0.000 000 000 000 000 000 008 534 3 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 000 000 008 534 3(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
0.000 000 000 000 000 000 008 534 3(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: 0.000 000 000 000 000 000 008 534 3.

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) 0.000 000 000 000 000 000 008 534 3 × 2 = 0 + 0.000 000 000 000 000 000 017 068 6;
  • 2) 0.000 000 000 000 000 000 017 068 6 × 2 = 0 + 0.000 000 000 000 000 000 034 137 2;
  • 3) 0.000 000 000 000 000 000 034 137 2 × 2 = 0 + 0.000 000 000 000 000 000 068 274 4;
  • 4) 0.000 000 000 000 000 000 068 274 4 × 2 = 0 + 0.000 000 000 000 000 000 136 548 8;
  • 5) 0.000 000 000 000 000 000 136 548 8 × 2 = 0 + 0.000 000 000 000 000 000 273 097 6;
  • 6) 0.000 000 000 000 000 000 273 097 6 × 2 = 0 + 0.000 000 000 000 000 000 546 195 2;
  • 7) 0.000 000 000 000 000 000 546 195 2 × 2 = 0 + 0.000 000 000 000 000 001 092 390 4;
  • 8) 0.000 000 000 000 000 001 092 390 4 × 2 = 0 + 0.000 000 000 000 000 002 184 780 8;
  • 9) 0.000 000 000 000 000 002 184 780 8 × 2 = 0 + 0.000 000 000 000 000 004 369 561 6;
  • 10) 0.000 000 000 000 000 004 369 561 6 × 2 = 0 + 0.000 000 000 000 000 008 739 123 2;
  • 11) 0.000 000 000 000 000 008 739 123 2 × 2 = 0 + 0.000 000 000 000 000 017 478 246 4;
  • 12) 0.000 000 000 000 000 017 478 246 4 × 2 = 0 + 0.000 000 000 000 000 034 956 492 8;
  • 13) 0.000 000 000 000 000 034 956 492 8 × 2 = 0 + 0.000 000 000 000 000 069 912 985 6;
  • 14) 0.000 000 000 000 000 069 912 985 6 × 2 = 0 + 0.000 000 000 000 000 139 825 971 2;
  • 15) 0.000 000 000 000 000 139 825 971 2 × 2 = 0 + 0.000 000 000 000 000 279 651 942 4;
  • 16) 0.000 000 000 000 000 279 651 942 4 × 2 = 0 + 0.000 000 000 000 000 559 303 884 8;
  • 17) 0.000 000 000 000 000 559 303 884 8 × 2 = 0 + 0.000 000 000 000 001 118 607 769 6;
  • 18) 0.000 000 000 000 001 118 607 769 6 × 2 = 0 + 0.000 000 000 000 002 237 215 539 2;
  • 19) 0.000 000 000 000 002 237 215 539 2 × 2 = 0 + 0.000 000 000 000 004 474 431 078 4;
  • 20) 0.000 000 000 000 004 474 431 078 4 × 2 = 0 + 0.000 000 000 000 008 948 862 156 8;
  • 21) 0.000 000 000 000 008 948 862 156 8 × 2 = 0 + 0.000 000 000 000 017 897 724 313 6;
  • 22) 0.000 000 000 000 017 897 724 313 6 × 2 = 0 + 0.000 000 000 000 035 795 448 627 2;
  • 23) 0.000 000 000 000 035 795 448 627 2 × 2 = 0 + 0.000 000 000 000 071 590 897 254 4;
  • 24) 0.000 000 000 000 071 590 897 254 4 × 2 = 0 + 0.000 000 000 000 143 181 794 508 8;
  • 25) 0.000 000 000 000 143 181 794 508 8 × 2 = 0 + 0.000 000 000 000 286 363 589 017 6;
  • 26) 0.000 000 000 000 286 363 589 017 6 × 2 = 0 + 0.000 000 000 000 572 727 178 035 2;
  • 27) 0.000 000 000 000 572 727 178 035 2 × 2 = 0 + 0.000 000 000 001 145 454 356 070 4;
  • 28) 0.000 000 000 001 145 454 356 070 4 × 2 = 0 + 0.000 000 000 002 290 908 712 140 8;
  • 29) 0.000 000 000 002 290 908 712 140 8 × 2 = 0 + 0.000 000 000 004 581 817 424 281 6;
  • 30) 0.000 000 000 004 581 817 424 281 6 × 2 = 0 + 0.000 000 000 009 163 634 848 563 2;
  • 31) 0.000 000 000 009 163 634 848 563 2 × 2 = 0 + 0.000 000 000 018 327 269 697 126 4;
  • 32) 0.000 000 000 018 327 269 697 126 4 × 2 = 0 + 0.000 000 000 036 654 539 394 252 8;
  • 33) 0.000 000 000 036 654 539 394 252 8 × 2 = 0 + 0.000 000 000 073 309 078 788 505 6;
  • 34) 0.000 000 000 073 309 078 788 505 6 × 2 = 0 + 0.000 000 000 146 618 157 577 011 2;
  • 35) 0.000 000 000 146 618 157 577 011 2 × 2 = 0 + 0.000 000 000 293 236 315 154 022 4;
  • 36) 0.000 000 000 293 236 315 154 022 4 × 2 = 0 + 0.000 000 000 586 472 630 308 044 8;
  • 37) 0.000 000 000 586 472 630 308 044 8 × 2 = 0 + 0.000 000 001 172 945 260 616 089 6;
  • 38) 0.000 000 001 172 945 260 616 089 6 × 2 = 0 + 0.000 000 002 345 890 521 232 179 2;
  • 39) 0.000 000 002 345 890 521 232 179 2 × 2 = 0 + 0.000 000 004 691 781 042 464 358 4;
  • 40) 0.000 000 004 691 781 042 464 358 4 × 2 = 0 + 0.000 000 009 383 562 084 928 716 8;
  • 41) 0.000 000 009 383 562 084 928 716 8 × 2 = 0 + 0.000 000 018 767 124 169 857 433 6;
  • 42) 0.000 000 018 767 124 169 857 433 6 × 2 = 0 + 0.000 000 037 534 248 339 714 867 2;
  • 43) 0.000 000 037 534 248 339 714 867 2 × 2 = 0 + 0.000 000 075 068 496 679 429 734 4;
  • 44) 0.000 000 075 068 496 679 429 734 4 × 2 = 0 + 0.000 000 150 136 993 358 859 468 8;
  • 45) 0.000 000 150 136 993 358 859 468 8 × 2 = 0 + 0.000 000 300 273 986 717 718 937 6;
  • 46) 0.000 000 300 273 986 717 718 937 6 × 2 = 0 + 0.000 000 600 547 973 435 437 875 2;
  • 47) 0.000 000 600 547 973 435 437 875 2 × 2 = 0 + 0.000 001 201 095 946 870 875 750 4;
  • 48) 0.000 001 201 095 946 870 875 750 4 × 2 = 0 + 0.000 002 402 191 893 741 751 500 8;
  • 49) 0.000 002 402 191 893 741 751 500 8 × 2 = 0 + 0.000 004 804 383 787 483 503 001 6;
  • 50) 0.000 004 804 383 787 483 503 001 6 × 2 = 0 + 0.000 009 608 767 574 967 006 003 2;
  • 51) 0.000 009 608 767 574 967 006 003 2 × 2 = 0 + 0.000 019 217 535 149 934 012 006 4;
  • 52) 0.000 019 217 535 149 934 012 006 4 × 2 = 0 + 0.000 038 435 070 299 868 024 012 8;
  • 53) 0.000 038 435 070 299 868 024 012 8 × 2 = 0 + 0.000 076 870 140 599 736 048 025 6;
  • 54) 0.000 076 870 140 599 736 048 025 6 × 2 = 0 + 0.000 153 740 281 199 472 096 051 2;
  • 55) 0.000 153 740 281 199 472 096 051 2 × 2 = 0 + 0.000 307 480 562 398 944 192 102 4;
  • 56) 0.000 307 480 562 398 944 192 102 4 × 2 = 0 + 0.000 614 961 124 797 888 384 204 8;
  • 57) 0.000 614 961 124 797 888 384 204 8 × 2 = 0 + 0.001 229 922 249 595 776 768 409 6;
  • 58) 0.001 229 922 249 595 776 768 409 6 × 2 = 0 + 0.002 459 844 499 191 553 536 819 2;
  • 59) 0.002 459 844 499 191 553 536 819 2 × 2 = 0 + 0.004 919 688 998 383 107 073 638 4;
  • 60) 0.004 919 688 998 383 107 073 638 4 × 2 = 0 + 0.009 839 377 996 766 214 147 276 8;
  • 61) 0.009 839 377 996 766 214 147 276 8 × 2 = 0 + 0.019 678 755 993 532 428 294 553 6;
  • 62) 0.019 678 755 993 532 428 294 553 6 × 2 = 0 + 0.039 357 511 987 064 856 589 107 2;
  • 63) 0.039 357 511 987 064 856 589 107 2 × 2 = 0 + 0.078 715 023 974 129 713 178 214 4;
  • 64) 0.078 715 023 974 129 713 178 214 4 × 2 = 0 + 0.157 430 047 948 259 426 356 428 8;
  • 65) 0.157 430 047 948 259 426 356 428 8 × 2 = 0 + 0.314 860 095 896 518 852 712 857 6;
  • 66) 0.314 860 095 896 518 852 712 857 6 × 2 = 0 + 0.629 720 191 793 037 705 425 715 2;
  • 67) 0.629 720 191 793 037 705 425 715 2 × 2 = 1 + 0.259 440 383 586 075 410 851 430 4;
  • 68) 0.259 440 383 586 075 410 851 430 4 × 2 = 0 + 0.518 880 767 172 150 821 702 860 8;
  • 69) 0.518 880 767 172 150 821 702 860 8 × 2 = 1 + 0.037 761 534 344 301 643 405 721 6;
  • 70) 0.037 761 534 344 301 643 405 721 6 × 2 = 0 + 0.075 523 068 688 603 286 811 443 2;
  • 71) 0.075 523 068 688 603 286 811 443 2 × 2 = 0 + 0.151 046 137 377 206 573 622 886 4;
  • 72) 0.151 046 137 377 206 573 622 886 4 × 2 = 0 + 0.302 092 274 754 413 147 245 772 8;
  • 73) 0.302 092 274 754 413 147 245 772 8 × 2 = 0 + 0.604 184 549 508 826 294 491 545 6;
  • 74) 0.604 184 549 508 826 294 491 545 6 × 2 = 1 + 0.208 369 099 017 652 588 983 091 2;
  • 75) 0.208 369 099 017 652 588 983 091 2 × 2 = 0 + 0.416 738 198 035 305 177 966 182 4;
  • 76) 0.416 738 198 035 305 177 966 182 4 × 2 = 0 + 0.833 476 396 070 610 355 932 364 8;
  • 77) 0.833 476 396 070 610 355 932 364 8 × 2 = 1 + 0.666 952 792 141 220 711 864 729 6;
  • 78) 0.666 952 792 141 220 711 864 729 6 × 2 = 1 + 0.333 905 584 282 441 423 729 459 2;
  • 79) 0.333 905 584 282 441 423 729 459 2 × 2 = 0 + 0.667 811 168 564 882 847 458 918 4;
  • 80) 0.667 811 168 564 882 847 458 918 4 × 2 = 1 + 0.335 622 337 129 765 694 917 836 8;
  • 81) 0.335 622 337 129 765 694 917 836 8 × 2 = 0 + 0.671 244 674 259 531 389 835 673 6;
  • 82) 0.671 244 674 259 531 389 835 673 6 × 2 = 1 + 0.342 489 348 519 062 779 671 347 2;
  • 83) 0.342 489 348 519 062 779 671 347 2 × 2 = 0 + 0.684 978 697 038 125 559 342 694 4;
  • 84) 0.684 978 697 038 125 559 342 694 4 × 2 = 1 + 0.369 957 394 076 251 118 685 388 8;
  • 85) 0.369 957 394 076 251 118 685 388 8 × 2 = 0 + 0.739 914 788 152 502 237 370 777 6;
  • 86) 0.739 914 788 152 502 237 370 777 6 × 2 = 1 + 0.479 829 576 305 004 474 741 555 2;
  • 87) 0.479 829 576 305 004 474 741 555 2 × 2 = 0 + 0.959 659 152 610 008 949 483 110 4;
  • 88) 0.959 659 152 610 008 949 483 110 4 × 2 = 1 + 0.919 318 305 220 017 898 966 220 8;
  • 89) 0.919 318 305 220 017 898 966 220 8 × 2 = 1 + 0.838 636 610 440 035 797 932 441 6;
  • 90) 0.838 636 610 440 035 797 932 441 6 × 2 = 1 + 0.677 273 220 880 071 595 864 883 2;
  • 91) 0.677 273 220 880 071 595 864 883 2 × 2 = 1 + 0.354 546 441 760 143 191 729 766 4;
  • 92) 0.354 546 441 760 143 191 729 766 4 × 2 = 0 + 0.709 092 883 520 286 383 459 532 8;
  • 93) 0.709 092 883 520 286 383 459 532 8 × 2 = 1 + 0.418 185 767 040 572 766 919 065 6;
  • 94) 0.418 185 767 040 572 766 919 065 6 × 2 = 0 + 0.836 371 534 081 145 533 838 131 2;
  • 95) 0.836 371 534 081 145 533 838 131 2 × 2 = 1 + 0.672 743 068 162 291 067 676 262 4;
  • 96) 0.672 743 068 162 291 067 676 262 4 × 2 = 1 + 0.345 486 136 324 582 135 352 524 8;
  • 97) 0.345 486 136 324 582 135 352 524 8 × 2 = 0 + 0.690 972 272 649 164 270 705 049 6;
  • 98) 0.690 972 272 649 164 270 705 049 6 × 2 = 1 + 0.381 944 545 298 328 541 410 099 2;
  • 99) 0.381 944 545 298 328 541 410 099 2 × 2 = 0 + 0.763 889 090 596 657 082 820 198 4;
  • 100) 0.763 889 090 596 657 082 820 198 4 × 2 = 1 + 0.527 778 181 193 314 165 640 396 8;
  • 101) 0.527 778 181 193 314 165 640 396 8 × 2 = 1 + 0.055 556 362 386 628 331 280 793 6;
  • 102) 0.055 556 362 386 628 331 280 793 6 × 2 = 0 + 0.111 112 724 773 256 662 561 587 2;
  • 103) 0.111 112 724 773 256 662 561 587 2 × 2 = 0 + 0.222 225 449 546 513 325 123 174 4;
  • 104) 0.222 225 449 546 513 325 123 174 4 × 2 = 0 + 0.444 450 899 093 026 650 246 348 8;
  • 105) 0.444 450 899 093 026 650 246 348 8 × 2 = 0 + 0.888 901 798 186 053 300 492 697 6;
  • 106) 0.888 901 798 186 053 300 492 697 6 × 2 = 1 + 0.777 803 596 372 106 600 985 395 2;
  • 107) 0.777 803 596 372 106 600 985 395 2 × 2 = 1 + 0.555 607 192 744 213 201 970 790 4;
  • 108) 0.555 607 192 744 213 201 970 790 4 × 2 = 1 + 0.111 214 385 488 426 403 941 580 8;
  • 109) 0.111 214 385 488 426 403 941 580 8 × 2 = 0 + 0.222 428 770 976 852 807 883 161 6;
  • 110) 0.222 428 770 976 852 807 883 161 6 × 2 = 0 + 0.444 857 541 953 705 615 766 323 2;
  • 111) 0.444 857 541 953 705 615 766 323 2 × 2 = 0 + 0.889 715 083 907 411 231 532 646 4;
  • 112) 0.889 715 083 907 411 231 532 646 4 × 2 = 1 + 0.779 430 167 814 822 463 065 292 8;
  • 113) 0.779 430 167 814 822 463 065 292 8 × 2 = 1 + 0.558 860 335 629 644 926 130 585 6;
  • 114) 0.558 860 335 629 644 926 130 585 6 × 2 = 1 + 0.117 720 671 259 289 852 261 171 2;
  • 115) 0.117 720 671 259 289 852 261 171 2 × 2 = 0 + 0.235 441 342 518 579 704 522 342 4;
  • 116) 0.235 441 342 518 579 704 522 342 4 × 2 = 0 + 0.470 882 685 037 159 409 044 684 8;
  • 117) 0.470 882 685 037 159 409 044 684 8 × 2 = 0 + 0.941 765 370 074 318 818 089 369 6;
  • 118) 0.941 765 370 074 318 818 089 369 6 × 2 = 1 + 0.883 530 740 148 637 636 178 739 2;
  • 119) 0.883 530 740 148 637 636 178 739 2 × 2 = 1 + 0.767 061 480 297 275 272 357 478 4;

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:


0.000 000 000 000 000 000 008 534 3(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1101 0101 0101 1110 1011 0101 1000 0111 0001 1100 011(2)

5. Positive number before normalization:

0.000 000 000 000 000 000 008 534 3(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1101 0101 0101 1110 1011 0101 1000 0111 0001 1100 011(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:


0.000 000 000 000 000 000 008 534 3(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1101 0101 0101 1110 1011 0101 1000 0111 0001 1100 011(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1101 0101 0101 1110 1011 0101 1000 0111 0001 1100 011(2) × 20 =

1.0100 0010 0110 1010 1010 1111 0101 1010 1100 0011 1000 1110 0011(2) × 2-67


7. Up to this moment, there are the following elements that would feed into the 64 bit double precision IEEE 754 binary floating point representation:

Sign 0 (a positive number)

Exponent (unadjusted): -67

Mantissa (not normalized):
1.0100 0010 0110 1010 1010 1111 0101 1010 1100 0011 1000 1110 0011


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

Exponent (unadjusted) + 2(11-1) - 1 =

-67 + 2(11-1) - 1 =

(-67 + 1 023)(10) =

956(10)


9. Convert the adjusted exponent from the decimal (base 10) to 11 bit binary.

Use the same technique of repeatedly dividing by 2:


  • division = quotient + remainder;
  • 956 ÷ 2 = 478 + 0;
  • 478 ÷ 2 = 239 + 0;
  • 239 ÷ 2 = 119 + 1;
  • 119 ÷ 2 = 59 + 1;
  • 59 ÷ 2 = 29 + 1;
  • 29 ÷ 2 = 14 + 1;
  • 14 ÷ 2 = 7 + 0;
  • 7 ÷ 2 = 3 + 1;
  • 3 ÷ 2 = 1 + 1;
  • 1 ÷ 2 = 0 + 1;

10. Construct the base 2 representation of the adjusted exponent.

Take all the remainders starting from the bottom of the list constructed above.


Exponent (adjusted) =

956(10) =

011 1011 1100(2)


11. Normalize the mantissa.

a) Remove the leading (the leftmost) bit, since it's allways 1, and the decimal point, if the case.

b) Adjust its length to 52 bits, only if necessary (not the case here).


Mantissa (normalized) =

1. 0100 0010 0110 1010 1010 1111 0101 1010 1100 0011 1000 1110 0011 =

0100 0010 0110 1010 1010 1111 0101 1010 1100 0011 1000 1110 0011


12. The three elements that make up the number's 64 bit double precision IEEE 754 binary floating point representation:

Sign (1 bit) =
0 (a positive number)

Exponent (11 bits) =
011 1011 1100

Mantissa (52 bits) =
0100 0010 0110 1010 1010 1111 0101 1010 1100 0011 1000 1110 0011


Decimal number 0.000 000 000 000 000 000 008 534 3 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1011 1100 - 0100 0010 0110 1010 1010 1111 0101 1010 1100 0011 1000 1110 0011

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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