0.000 000 000 000 000 000 008 536 44 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 000 000 008 536 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
0.000 000 000 000 000 000 008 536 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: 0.000 000 000 000 000 000 008 536 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) 0.000 000 000 000 000 000 008 536 44 × 2 = 0 + 0.000 000 000 000 000 000 017 072 88;
  • 2) 0.000 000 000 000 000 000 017 072 88 × 2 = 0 + 0.000 000 000 000 000 000 034 145 76;
  • 3) 0.000 000 000 000 000 000 034 145 76 × 2 = 0 + 0.000 000 000 000 000 000 068 291 52;
  • 4) 0.000 000 000 000 000 000 068 291 52 × 2 = 0 + 0.000 000 000 000 000 000 136 583 04;
  • 5) 0.000 000 000 000 000 000 136 583 04 × 2 = 0 + 0.000 000 000 000 000 000 273 166 08;
  • 6) 0.000 000 000 000 000 000 273 166 08 × 2 = 0 + 0.000 000 000 000 000 000 546 332 16;
  • 7) 0.000 000 000 000 000 000 546 332 16 × 2 = 0 + 0.000 000 000 000 000 001 092 664 32;
  • 8) 0.000 000 000 000 000 001 092 664 32 × 2 = 0 + 0.000 000 000 000 000 002 185 328 64;
  • 9) 0.000 000 000 000 000 002 185 328 64 × 2 = 0 + 0.000 000 000 000 000 004 370 657 28;
  • 10) 0.000 000 000 000 000 004 370 657 28 × 2 = 0 + 0.000 000 000 000 000 008 741 314 56;
  • 11) 0.000 000 000 000 000 008 741 314 56 × 2 = 0 + 0.000 000 000 000 000 017 482 629 12;
  • 12) 0.000 000 000 000 000 017 482 629 12 × 2 = 0 + 0.000 000 000 000 000 034 965 258 24;
  • 13) 0.000 000 000 000 000 034 965 258 24 × 2 = 0 + 0.000 000 000 000 000 069 930 516 48;
  • 14) 0.000 000 000 000 000 069 930 516 48 × 2 = 0 + 0.000 000 000 000 000 139 861 032 96;
  • 15) 0.000 000 000 000 000 139 861 032 96 × 2 = 0 + 0.000 000 000 000 000 279 722 065 92;
  • 16) 0.000 000 000 000 000 279 722 065 92 × 2 = 0 + 0.000 000 000 000 000 559 444 131 84;
  • 17) 0.000 000 000 000 000 559 444 131 84 × 2 = 0 + 0.000 000 000 000 001 118 888 263 68;
  • 18) 0.000 000 000 000 001 118 888 263 68 × 2 = 0 + 0.000 000 000 000 002 237 776 527 36;
  • 19) 0.000 000 000 000 002 237 776 527 36 × 2 = 0 + 0.000 000 000 000 004 475 553 054 72;
  • 20) 0.000 000 000 000 004 475 553 054 72 × 2 = 0 + 0.000 000 000 000 008 951 106 109 44;
  • 21) 0.000 000 000 000 008 951 106 109 44 × 2 = 0 + 0.000 000 000 000 017 902 212 218 88;
  • 22) 0.000 000 000 000 017 902 212 218 88 × 2 = 0 + 0.000 000 000 000 035 804 424 437 76;
  • 23) 0.000 000 000 000 035 804 424 437 76 × 2 = 0 + 0.000 000 000 000 071 608 848 875 52;
  • 24) 0.000 000 000 000 071 608 848 875 52 × 2 = 0 + 0.000 000 000 000 143 217 697 751 04;
  • 25) 0.000 000 000 000 143 217 697 751 04 × 2 = 0 + 0.000 000 000 000 286 435 395 502 08;
  • 26) 0.000 000 000 000 286 435 395 502 08 × 2 = 0 + 0.000 000 000 000 572 870 791 004 16;
  • 27) 0.000 000 000 000 572 870 791 004 16 × 2 = 0 + 0.000 000 000 001 145 741 582 008 32;
  • 28) 0.000 000 000 001 145 741 582 008 32 × 2 = 0 + 0.000 000 000 002 291 483 164 016 64;
  • 29) 0.000 000 000 002 291 483 164 016 64 × 2 = 0 + 0.000 000 000 004 582 966 328 033 28;
  • 30) 0.000 000 000 004 582 966 328 033 28 × 2 = 0 + 0.000 000 000 009 165 932 656 066 56;
  • 31) 0.000 000 000 009 165 932 656 066 56 × 2 = 0 + 0.000 000 000 018 331 865 312 133 12;
  • 32) 0.000 000 000 018 331 865 312 133 12 × 2 = 0 + 0.000 000 000 036 663 730 624 266 24;
  • 33) 0.000 000 000 036 663 730 624 266 24 × 2 = 0 + 0.000 000 000 073 327 461 248 532 48;
  • 34) 0.000 000 000 073 327 461 248 532 48 × 2 = 0 + 0.000 000 000 146 654 922 497 064 96;
  • 35) 0.000 000 000 146 654 922 497 064 96 × 2 = 0 + 0.000 000 000 293 309 844 994 129 92;
  • 36) 0.000 000 000 293 309 844 994 129 92 × 2 = 0 + 0.000 000 000 586 619 689 988 259 84;
  • 37) 0.000 000 000 586 619 689 988 259 84 × 2 = 0 + 0.000 000 001 173 239 379 976 519 68;
  • 38) 0.000 000 001 173 239 379 976 519 68 × 2 = 0 + 0.000 000 002 346 478 759 953 039 36;
  • 39) 0.000 000 002 346 478 759 953 039 36 × 2 = 0 + 0.000 000 004 692 957 519 906 078 72;
  • 40) 0.000 000 004 692 957 519 906 078 72 × 2 = 0 + 0.000 000 009 385 915 039 812 157 44;
  • 41) 0.000 000 009 385 915 039 812 157 44 × 2 = 0 + 0.000 000 018 771 830 079 624 314 88;
  • 42) 0.000 000 018 771 830 079 624 314 88 × 2 = 0 + 0.000 000 037 543 660 159 248 629 76;
  • 43) 0.000 000 037 543 660 159 248 629 76 × 2 = 0 + 0.000 000 075 087 320 318 497 259 52;
  • 44) 0.000 000 075 087 320 318 497 259 52 × 2 = 0 + 0.000 000 150 174 640 636 994 519 04;
  • 45) 0.000 000 150 174 640 636 994 519 04 × 2 = 0 + 0.000 000 300 349 281 273 989 038 08;
  • 46) 0.000 000 300 349 281 273 989 038 08 × 2 = 0 + 0.000 000 600 698 562 547 978 076 16;
  • 47) 0.000 000 600 698 562 547 978 076 16 × 2 = 0 + 0.000 001 201 397 125 095 956 152 32;
  • 48) 0.000 001 201 397 125 095 956 152 32 × 2 = 0 + 0.000 002 402 794 250 191 912 304 64;
  • 49) 0.000 002 402 794 250 191 912 304 64 × 2 = 0 + 0.000 004 805 588 500 383 824 609 28;
  • 50) 0.000 004 805 588 500 383 824 609 28 × 2 = 0 + 0.000 009 611 177 000 767 649 218 56;
  • 51) 0.000 009 611 177 000 767 649 218 56 × 2 = 0 + 0.000 019 222 354 001 535 298 437 12;
  • 52) 0.000 019 222 354 001 535 298 437 12 × 2 = 0 + 0.000 038 444 708 003 070 596 874 24;
  • 53) 0.000 038 444 708 003 070 596 874 24 × 2 = 0 + 0.000 076 889 416 006 141 193 748 48;
  • 54) 0.000 076 889 416 006 141 193 748 48 × 2 = 0 + 0.000 153 778 832 012 282 387 496 96;
  • 55) 0.000 153 778 832 012 282 387 496 96 × 2 = 0 + 0.000 307 557 664 024 564 774 993 92;
  • 56) 0.000 307 557 664 024 564 774 993 92 × 2 = 0 + 0.000 615 115 328 049 129 549 987 84;
  • 57) 0.000 615 115 328 049 129 549 987 84 × 2 = 0 + 0.001 230 230 656 098 259 099 975 68;
  • 58) 0.001 230 230 656 098 259 099 975 68 × 2 = 0 + 0.002 460 461 312 196 518 199 951 36;
  • 59) 0.002 460 461 312 196 518 199 951 36 × 2 = 0 + 0.004 920 922 624 393 036 399 902 72;
  • 60) 0.004 920 922 624 393 036 399 902 72 × 2 = 0 + 0.009 841 845 248 786 072 799 805 44;
  • 61) 0.009 841 845 248 786 072 799 805 44 × 2 = 0 + 0.019 683 690 497 572 145 599 610 88;
  • 62) 0.019 683 690 497 572 145 599 610 88 × 2 = 0 + 0.039 367 380 995 144 291 199 221 76;
  • 63) 0.039 367 380 995 144 291 199 221 76 × 2 = 0 + 0.078 734 761 990 288 582 398 443 52;
  • 64) 0.078 734 761 990 288 582 398 443 52 × 2 = 0 + 0.157 469 523 980 577 164 796 887 04;
  • 65) 0.157 469 523 980 577 164 796 887 04 × 2 = 0 + 0.314 939 047 961 154 329 593 774 08;
  • 66) 0.314 939 047 961 154 329 593 774 08 × 2 = 0 + 0.629 878 095 922 308 659 187 548 16;
  • 67) 0.629 878 095 922 308 659 187 548 16 × 2 = 1 + 0.259 756 191 844 617 318 375 096 32;
  • 68) 0.259 756 191 844 617 318 375 096 32 × 2 = 0 + 0.519 512 383 689 234 636 750 192 64;
  • 69) 0.519 512 383 689 234 636 750 192 64 × 2 = 1 + 0.039 024 767 378 469 273 500 385 28;
  • 70) 0.039 024 767 378 469 273 500 385 28 × 2 = 0 + 0.078 049 534 756 938 547 000 770 56;
  • 71) 0.078 049 534 756 938 547 000 770 56 × 2 = 0 + 0.156 099 069 513 877 094 001 541 12;
  • 72) 0.156 099 069 513 877 094 001 541 12 × 2 = 0 + 0.312 198 139 027 754 188 003 082 24;
  • 73) 0.312 198 139 027 754 188 003 082 24 × 2 = 0 + 0.624 396 278 055 508 376 006 164 48;
  • 74) 0.624 396 278 055 508 376 006 164 48 × 2 = 1 + 0.248 792 556 111 016 752 012 328 96;
  • 75) 0.248 792 556 111 016 752 012 328 96 × 2 = 0 + 0.497 585 112 222 033 504 024 657 92;
  • 76) 0.497 585 112 222 033 504 024 657 92 × 2 = 0 + 0.995 170 224 444 067 008 049 315 84;
  • 77) 0.995 170 224 444 067 008 049 315 84 × 2 = 1 + 0.990 340 448 888 134 016 098 631 68;
  • 78) 0.990 340 448 888 134 016 098 631 68 × 2 = 1 + 0.980 680 897 776 268 032 197 263 36;
  • 79) 0.980 680 897 776 268 032 197 263 36 × 2 = 1 + 0.961 361 795 552 536 064 394 526 72;
  • 80) 0.961 361 795 552 536 064 394 526 72 × 2 = 1 + 0.922 723 591 105 072 128 789 053 44;
  • 81) 0.922 723 591 105 072 128 789 053 44 × 2 = 1 + 0.845 447 182 210 144 257 578 106 88;
  • 82) 0.845 447 182 210 144 257 578 106 88 × 2 = 1 + 0.690 894 364 420 288 515 156 213 76;
  • 83) 0.690 894 364 420 288 515 156 213 76 × 2 = 1 + 0.381 788 728 840 577 030 312 427 52;
  • 84) 0.381 788 728 840 577 030 312 427 52 × 2 = 0 + 0.763 577 457 681 154 060 624 855 04;
  • 85) 0.763 577 457 681 154 060 624 855 04 × 2 = 1 + 0.527 154 915 362 308 121 249 710 08;
  • 86) 0.527 154 915 362 308 121 249 710 08 × 2 = 1 + 0.054 309 830 724 616 242 499 420 16;
  • 87) 0.054 309 830 724 616 242 499 420 16 × 2 = 0 + 0.108 619 661 449 232 484 998 840 32;
  • 88) 0.108 619 661 449 232 484 998 840 32 × 2 = 0 + 0.217 239 322 898 464 969 997 680 64;
  • 89) 0.217 239 322 898 464 969 997 680 64 × 2 = 0 + 0.434 478 645 796 929 939 995 361 28;
  • 90) 0.434 478 645 796 929 939 995 361 28 × 2 = 0 + 0.868 957 291 593 859 879 990 722 56;
  • 91) 0.868 957 291 593 859 879 990 722 56 × 2 = 1 + 0.737 914 583 187 719 759 981 445 12;
  • 92) 0.737 914 583 187 719 759 981 445 12 × 2 = 1 + 0.475 829 166 375 439 519 962 890 24;
  • 93) 0.475 829 166 375 439 519 962 890 24 × 2 = 0 + 0.951 658 332 750 879 039 925 780 48;
  • 94) 0.951 658 332 750 879 039 925 780 48 × 2 = 1 + 0.903 316 665 501 758 079 851 560 96;
  • 95) 0.903 316 665 501 758 079 851 560 96 × 2 = 1 + 0.806 633 331 003 516 159 703 121 92;
  • 96) 0.806 633 331 003 516 159 703 121 92 × 2 = 1 + 0.613 266 662 007 032 319 406 243 84;
  • 97) 0.613 266 662 007 032 319 406 243 84 × 2 = 1 + 0.226 533 324 014 064 638 812 487 68;
  • 98) 0.226 533 324 014 064 638 812 487 68 × 2 = 0 + 0.453 066 648 028 129 277 624 975 36;
  • 99) 0.453 066 648 028 129 277 624 975 36 × 2 = 0 + 0.906 133 296 056 258 555 249 950 72;
  • 100) 0.906 133 296 056 258 555 249 950 72 × 2 = 1 + 0.812 266 592 112 517 110 499 901 44;
  • 101) 0.812 266 592 112 517 110 499 901 44 × 2 = 1 + 0.624 533 184 225 034 220 999 802 88;
  • 102) 0.624 533 184 225 034 220 999 802 88 × 2 = 1 + 0.249 066 368 450 068 441 999 605 76;
  • 103) 0.249 066 368 450 068 441 999 605 76 × 2 = 0 + 0.498 132 736 900 136 883 999 211 52;
  • 104) 0.498 132 736 900 136 883 999 211 52 × 2 = 0 + 0.996 265 473 800 273 767 998 423 04;
  • 105) 0.996 265 473 800 273 767 998 423 04 × 2 = 1 + 0.992 530 947 600 547 535 996 846 08;
  • 106) 0.992 530 947 600 547 535 996 846 08 × 2 = 1 + 0.985 061 895 201 095 071 993 692 16;
  • 107) 0.985 061 895 201 095 071 993 692 16 × 2 = 1 + 0.970 123 790 402 190 143 987 384 32;
  • 108) 0.970 123 790 402 190 143 987 384 32 × 2 = 1 + 0.940 247 580 804 380 287 974 768 64;
  • 109) 0.940 247 580 804 380 287 974 768 64 × 2 = 1 + 0.880 495 161 608 760 575 949 537 28;
  • 110) 0.880 495 161 608 760 575 949 537 28 × 2 = 1 + 0.760 990 323 217 521 151 899 074 56;
  • 111) 0.760 990 323 217 521 151 899 074 56 × 2 = 1 + 0.521 980 646 435 042 303 798 149 12;
  • 112) 0.521 980 646 435 042 303 798 149 12 × 2 = 1 + 0.043 961 292 870 084 607 596 298 24;
  • 113) 0.043 961 292 870 084 607 596 298 24 × 2 = 0 + 0.087 922 585 740 169 215 192 596 48;
  • 114) 0.087 922 585 740 169 215 192 596 48 × 2 = 0 + 0.175 845 171 480 338 430 385 192 96;
  • 115) 0.175 845 171 480 338 430 385 192 96 × 2 = 0 + 0.351 690 342 960 676 860 770 385 92;
  • 116) 0.351 690 342 960 676 860 770 385 92 × 2 = 0 + 0.703 380 685 921 353 721 540 771 84;
  • 117) 0.703 380 685 921 353 721 540 771 84 × 2 = 1 + 0.406 761 371 842 707 443 081 543 68;
  • 118) 0.406 761 371 842 707 443 081 543 68 × 2 = 0 + 0.813 522 743 685 414 886 163 087 36;
  • 119) 0.813 522 743 685 414 886 163 087 36 × 2 = 1 + 0.627 045 487 370 829 772 326 174 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:


0.000 000 000 000 000 000 008 536 44(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1111 1110 1100 0011 0111 1001 1100 1111 1111 0000 101(2)

5. Positive number before normalization:

0.000 000 000 000 000 000 008 536 44(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1111 1110 1100 0011 0111 1001 1100 1111 1111 0000 101(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:


0.000 000 000 000 000 000 008 536 44(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1111 1110 1100 0011 0111 1001 1100 1111 1111 0000 101(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1111 1110 1100 0011 0111 1001 1100 1111 1111 0000 101(2) × 20 =

1.0100 0010 0111 1111 0110 0001 1011 1100 1110 0111 1111 1000 0101(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 0111 1111 0110 0001 1011 1100 1110 0111 1111 1000 0101


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 0111 1111 0110 0001 1011 1100 1110 0111 1111 1000 0101 =

0100 0010 0111 1111 0110 0001 1011 1100 1110 0111 1111 1000 0101


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 0111 1111 0110 0001 1011 1100 1110 0111 1111 1000 0101


Decimal number 0.000 000 000 000 000 000 008 536 44 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1011 1100 - 0100 0010 0111 1111 0110 0001 1011 1100 1110 0111 1111 1000 0101

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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