0.000 000 000 000 000 000 008 531 44 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 000 000 008 531 44(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
0.000 000 000 000 000 000 008 531 44(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: 0.000 000 000 000 000 000 008 531 44.

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) 0.000 000 000 000 000 000 008 531 44 × 2 = 0 + 0.000 000 000 000 000 000 017 062 88;
  • 2) 0.000 000 000 000 000 000 017 062 88 × 2 = 0 + 0.000 000 000 000 000 000 034 125 76;
  • 3) 0.000 000 000 000 000 000 034 125 76 × 2 = 0 + 0.000 000 000 000 000 000 068 251 52;
  • 4) 0.000 000 000 000 000 000 068 251 52 × 2 = 0 + 0.000 000 000 000 000 000 136 503 04;
  • 5) 0.000 000 000 000 000 000 136 503 04 × 2 = 0 + 0.000 000 000 000 000 000 273 006 08;
  • 6) 0.000 000 000 000 000 000 273 006 08 × 2 = 0 + 0.000 000 000 000 000 000 546 012 16;
  • 7) 0.000 000 000 000 000 000 546 012 16 × 2 = 0 + 0.000 000 000 000 000 001 092 024 32;
  • 8) 0.000 000 000 000 000 001 092 024 32 × 2 = 0 + 0.000 000 000 000 000 002 184 048 64;
  • 9) 0.000 000 000 000 000 002 184 048 64 × 2 = 0 + 0.000 000 000 000 000 004 368 097 28;
  • 10) 0.000 000 000 000 000 004 368 097 28 × 2 = 0 + 0.000 000 000 000 000 008 736 194 56;
  • 11) 0.000 000 000 000 000 008 736 194 56 × 2 = 0 + 0.000 000 000 000 000 017 472 389 12;
  • 12) 0.000 000 000 000 000 017 472 389 12 × 2 = 0 + 0.000 000 000 000 000 034 944 778 24;
  • 13) 0.000 000 000 000 000 034 944 778 24 × 2 = 0 + 0.000 000 000 000 000 069 889 556 48;
  • 14) 0.000 000 000 000 000 069 889 556 48 × 2 = 0 + 0.000 000 000 000 000 139 779 112 96;
  • 15) 0.000 000 000 000 000 139 779 112 96 × 2 = 0 + 0.000 000 000 000 000 279 558 225 92;
  • 16) 0.000 000 000 000 000 279 558 225 92 × 2 = 0 + 0.000 000 000 000 000 559 116 451 84;
  • 17) 0.000 000 000 000 000 559 116 451 84 × 2 = 0 + 0.000 000 000 000 001 118 232 903 68;
  • 18) 0.000 000 000 000 001 118 232 903 68 × 2 = 0 + 0.000 000 000 000 002 236 465 807 36;
  • 19) 0.000 000 000 000 002 236 465 807 36 × 2 = 0 + 0.000 000 000 000 004 472 931 614 72;
  • 20) 0.000 000 000 000 004 472 931 614 72 × 2 = 0 + 0.000 000 000 000 008 945 863 229 44;
  • 21) 0.000 000 000 000 008 945 863 229 44 × 2 = 0 + 0.000 000 000 000 017 891 726 458 88;
  • 22) 0.000 000 000 000 017 891 726 458 88 × 2 = 0 + 0.000 000 000 000 035 783 452 917 76;
  • 23) 0.000 000 000 000 035 783 452 917 76 × 2 = 0 + 0.000 000 000 000 071 566 905 835 52;
  • 24) 0.000 000 000 000 071 566 905 835 52 × 2 = 0 + 0.000 000 000 000 143 133 811 671 04;
  • 25) 0.000 000 000 000 143 133 811 671 04 × 2 = 0 + 0.000 000 000 000 286 267 623 342 08;
  • 26) 0.000 000 000 000 286 267 623 342 08 × 2 = 0 + 0.000 000 000 000 572 535 246 684 16;
  • 27) 0.000 000 000 000 572 535 246 684 16 × 2 = 0 + 0.000 000 000 001 145 070 493 368 32;
  • 28) 0.000 000 000 001 145 070 493 368 32 × 2 = 0 + 0.000 000 000 002 290 140 986 736 64;
  • 29) 0.000 000 000 002 290 140 986 736 64 × 2 = 0 + 0.000 000 000 004 580 281 973 473 28;
  • 30) 0.000 000 000 004 580 281 973 473 28 × 2 = 0 + 0.000 000 000 009 160 563 946 946 56;
  • 31) 0.000 000 000 009 160 563 946 946 56 × 2 = 0 + 0.000 000 000 018 321 127 893 893 12;
  • 32) 0.000 000 000 018 321 127 893 893 12 × 2 = 0 + 0.000 000 000 036 642 255 787 786 24;
  • 33) 0.000 000 000 036 642 255 787 786 24 × 2 = 0 + 0.000 000 000 073 284 511 575 572 48;
  • 34) 0.000 000 000 073 284 511 575 572 48 × 2 = 0 + 0.000 000 000 146 569 023 151 144 96;
  • 35) 0.000 000 000 146 569 023 151 144 96 × 2 = 0 + 0.000 000 000 293 138 046 302 289 92;
  • 36) 0.000 000 000 293 138 046 302 289 92 × 2 = 0 + 0.000 000 000 586 276 092 604 579 84;
  • 37) 0.000 000 000 586 276 092 604 579 84 × 2 = 0 + 0.000 000 001 172 552 185 209 159 68;
  • 38) 0.000 000 001 172 552 185 209 159 68 × 2 = 0 + 0.000 000 002 345 104 370 418 319 36;
  • 39) 0.000 000 002 345 104 370 418 319 36 × 2 = 0 + 0.000 000 004 690 208 740 836 638 72;
  • 40) 0.000 000 004 690 208 740 836 638 72 × 2 = 0 + 0.000 000 009 380 417 481 673 277 44;
  • 41) 0.000 000 009 380 417 481 673 277 44 × 2 = 0 + 0.000 000 018 760 834 963 346 554 88;
  • 42) 0.000 000 018 760 834 963 346 554 88 × 2 = 0 + 0.000 000 037 521 669 926 693 109 76;
  • 43) 0.000 000 037 521 669 926 693 109 76 × 2 = 0 + 0.000 000 075 043 339 853 386 219 52;
  • 44) 0.000 000 075 043 339 853 386 219 52 × 2 = 0 + 0.000 000 150 086 679 706 772 439 04;
  • 45) 0.000 000 150 086 679 706 772 439 04 × 2 = 0 + 0.000 000 300 173 359 413 544 878 08;
  • 46) 0.000 000 300 173 359 413 544 878 08 × 2 = 0 + 0.000 000 600 346 718 827 089 756 16;
  • 47) 0.000 000 600 346 718 827 089 756 16 × 2 = 0 + 0.000 001 200 693 437 654 179 512 32;
  • 48) 0.000 001 200 693 437 654 179 512 32 × 2 = 0 + 0.000 002 401 386 875 308 359 024 64;
  • 49) 0.000 002 401 386 875 308 359 024 64 × 2 = 0 + 0.000 004 802 773 750 616 718 049 28;
  • 50) 0.000 004 802 773 750 616 718 049 28 × 2 = 0 + 0.000 009 605 547 501 233 436 098 56;
  • 51) 0.000 009 605 547 501 233 436 098 56 × 2 = 0 + 0.000 019 211 095 002 466 872 197 12;
  • 52) 0.000 019 211 095 002 466 872 197 12 × 2 = 0 + 0.000 038 422 190 004 933 744 394 24;
  • 53) 0.000 038 422 190 004 933 744 394 24 × 2 = 0 + 0.000 076 844 380 009 867 488 788 48;
  • 54) 0.000 076 844 380 009 867 488 788 48 × 2 = 0 + 0.000 153 688 760 019 734 977 576 96;
  • 55) 0.000 153 688 760 019 734 977 576 96 × 2 = 0 + 0.000 307 377 520 039 469 955 153 92;
  • 56) 0.000 307 377 520 039 469 955 153 92 × 2 = 0 + 0.000 614 755 040 078 939 910 307 84;
  • 57) 0.000 614 755 040 078 939 910 307 84 × 2 = 0 + 0.001 229 510 080 157 879 820 615 68;
  • 58) 0.001 229 510 080 157 879 820 615 68 × 2 = 0 + 0.002 459 020 160 315 759 641 231 36;
  • 59) 0.002 459 020 160 315 759 641 231 36 × 2 = 0 + 0.004 918 040 320 631 519 282 462 72;
  • 60) 0.004 918 040 320 631 519 282 462 72 × 2 = 0 + 0.009 836 080 641 263 038 564 925 44;
  • 61) 0.009 836 080 641 263 038 564 925 44 × 2 = 0 + 0.019 672 161 282 526 077 129 850 88;
  • 62) 0.019 672 161 282 526 077 129 850 88 × 2 = 0 + 0.039 344 322 565 052 154 259 701 76;
  • 63) 0.039 344 322 565 052 154 259 701 76 × 2 = 0 + 0.078 688 645 130 104 308 519 403 52;
  • 64) 0.078 688 645 130 104 308 519 403 52 × 2 = 0 + 0.157 377 290 260 208 617 038 807 04;
  • 65) 0.157 377 290 260 208 617 038 807 04 × 2 = 0 + 0.314 754 580 520 417 234 077 614 08;
  • 66) 0.314 754 580 520 417 234 077 614 08 × 2 = 0 + 0.629 509 161 040 834 468 155 228 16;
  • 67) 0.629 509 161 040 834 468 155 228 16 × 2 = 1 + 0.259 018 322 081 668 936 310 456 32;
  • 68) 0.259 018 322 081 668 936 310 456 32 × 2 = 0 + 0.518 036 644 163 337 872 620 912 64;
  • 69) 0.518 036 644 163 337 872 620 912 64 × 2 = 1 + 0.036 073 288 326 675 745 241 825 28;
  • 70) 0.036 073 288 326 675 745 241 825 28 × 2 = 0 + 0.072 146 576 653 351 490 483 650 56;
  • 71) 0.072 146 576 653 351 490 483 650 56 × 2 = 0 + 0.144 293 153 306 702 980 967 301 12;
  • 72) 0.144 293 153 306 702 980 967 301 12 × 2 = 0 + 0.288 586 306 613 405 961 934 602 24;
  • 73) 0.288 586 306 613 405 961 934 602 24 × 2 = 0 + 0.577 172 613 226 811 923 869 204 48;
  • 74) 0.577 172 613 226 811 923 869 204 48 × 2 = 1 + 0.154 345 226 453 623 847 738 408 96;
  • 75) 0.154 345 226 453 623 847 738 408 96 × 2 = 0 + 0.308 690 452 907 247 695 476 817 92;
  • 76) 0.308 690 452 907 247 695 476 817 92 × 2 = 0 + 0.617 380 905 814 495 390 953 635 84;
  • 77) 0.617 380 905 814 495 390 953 635 84 × 2 = 1 + 0.234 761 811 628 990 781 907 271 68;
  • 78) 0.234 761 811 628 990 781 907 271 68 × 2 = 0 + 0.469 523 623 257 981 563 814 543 36;
  • 79) 0.469 523 623 257 981 563 814 543 36 × 2 = 0 + 0.939 047 246 515 963 127 629 086 72;
  • 80) 0.939 047 246 515 963 127 629 086 72 × 2 = 1 + 0.878 094 493 031 926 255 258 173 44;
  • 81) 0.878 094 493 031 926 255 258 173 44 × 2 = 1 + 0.756 188 986 063 852 510 516 346 88;
  • 82) 0.756 188 986 063 852 510 516 346 88 × 2 = 1 + 0.512 377 972 127 705 021 032 693 76;
  • 83) 0.512 377 972 127 705 021 032 693 76 × 2 = 1 + 0.024 755 944 255 410 042 065 387 52;
  • 84) 0.024 755 944 255 410 042 065 387 52 × 2 = 0 + 0.049 511 888 510 820 084 130 775 04;
  • 85) 0.049 511 888 510 820 084 130 775 04 × 2 = 0 + 0.099 023 777 021 640 168 261 550 08;
  • 86) 0.099 023 777 021 640 168 261 550 08 × 2 = 0 + 0.198 047 554 043 280 336 523 100 16;
  • 87) 0.198 047 554 043 280 336 523 100 16 × 2 = 0 + 0.396 095 108 086 560 673 046 200 32;
  • 88) 0.396 095 108 086 560 673 046 200 32 × 2 = 0 + 0.792 190 216 173 121 346 092 400 64;
  • 89) 0.792 190 216 173 121 346 092 400 64 × 2 = 1 + 0.584 380 432 346 242 692 184 801 28;
  • 90) 0.584 380 432 346 242 692 184 801 28 × 2 = 1 + 0.168 760 864 692 485 384 369 602 56;
  • 91) 0.168 760 864 692 485 384 369 602 56 × 2 = 0 + 0.337 521 729 384 970 768 739 205 12;
  • 92) 0.337 521 729 384 970 768 739 205 12 × 2 = 0 + 0.675 043 458 769 941 537 478 410 24;
  • 93) 0.675 043 458 769 941 537 478 410 24 × 2 = 1 + 0.350 086 917 539 883 074 956 820 48;
  • 94) 0.350 086 917 539 883 074 956 820 48 × 2 = 0 + 0.700 173 835 079 766 149 913 640 96;
  • 95) 0.700 173 835 079 766 149 913 640 96 × 2 = 1 + 0.400 347 670 159 532 299 827 281 92;
  • 96) 0.400 347 670 159 532 299 827 281 92 × 2 = 0 + 0.800 695 340 319 064 599 654 563 84;
  • 97) 0.800 695 340 319 064 599 654 563 84 × 2 = 1 + 0.601 390 680 638 129 199 309 127 68;
  • 98) 0.601 390 680 638 129 199 309 127 68 × 2 = 1 + 0.202 781 361 276 258 398 618 255 36;
  • 99) 0.202 781 361 276 258 398 618 255 36 × 2 = 0 + 0.405 562 722 552 516 797 236 510 72;
  • 100) 0.405 562 722 552 516 797 236 510 72 × 2 = 0 + 0.811 125 445 105 033 594 473 021 44;
  • 101) 0.811 125 445 105 033 594 473 021 44 × 2 = 1 + 0.622 250 890 210 067 188 946 042 88;
  • 102) 0.622 250 890 210 067 188 946 042 88 × 2 = 1 + 0.244 501 780 420 134 377 892 085 76;
  • 103) 0.244 501 780 420 134 377 892 085 76 × 2 = 0 + 0.489 003 560 840 268 755 784 171 52;
  • 104) 0.489 003 560 840 268 755 784 171 52 × 2 = 0 + 0.978 007 121 680 537 511 568 343 04;
  • 105) 0.978 007 121 680 537 511 568 343 04 × 2 = 1 + 0.956 014 243 361 075 023 136 686 08;
  • 106) 0.956 014 243 361 075 023 136 686 08 × 2 = 1 + 0.912 028 486 722 150 046 273 372 16;
  • 107) 0.912 028 486 722 150 046 273 372 16 × 2 = 1 + 0.824 056 973 444 300 092 546 744 32;
  • 108) 0.824 056 973 444 300 092 546 744 32 × 2 = 1 + 0.648 113 946 888 600 185 093 488 64;
  • 109) 0.648 113 946 888 600 185 093 488 64 × 2 = 1 + 0.296 227 893 777 200 370 186 977 28;
  • 110) 0.296 227 893 777 200 370 186 977 28 × 2 = 0 + 0.592 455 787 554 400 740 373 954 56;
  • 111) 0.592 455 787 554 400 740 373 954 56 × 2 = 1 + 0.184 911 575 108 801 480 747 909 12;
  • 112) 0.184 911 575 108 801 480 747 909 12 × 2 = 0 + 0.369 823 150 217 602 961 495 818 24;
  • 113) 0.369 823 150 217 602 961 495 818 24 × 2 = 0 + 0.739 646 300 435 205 922 991 636 48;
  • 114) 0.739 646 300 435 205 922 991 636 48 × 2 = 1 + 0.479 292 600 870 411 845 983 272 96;
  • 115) 0.479 292 600 870 411 845 983 272 96 × 2 = 0 + 0.958 585 201 740 823 691 966 545 92;
  • 116) 0.958 585 201 740 823 691 966 545 92 × 2 = 1 + 0.917 170 403 481 647 383 933 091 84;
  • 117) 0.917 170 403 481 647 383 933 091 84 × 2 = 1 + 0.834 340 806 963 294 767 866 183 68;
  • 118) 0.834 340 806 963 294 767 866 183 68 × 2 = 1 + 0.668 681 613 926 589 535 732 367 36;
  • 119) 0.668 681 613 926 589 535 732 367 36 × 2 = 1 + 0.337 363 227 853 179 071 464 734 72;

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:


0.000 000 000 000 000 000 008 531 44(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1001 1110 0000 1100 1010 1100 1100 1111 1010 0101 111(2)

5. Positive number before normalization:

0.000 000 000 000 000 000 008 531 44(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1001 1110 0000 1100 1010 1100 1100 1111 1010 0101 111(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:


0.000 000 000 000 000 000 008 531 44(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1001 1110 0000 1100 1010 1100 1100 1111 1010 0101 111(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1001 1110 0000 1100 1010 1100 1100 1111 1010 0101 111(2) × 20 =

1.0100 0010 0100 1111 0000 0110 0101 0110 0110 0111 1101 0010 1111(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 0100 1111 0000 0110 0101 0110 0110 0111 1101 0010 1111


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 0100 1111 0000 0110 0101 0110 0110 0111 1101 0010 1111 =

0100 0010 0100 1111 0000 0110 0101 0110 0110 0111 1101 0010 1111


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 0100 1111 0000 0110 0101 0110 0110 0111 1101 0010 1111


Decimal number 0.000 000 000 000 000 000 008 531 44 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1011 1100 - 0100 0010 0100 1111 0000 0110 0101 0110 0110 0111 1101 0010 1111

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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