0.000 000 000 000 000 000 008 568 2 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 000 000 008 568 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
0.000 000 000 000 000 000 008 568 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: 0.000 000 000 000 000 000 008 568 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) 0.000 000 000 000 000 000 008 568 2 × 2 = 0 + 0.000 000 000 000 000 000 017 136 4;
  • 2) 0.000 000 000 000 000 000 017 136 4 × 2 = 0 + 0.000 000 000 000 000 000 034 272 8;
  • 3) 0.000 000 000 000 000 000 034 272 8 × 2 = 0 + 0.000 000 000 000 000 000 068 545 6;
  • 4) 0.000 000 000 000 000 000 068 545 6 × 2 = 0 + 0.000 000 000 000 000 000 137 091 2;
  • 5) 0.000 000 000 000 000 000 137 091 2 × 2 = 0 + 0.000 000 000 000 000 000 274 182 4;
  • 6) 0.000 000 000 000 000 000 274 182 4 × 2 = 0 + 0.000 000 000 000 000 000 548 364 8;
  • 7) 0.000 000 000 000 000 000 548 364 8 × 2 = 0 + 0.000 000 000 000 000 001 096 729 6;
  • 8) 0.000 000 000 000 000 001 096 729 6 × 2 = 0 + 0.000 000 000 000 000 002 193 459 2;
  • 9) 0.000 000 000 000 000 002 193 459 2 × 2 = 0 + 0.000 000 000 000 000 004 386 918 4;
  • 10) 0.000 000 000 000 000 004 386 918 4 × 2 = 0 + 0.000 000 000 000 000 008 773 836 8;
  • 11) 0.000 000 000 000 000 008 773 836 8 × 2 = 0 + 0.000 000 000 000 000 017 547 673 6;
  • 12) 0.000 000 000 000 000 017 547 673 6 × 2 = 0 + 0.000 000 000 000 000 035 095 347 2;
  • 13) 0.000 000 000 000 000 035 095 347 2 × 2 = 0 + 0.000 000 000 000 000 070 190 694 4;
  • 14) 0.000 000 000 000 000 070 190 694 4 × 2 = 0 + 0.000 000 000 000 000 140 381 388 8;
  • 15) 0.000 000 000 000 000 140 381 388 8 × 2 = 0 + 0.000 000 000 000 000 280 762 777 6;
  • 16) 0.000 000 000 000 000 280 762 777 6 × 2 = 0 + 0.000 000 000 000 000 561 525 555 2;
  • 17) 0.000 000 000 000 000 561 525 555 2 × 2 = 0 + 0.000 000 000 000 001 123 051 110 4;
  • 18) 0.000 000 000 000 001 123 051 110 4 × 2 = 0 + 0.000 000 000 000 002 246 102 220 8;
  • 19) 0.000 000 000 000 002 246 102 220 8 × 2 = 0 + 0.000 000 000 000 004 492 204 441 6;
  • 20) 0.000 000 000 000 004 492 204 441 6 × 2 = 0 + 0.000 000 000 000 008 984 408 883 2;
  • 21) 0.000 000 000 000 008 984 408 883 2 × 2 = 0 + 0.000 000 000 000 017 968 817 766 4;
  • 22) 0.000 000 000 000 017 968 817 766 4 × 2 = 0 + 0.000 000 000 000 035 937 635 532 8;
  • 23) 0.000 000 000 000 035 937 635 532 8 × 2 = 0 + 0.000 000 000 000 071 875 271 065 6;
  • 24) 0.000 000 000 000 071 875 271 065 6 × 2 = 0 + 0.000 000 000 000 143 750 542 131 2;
  • 25) 0.000 000 000 000 143 750 542 131 2 × 2 = 0 + 0.000 000 000 000 287 501 084 262 4;
  • 26) 0.000 000 000 000 287 501 084 262 4 × 2 = 0 + 0.000 000 000 000 575 002 168 524 8;
  • 27) 0.000 000 000 000 575 002 168 524 8 × 2 = 0 + 0.000 000 000 001 150 004 337 049 6;
  • 28) 0.000 000 000 001 150 004 337 049 6 × 2 = 0 + 0.000 000 000 002 300 008 674 099 2;
  • 29) 0.000 000 000 002 300 008 674 099 2 × 2 = 0 + 0.000 000 000 004 600 017 348 198 4;
  • 30) 0.000 000 000 004 600 017 348 198 4 × 2 = 0 + 0.000 000 000 009 200 034 696 396 8;
  • 31) 0.000 000 000 009 200 034 696 396 8 × 2 = 0 + 0.000 000 000 018 400 069 392 793 6;
  • 32) 0.000 000 000 018 400 069 392 793 6 × 2 = 0 + 0.000 000 000 036 800 138 785 587 2;
  • 33) 0.000 000 000 036 800 138 785 587 2 × 2 = 0 + 0.000 000 000 073 600 277 571 174 4;
  • 34) 0.000 000 000 073 600 277 571 174 4 × 2 = 0 + 0.000 000 000 147 200 555 142 348 8;
  • 35) 0.000 000 000 147 200 555 142 348 8 × 2 = 0 + 0.000 000 000 294 401 110 284 697 6;
  • 36) 0.000 000 000 294 401 110 284 697 6 × 2 = 0 + 0.000 000 000 588 802 220 569 395 2;
  • 37) 0.000 000 000 588 802 220 569 395 2 × 2 = 0 + 0.000 000 001 177 604 441 138 790 4;
  • 38) 0.000 000 001 177 604 441 138 790 4 × 2 = 0 + 0.000 000 002 355 208 882 277 580 8;
  • 39) 0.000 000 002 355 208 882 277 580 8 × 2 = 0 + 0.000 000 004 710 417 764 555 161 6;
  • 40) 0.000 000 004 710 417 764 555 161 6 × 2 = 0 + 0.000 000 009 420 835 529 110 323 2;
  • 41) 0.000 000 009 420 835 529 110 323 2 × 2 = 0 + 0.000 000 018 841 671 058 220 646 4;
  • 42) 0.000 000 018 841 671 058 220 646 4 × 2 = 0 + 0.000 000 037 683 342 116 441 292 8;
  • 43) 0.000 000 037 683 342 116 441 292 8 × 2 = 0 + 0.000 000 075 366 684 232 882 585 6;
  • 44) 0.000 000 075 366 684 232 882 585 6 × 2 = 0 + 0.000 000 150 733 368 465 765 171 2;
  • 45) 0.000 000 150 733 368 465 765 171 2 × 2 = 0 + 0.000 000 301 466 736 931 530 342 4;
  • 46) 0.000 000 301 466 736 931 530 342 4 × 2 = 0 + 0.000 000 602 933 473 863 060 684 8;
  • 47) 0.000 000 602 933 473 863 060 684 8 × 2 = 0 + 0.000 001 205 866 947 726 121 369 6;
  • 48) 0.000 001 205 866 947 726 121 369 6 × 2 = 0 + 0.000 002 411 733 895 452 242 739 2;
  • 49) 0.000 002 411 733 895 452 242 739 2 × 2 = 0 + 0.000 004 823 467 790 904 485 478 4;
  • 50) 0.000 004 823 467 790 904 485 478 4 × 2 = 0 + 0.000 009 646 935 581 808 970 956 8;
  • 51) 0.000 009 646 935 581 808 970 956 8 × 2 = 0 + 0.000 019 293 871 163 617 941 913 6;
  • 52) 0.000 019 293 871 163 617 941 913 6 × 2 = 0 + 0.000 038 587 742 327 235 883 827 2;
  • 53) 0.000 038 587 742 327 235 883 827 2 × 2 = 0 + 0.000 077 175 484 654 471 767 654 4;
  • 54) 0.000 077 175 484 654 471 767 654 4 × 2 = 0 + 0.000 154 350 969 308 943 535 308 8;
  • 55) 0.000 154 350 969 308 943 535 308 8 × 2 = 0 + 0.000 308 701 938 617 887 070 617 6;
  • 56) 0.000 308 701 938 617 887 070 617 6 × 2 = 0 + 0.000 617 403 877 235 774 141 235 2;
  • 57) 0.000 617 403 877 235 774 141 235 2 × 2 = 0 + 0.001 234 807 754 471 548 282 470 4;
  • 58) 0.001 234 807 754 471 548 282 470 4 × 2 = 0 + 0.002 469 615 508 943 096 564 940 8;
  • 59) 0.002 469 615 508 943 096 564 940 8 × 2 = 0 + 0.004 939 231 017 886 193 129 881 6;
  • 60) 0.004 939 231 017 886 193 129 881 6 × 2 = 0 + 0.009 878 462 035 772 386 259 763 2;
  • 61) 0.009 878 462 035 772 386 259 763 2 × 2 = 0 + 0.019 756 924 071 544 772 519 526 4;
  • 62) 0.019 756 924 071 544 772 519 526 4 × 2 = 0 + 0.039 513 848 143 089 545 039 052 8;
  • 63) 0.039 513 848 143 089 545 039 052 8 × 2 = 0 + 0.079 027 696 286 179 090 078 105 6;
  • 64) 0.079 027 696 286 179 090 078 105 6 × 2 = 0 + 0.158 055 392 572 358 180 156 211 2;
  • 65) 0.158 055 392 572 358 180 156 211 2 × 2 = 0 + 0.316 110 785 144 716 360 312 422 4;
  • 66) 0.316 110 785 144 716 360 312 422 4 × 2 = 0 + 0.632 221 570 289 432 720 624 844 8;
  • 67) 0.632 221 570 289 432 720 624 844 8 × 2 = 1 + 0.264 443 140 578 865 441 249 689 6;
  • 68) 0.264 443 140 578 865 441 249 689 6 × 2 = 0 + 0.528 886 281 157 730 882 499 379 2;
  • 69) 0.528 886 281 157 730 882 499 379 2 × 2 = 1 + 0.057 772 562 315 461 764 998 758 4;
  • 70) 0.057 772 562 315 461 764 998 758 4 × 2 = 0 + 0.115 545 124 630 923 529 997 516 8;
  • 71) 0.115 545 124 630 923 529 997 516 8 × 2 = 0 + 0.231 090 249 261 847 059 995 033 6;
  • 72) 0.231 090 249 261 847 059 995 033 6 × 2 = 0 + 0.462 180 498 523 694 119 990 067 2;
  • 73) 0.462 180 498 523 694 119 990 067 2 × 2 = 0 + 0.924 360 997 047 388 239 980 134 4;
  • 74) 0.924 360 997 047 388 239 980 134 4 × 2 = 1 + 0.848 721 994 094 776 479 960 268 8;
  • 75) 0.848 721 994 094 776 479 960 268 8 × 2 = 1 + 0.697 443 988 189 552 959 920 537 6;
  • 76) 0.697 443 988 189 552 959 920 537 6 × 2 = 1 + 0.394 887 976 379 105 919 841 075 2;
  • 77) 0.394 887 976 379 105 919 841 075 2 × 2 = 0 + 0.789 775 952 758 211 839 682 150 4;
  • 78) 0.789 775 952 758 211 839 682 150 4 × 2 = 1 + 0.579 551 905 516 423 679 364 300 8;
  • 79) 0.579 551 905 516 423 679 364 300 8 × 2 = 1 + 0.159 103 811 032 847 358 728 601 6;
  • 80) 0.159 103 811 032 847 358 728 601 6 × 2 = 0 + 0.318 207 622 065 694 717 457 203 2;
  • 81) 0.318 207 622 065 694 717 457 203 2 × 2 = 0 + 0.636 415 244 131 389 434 914 406 4;
  • 82) 0.636 415 244 131 389 434 914 406 4 × 2 = 1 + 0.272 830 488 262 778 869 828 812 8;
  • 83) 0.272 830 488 262 778 869 828 812 8 × 2 = 0 + 0.545 660 976 525 557 739 657 625 6;
  • 84) 0.545 660 976 525 557 739 657 625 6 × 2 = 1 + 0.091 321 953 051 115 479 315 251 2;
  • 85) 0.091 321 953 051 115 479 315 251 2 × 2 = 0 + 0.182 643 906 102 230 958 630 502 4;
  • 86) 0.182 643 906 102 230 958 630 502 4 × 2 = 0 + 0.365 287 812 204 461 917 261 004 8;
  • 87) 0.365 287 812 204 461 917 261 004 8 × 2 = 0 + 0.730 575 624 408 923 834 522 009 6;
  • 88) 0.730 575 624 408 923 834 522 009 6 × 2 = 1 + 0.461 151 248 817 847 669 044 019 2;
  • 89) 0.461 151 248 817 847 669 044 019 2 × 2 = 0 + 0.922 302 497 635 695 338 088 038 4;
  • 90) 0.922 302 497 635 695 338 088 038 4 × 2 = 1 + 0.844 604 995 271 390 676 176 076 8;
  • 91) 0.844 604 995 271 390 676 176 076 8 × 2 = 1 + 0.689 209 990 542 781 352 352 153 6;
  • 92) 0.689 209 990 542 781 352 352 153 6 × 2 = 1 + 0.378 419 981 085 562 704 704 307 2;
  • 93) 0.378 419 981 085 562 704 704 307 2 × 2 = 0 + 0.756 839 962 171 125 409 408 614 4;
  • 94) 0.756 839 962 171 125 409 408 614 4 × 2 = 1 + 0.513 679 924 342 250 818 817 228 8;
  • 95) 0.513 679 924 342 250 818 817 228 8 × 2 = 1 + 0.027 359 848 684 501 637 634 457 6;
  • 96) 0.027 359 848 684 501 637 634 457 6 × 2 = 0 + 0.054 719 697 369 003 275 268 915 2;
  • 97) 0.054 719 697 369 003 275 268 915 2 × 2 = 0 + 0.109 439 394 738 006 550 537 830 4;
  • 98) 0.109 439 394 738 006 550 537 830 4 × 2 = 0 + 0.218 878 789 476 013 101 075 660 8;
  • 99) 0.218 878 789 476 013 101 075 660 8 × 2 = 0 + 0.437 757 578 952 026 202 151 321 6;
  • 100) 0.437 757 578 952 026 202 151 321 6 × 2 = 0 + 0.875 515 157 904 052 404 302 643 2;
  • 101) 0.875 515 157 904 052 404 302 643 2 × 2 = 1 + 0.751 030 315 808 104 808 605 286 4;
  • 102) 0.751 030 315 808 104 808 605 286 4 × 2 = 1 + 0.502 060 631 616 209 617 210 572 8;
  • 103) 0.502 060 631 616 209 617 210 572 8 × 2 = 1 + 0.004 121 263 232 419 234 421 145 6;
  • 104) 0.004 121 263 232 419 234 421 145 6 × 2 = 0 + 0.008 242 526 464 838 468 842 291 2;
  • 105) 0.008 242 526 464 838 468 842 291 2 × 2 = 0 + 0.016 485 052 929 676 937 684 582 4;
  • 106) 0.016 485 052 929 676 937 684 582 4 × 2 = 0 + 0.032 970 105 859 353 875 369 164 8;
  • 107) 0.032 970 105 859 353 875 369 164 8 × 2 = 0 + 0.065 940 211 718 707 750 738 329 6;
  • 108) 0.065 940 211 718 707 750 738 329 6 × 2 = 0 + 0.131 880 423 437 415 501 476 659 2;
  • 109) 0.131 880 423 437 415 501 476 659 2 × 2 = 0 + 0.263 760 846 874 831 002 953 318 4;
  • 110) 0.263 760 846 874 831 002 953 318 4 × 2 = 0 + 0.527 521 693 749 662 005 906 636 8;
  • 111) 0.527 521 693 749 662 005 906 636 8 × 2 = 1 + 0.055 043 387 499 324 011 813 273 6;
  • 112) 0.055 043 387 499 324 011 813 273 6 × 2 = 0 + 0.110 086 774 998 648 023 626 547 2;
  • 113) 0.110 086 774 998 648 023 626 547 2 × 2 = 0 + 0.220 173 549 997 296 047 253 094 4;
  • 114) 0.220 173 549 997 296 047 253 094 4 × 2 = 0 + 0.440 347 099 994 592 094 506 188 8;
  • 115) 0.440 347 099 994 592 094 506 188 8 × 2 = 0 + 0.880 694 199 989 184 189 012 377 6;
  • 116) 0.880 694 199 989 184 189 012 377 6 × 2 = 1 + 0.761 388 399 978 368 378 024 755 2;
  • 117) 0.761 388 399 978 368 378 024 755 2 × 2 = 1 + 0.522 776 799 956 736 756 049 510 4;
  • 118) 0.522 776 799 956 736 756 049 510 4 × 2 = 1 + 0.045 553 599 913 473 512 099 020 8;
  • 119) 0.045 553 599 913 473 512 099 020 8 × 2 = 0 + 0.091 107 199 826 947 024 198 041 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:


0.000 000 000 000 000 000 008 568 2(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0111 0110 0101 0001 0111 0110 0000 1110 0000 0010 0001 110(2)

5. Positive number before normalization:

0.000 000 000 000 000 000 008 568 2(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0111 0110 0101 0001 0111 0110 0000 1110 0000 0010 0001 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:


0.000 000 000 000 000 000 008 568 2(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0111 0110 0101 0001 0111 0110 0000 1110 0000 0010 0001 110(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0111 0110 0101 0001 0111 0110 0000 1110 0000 0010 0001 110(2) × 20 =

1.0100 0011 1011 0010 1000 1011 1011 0000 0111 0000 0001 0000 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 0011 1011 0010 1000 1011 1011 0000 0111 0000 0001 0000 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 0011 1011 0010 1000 1011 1011 0000 0111 0000 0001 0000 1110 =

0100 0011 1011 0010 1000 1011 1011 0000 0111 0000 0001 0000 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 0011 1011 0010 1000 1011 1011 0000 0111 0000 0001 0000 1110


Decimal number 0.000 000 000 000 000 000 008 568 2 converted to 64 bit double precision IEEE 754 binary floating point representation:

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