 000 000 000 000 000 135 472;
  • 5) 0.000 000 000 000 000 000 135 472 × 2 = 0 + 0.000 000 000 000 000 000 270 944;
  • 6) 0.000 000 000 000 000 000 270 944 × 2 = 0 + 0.000 000 000 000 000 000 541 888;
  • 7) 0.000 000 000 000 000 000 541 888 × 2 = 0 + 0.000 000 000 000 000 001 083 776;
  • 8) 0.000 000 000 000 000 001 083 776 × 2 = 0 + 0.000 000 000 000 000 002 167 552;
  • 9) 0.000 000 000 000 000 002 167 552 × 2 = 0 + 0.000 000 000 000 000 004 335 104;
  • 10) 0.000 000 000 000 000 004 335 104 × 2 = 0 + 0.000 000 000 000 000 008 670 208;
  • 11) 0.000 000 000 000 000 008 670 208 × 2 = 0 + 0.000 000 000 000 000 017 340 416;
  • 12) 0.000 000 000 000 000 017 340 416 × 2 = 0 + 0.000 000 000 000 000 034 680 832;
  • 13) 0.000 000 000 000 000 034 680 832 × 2 = 0 + 0.000 000 000 000 000 069 361 664;
  • 14) 0.000 000 000 000 000 069 361 664 × 2 = 0 + 0.000 000 000 000 000 138 723 328;
  • 15) 0.000 000 000 000 000 138 723 328 × 2 = 0 + 0.000 000 000 000 000 277 446 656;
  • 16) 0.000 000 000 000 000 277 446 656 × 2 = 0 + 0.000 000 000 000 000 554 893 312;
  • 17) 0.000 000 000 000 000 554 893 312 × 2 = 0 + 0.000 000 000 000 001 109 786 624;
  • 18) 0.000 000 000 000 001 109 786 624 × 2 = 0 + 0.000 000 000 000 002 219 573 248;
  • 19) 0.000 000 000 000 002 219 573 248 × 2 = 0 + 0.000 000 000 000 004 439 146 496;
  • 20) 0.000 000 000 000 004 439 146 496 × 2 = 0 + 0.000 000 000 000 008 878 292 992;
  • 21) 0.000 000 000 000 008 878 292 992 × 2 = 0 + 0.000 000 000 000 017 756 585 984;
  • 22) 0.000 000 000 000 017 756 585 984 × 2 = 0 + 0.000 000 000 000 035 513 171 968;
  • 23) 0.000 000 000 000 035 513 171 968 × 2 = 0 + 0.000 000 000 000 071 026 343 936;
  • 24) 0.000 000 000 000 071 026 343 936 × 2 = 0 + 0.000 000 000 000 142 052 687 872;
  • 25) 0.000 000 000 000 142 052 687 872 × 2 = 0 + 0.000 000 000 000 284 105 375 744;
  • 26) 0.000 000 000 000 284 105 375 744 × 2 = 0 + 0.000 000 000 000 568 210 751 488;
  • 27) 0.000 000 000 000 568 210 751 488 × 2 = 0 + 0.000 000 000 001 136 421 502 976;
  • 28) 0.000 000 000 001 136 421 502 976 × 2 = 0 + 0.000 000 000 002 272 843 005 952;
  • 29) 0.000 000 000 002 272 843 005 952 × 2 = 0 + 0.000 000 000 004 545 686 011 904;
  • 30) 0.000 000 000 004 545 686 011 904 × 2 = 0 + 0.000 000 000 009 091 372 023 808;
  • 31) 0.000 000 000 009 091 372 023 808 × 2 = 0 + 0.000 000 000 018 182 744 047 616;
  • 32) 0.000 000 000 018 182 744 047 616 × 2 = 0 + 0.000 000 000 036 365 488 095 232;
  • 33) 0.000 000 000 036 365 488 095 232 × 2 = 0 + 0.000 000 000 072 730 976 190 464;
  • 34) 0.000 000 000 072 730 976 190 464 × 2 = 0 + 0.000 000 000 145 461 952 380 928;
  • 35) 0.000 000 000 145 461 952 380 928 × 2 = 0 + 0.000 000 000 290 923 904 761 856;
  • 36) 0.000 000 000 290 923 904 761 856 × 2 = 0 + 0.000 000 000 581 847 809 523 712;
  • 37) 0.000 000 000 581 847 809 523 712 × 2 = 0 + 0.000 000 001 163 695 619 047 424;
  • 38) 0.000 000 001 163 695 619 047 424 × 2 = 0 + 0.000 000 002 327 391 238 094 848;
  • 39) 0.000 000 002 327 391 238 094 848 × 2 = 0 + 0.000 000 004 654 782 476 189 696;
  • 40) 0.000 000 004 654 782 476 189 696 × 2 = 0 + 0.000 000 009 309 564 952 379 392;
  • 41) 0.000 000 009 309 564 952 379 392 × 2 = 0 + 0.000 000 018 619 129 904 758 784;
  • 42) 0.000 000 018 619 129 904 758 784 × 2 = 0 + 0.000 000 037 238 259 809 517 568;
  • 43) 0.000 000 037 238 259 809 517 568 × 2 = 0 + 0.000 000 074 476 519 619 035 136;
  • 44) 0.000 000 074 476 519 619 035 136 × 2 = 0 + 0.000 000 148 953 039 238 070 272;
  • 45) 0.000 000 148 953 039 238 070 272 × 2 = 0 + 0.000 000 297 906 078 476 140 544;
  • 46) 0.000 000 297 906 078 476 140 544 × 2 = 0 + 0.000 000 595 812 156 952 281 088;
  • 47) 0.000 000 595 812 156 952 281 088 × 2 = 0 + 0.000 001 191 624 313 904 562 176;
  • 48) 0.000 001 191 624 313 904 562 176 × 2 = 0 + 0.000 002 383 248 627 809 124 352;
  • 49) 0.000 002 383 248 627 809 124 352 × 2 = 0 + 0.000 004 766 497 255 618 248 704;
  • 50) 0.000 004 766 497 255 618 248 704 × 2 = 0 + 0.000 009 532 994 511 236 497 408;
  • 51) 0.000 009 532 994 511 236 497 408 × 2 = 0 + 0.000 019 065 989 022 472 994 816;
  • 52) 0.000 019 065 989 022 472 994 816 × 2 = 0 + 0.000 038 131 978 044 945 989 632;
  • 53) 0.000 038 131 978 044 945 989 632 × 2 = 0 + 0.000 076 263 956 089 891 979 264;
  • 54) 0.000 076 263 956 089 891 979 264 × 2 = 0 + 0.000 152 527 912 179 783 958 528;
  • 55) 0.000 152 527 912 179 783 958 528 × 2 = 0 + 0.000 305 055 824 359 567 917 056;
  • 56) 0.000 305 055 824 359 567 917 056 × 2 = 0 + 0.000 610 111 648 719 135 834 112;
  • 57) 0.000 610 111 648 719 135 834 112 × 2 = 0 + 0.001 220 223 297 438 271 668 224;
  • 58) 0.001 220 223 297 438 271 668 224 × 2 = 0 + 0.002 440 446 594 876 543 336 448;
  • 59) 0.002 440 446 594 876 543 336 448 × 2 = 0 + 0.004 880 893 189 753 086 672 896;
  • 60) 0.004 880 893 189 753 086 672 896 × 2 = 0 + 0.009 761 786 379 506 173 345 792;
  • 61) 0.009 761 786 379 506 173 345 792 × 2 = 0 + 0.019 523 572 759 012 346 691 584;
  • 62) 0.019 523 572 759 012 346 691 584 × 2 = 0 + 0.039 047 145 518 024 693 383 168;
  • 63) 0.039 047 145 518 024 693 383 168 × 2 = 0 + 0.078 094 291 036 049 386 766 336;
  • 64) 0.078 094 291 036 049 386 766 336 × 2 = 0 + 0.156 188 582 072 098 773 532 672;
  • 65) 0.156 188 582 072 098 773 532 672 × 2 = 0 + 0.312 377 164 144 197 547 065 344;
  • 66) 0.312 377 164 144 197 547 065 344 × 2 = 0 + 0.624 754 328 288 395 094 130 688;
  • 67) 0.624 754 328 288 395 094 130 688 × 2 = 1 + 0.249 508 656 576 790 188 261 376;
  • 68) 0.249 508 656 576 790 188 261 376 × 2 = 0 + 0.499 017 313 153 580 376 522 752;
  • 69) 0.499 017 313 153 580 376 522 752 × 2 = 0 + 0.998 034 626 307 160 753 045 504;
  • 70) 0.998 034 626 307 160 753 045 504 × 2 = 1 + 0.996 069 252 614 321 506 091 008;
  • 71) 0.996 069 252 614 321 506 091 008 × 2 = 1 + 0.992 138 505 228 643 012 182 016;
  • 72) 0.992 138 505 228 643 012 182 016 × 2 = 1 + 0.984 277 010 457 286 024 364 032;
  • 73) 0.984 277 010 457 286 024 364 032 × 2 = 1 + 0.968 554 020 914 572 048 728 064;
  • 74) 0.968 554 020 914 572 048 728 064 × 2 = 1 + 0.937 108 041 829 144 097 456 128;
  • 75) 0.937 108 041 829 144 097 456 128 × 2 = 1 + 0.874 216 083 658 288 194 912 256;
  • 76) 0.874 216 083 658 288 194 912 256 × 2 = 1 + 0.748 432 167 316 576 389 824 512;
  • 77) 0.748 432 167 316 576 389 824 512 × 2 = 1 + 0.496 864 334 633 152 779 649 024;
  • 78) 0.496 864 334 633 152 779 649 024 × 2 = 0 + 0.993 728 669 266 305 559 298 048;
  • 79) 0.993 728 669 266 305 559 298 048 × 2 = 1 + 0.987 457 338 532 611 118 596 096;
  • 80) 0.987 457 338 532 611 118 596 096 × 2 = 1 + 0.974 914 677 065 222 237 192 192;
  • 81) 0.974 914 677 065 222 237 192 192 × 2 = 1 + 0.949 829 354 130 444 474 384 384;
  • 82) 0.949 829 354 130 444 474 384 384 × 2 = 1 + 0.899 658 708 260 888 948 768 768;
  • 83) 0.899 658 708 260 888 948 768 768 × 2 = 1 + 0.799 317 416 521 777 897 537 536;
  • 84) 0.799 317 416 521 777 897 537 536 × 2 = 1 + 0.598 634 833 043 555 795 075 072;
  • 85) 0.598 634 833 043 555 795 075 072 × 2 = 1 + 0.197 269 666 087 111 590 150 144;
  • 86) 0.197 269 666 087 111 590 150 144 × 2 = 0 + 0.394 539 332 174 223 180 300 288;
  • 87) 0.394 539 332 174 223 180 300 288 × 2 = 0 + 0.789 078 664 348 446 360 600 576;
  • 88) 0.789 078 664 348 446 360 600 576 × 2 = 1 + 0.578 157 328 696 892 721 201 152;
  • 89) 0.578 157 328 696 892 721 201 152 × 2 = 1 + 0.156 314 657 393 785 442 402 304;
  • 90) 0.156 314 657 393 785 442 402 304 × 2 = 0 + 0.312 629 314 787 570 884 804 608;
  • 91) 0.312 629 314 787 570 884 804 608 × 2 = 0 + 0.625 258 629 575 141 769 609 216;
  • 92) 0.625 258 629 575 141 769 609 216 × 2 = 1 + 0.250 517 259 150 283 539 218 432;
  • 93) 0.250 517 259 150 283 539 218 432 × 2 = 0 + 0.501 034 518 300 567 078 436 864;
  • 94) 0.501 034 518 300 567 078 436 864 × 2 = 1 + 0.002 069 036 601 134 156 873 728;
  • 95) 0.002 069 036 601 134 156 873 728 × 2 = 0 + 0.004 138 073 202 268 313 747 456;
  • 96) 0.004 138 073 202 268 313 747 456 × 2 = 0 + 0.008 276 146 404 536 627 494 912;
  • 97) 0.008 276 146 404 536 627 494 912 × 2 = 0 + 0.016 552 292 809 073 254 989 824;
  • 98) 0.016 552 292 809 073 254 989 824 × 2 = 0 + 0.033 104 585 618 146 509 979 648;
  • 99) 0.033 104 585 618 146 509 979 648 × 2 = 0 + 0.066 209 171 236 293 019 959 296;
  • 100) 0.066 209 171 236 293 019 959 296 × 2 = 0 + 0.132 418 342 472 586 039 918 592;
  • 101) 0.132 418 342 472 586 039 918 592 × 2 = 0 + 0.264 836 684 945 172 079 837 184;
  • 102) 0.264 836 684 945 172 079 837 184 × 2 = 0 + 0.529 673 369 890 344 159 674 368;
  • 103) 0.529 673 369 890 344 159 674 368 × 2 = 1 + 0.059 346 739 780 688 319 348 736;
  • 104) 0.059 346 739 780 688 319 348 736 × 2 = 0 + 0.118 693 479 561 376 638 697 472;
  • 105) 0.118 693 479 561 376 638 697 472 × 2 = 0 + 0.237 386 959 122 753 277 394 944;
  • 106) 0.237 386 959 122 753 277 394 944 × 2 = 0 + 0.474 773 918 245 506 554 789 888;
  • 107) 0.474 773 918 245 506 554 789 888 × 2 = 0 + 0.949 547 836 491 013 109 579 776;
  • 108) 0.949 547 836 491 013 109 579 776 × 2 = 1 + 0.899 095 672 982 026 219 159 552;
  • 109) 0.899 095 672 982 026 219 159 552 × 2 = 1 + 0.798 191 345 964 052 438 319 104;
  • 110) 0.798 191 345 964 052 438 319 104 × 2 = 1 + 0.596 382 691 928 104 876 638 208;
  • 111) 0.596 382 691 928 104 876 638 208 × 2 = 1 + 0.192 765 383 856 209 753 276 416;
  • 112) 0.192 765 383 856 209 753 276 416 × 2 = 0 + 0.385 530 767 712 419 506 552 832;
  • 113) 0.385 530 767 712 419 506 552 832 × 2 = 0 + 0.771 061 535 424 839 013 105 664;
  • 114) 0.771 061 535 424 839 013 105 664 × 2 = 1 + 0.542 123 070 849 678 026 211 328;
  • 115) 0.542 123 070 849 678 026 211 328 × 2 = 1 + 0.084 246 141 699 356 052 422 656;
  • 116) 0.084 246 141 699 356 052 422 656 × 2 = 0 + 0.168 492 283 398 712 104 845 312;
  • 117) 0.168 492 283 398 712 104 845 312 × 2 = 0 + 0.336 984 566 797 424 209 690 624;
  • 118) 0.336 984 566 797 424 209 690 624 × 2 = 0 + 0.673 969 133 594 848 419 381 248;
  • 119) 0.673 969 133 594 848 419 381 248 × 2 = 1 + 0.347 938 267 189 696 838 762 496;

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:


0.000 000 000 000 000 000 008 467(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 0111 1111 1011 1111 1001 1001 0100 0000 0010 0001 1110 0110 001(2)

5. Positive number before normalization:

0.000 000 000 000 000 000 008 467(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 0111 1111 1011 1111 1001 1001 0100 0000 0010 0001 1110 0110 001(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:


0.000 000 000 000 000 000 008 467(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 0111 1111 1011 1111 1001 1001 0100 0000 0010 0001 1110 0110 001(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 0111 1111 1011 1111 1001 1001 0100 0000 0010 0001 1110 0110 001(2) × 20 =

1.0011 1111 1101 1111 1100 1100 1010 0000 0001 0000 1111 0011 0001(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):
1.0011 1111 1101 1111 1100 1100 1010 0000 0001 0000 1111 0011 0001


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. 0011 1111 1101 1111 1100 1100 1010 0000 0001 0000 1111 0011 0001 =

0011 1111 1101 1111 1100 1100 1010 0000 0001 0000 1111 0011 0001


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) =
0011 1111 1101 1111 1100 1100 1010 0000 0001 0000 1111 0011 0001


Decimal number 0.000 000 000 000 000 000 008 467 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1011 1100 - 0011 1111 1101 1111 1100 1100 1010 0000 0001 0000 1111 0011 0001

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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