0.000 000 000 000 000 000 008 536 35 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 000 000 008 536 35(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 35(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 35.

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 35 × 2 = 0 + 0.000 000 000 000 000 000 017 072 7;
  • 2) 0.000 000 000 000 000 000 017 072 7 × 2 = 0 + 0.000 000 000 000 000 000 034 145 4;
  • 3) 0.000 000 000 000 000 000 034 145 4 × 2 = 0 + 0.000 000 000 000 000 000 068 290 8;
  • 4) 0.000 000 000 000 000 000 068 290 8 × 2 = 0 + 0.000 000 000 000 000 000 136 581 6;
  • 5) 0.000 000 000 000 000 000 136 581 6 × 2 = 0 + 0.000 000 000 000 000 000 273 163 2;
  • 6) 0.000 000 000 000 000 000 273 163 2 × 2 = 0 + 0.000 000 000 000 000 000 546 326 4;
  • 7) 0.000 000 000 000 000 000 546 326 4 × 2 = 0 + 0.000 000 000 000 000 001 092 652 8;
  • 8) 0.000 000 000 000 000 001 092 652 8 × 2 = 0 + 0.000 000 000 000 000 002 185 305 6;
  • 9) 0.000 000 000 000 000 002 185 305 6 × 2 = 0 + 0.000 000 000 000 000 004 370 611 2;
  • 10) 0.000 000 000 000 000 004 370 611 2 × 2 = 0 + 0.000 000 000 000 000 008 741 222 4;
  • 11) 0.000 000 000 000 000 008 741 222 4 × 2 = 0 + 0.000 000 000 000 000 017 482 444 8;
  • 12) 0.000 000 000 000 000 017 482 444 8 × 2 = 0 + 0.000 000 000 000 000 034 964 889 6;
  • 13) 0.000 000 000 000 000 034 964 889 6 × 2 = 0 + 0.000 000 000 000 000 069 929 779 2;
  • 14) 0.000 000 000 000 000 069 929 779 2 × 2 = 0 + 0.000 000 000 000 000 139 859 558 4;
  • 15) 0.000 000 000 000 000 139 859 558 4 × 2 = 0 + 0.000 000 000 000 000 279 719 116 8;
  • 16) 0.000 000 000 000 000 279 719 116 8 × 2 = 0 + 0.000 000 000 000 000 559 438 233 6;
  • 17) 0.000 000 000 000 000 559 438 233 6 × 2 = 0 + 0.000 000 000 000 001 118 876 467 2;
  • 18) 0.000 000 000 000 001 118 876 467 2 × 2 = 0 + 0.000 000 000 000 002 237 752 934 4;
  • 19) 0.000 000 000 000 002 237 752 934 4 × 2 = 0 + 0.000 000 000 000 004 475 505 868 8;
  • 20) 0.000 000 000 000 004 475 505 868 8 × 2 = 0 + 0.000 000 000 000 008 951 011 737 6;
  • 21) 0.000 000 000 000 008 951 011 737 6 × 2 = 0 + 0.000 000 000 000 017 902 023 475 2;
  • 22) 0.000 000 000 000 017 902 023 475 2 × 2 = 0 + 0.000 000 000 000 035 804 046 950 4;
  • 23) 0.000 000 000 000 035 804 046 950 4 × 2 = 0 + 0.000 000 000 000 071 608 093 900 8;
  • 24) 0.000 000 000 000 071 608 093 900 8 × 2 = 0 + 0.000 000 000 000 143 216 187 801 6;
  • 25) 0.000 000 000 000 143 216 187 801 6 × 2 = 0 + 0.000 000 000 000 286 432 375 603 2;
  • 26) 0.000 000 000 000 286 432 375 603 2 × 2 = 0 + 0.000 000 000 000 572 864 751 206 4;
  • 27) 0.000 000 000 000 572 864 751 206 4 × 2 = 0 + 0.000 000 000 001 145 729 502 412 8;
  • 28) 0.000 000 000 001 145 729 502 412 8 × 2 = 0 + 0.000 000 000 002 291 459 004 825 6;
  • 29) 0.000 000 000 002 291 459 004 825 6 × 2 = 0 + 0.000 000 000 004 582 918 009 651 2;
  • 30) 0.000 000 000 004 582 918 009 651 2 × 2 = 0 + 0.000 000 000 009 165 836 019 302 4;
  • 31) 0.000 000 000 009 165 836 019 302 4 × 2 = 0 + 0.000 000 000 018 331 672 038 604 8;
  • 32) 0.000 000 000 018 331 672 038 604 8 × 2 = 0 + 0.000 000 000 036 663 344 077 209 6;
  • 33) 0.000 000 000 036 663 344 077 209 6 × 2 = 0 + 0.000 000 000 073 326 688 154 419 2;
  • 34) 0.000 000 000 073 326 688 154 419 2 × 2 = 0 + 0.000 000 000 146 653 376 308 838 4;
  • 35) 0.000 000 000 146 653 376 308 838 4 × 2 = 0 + 0.000 000 000 293 306 752 617 676 8;
  • 36) 0.000 000 000 293 306 752 617 676 8 × 2 = 0 + 0.000 000 000 586 613 505 235 353 6;
  • 37) 0.000 000 000 586 613 505 235 353 6 × 2 = 0 + 0.000 000 001 173 227 010 470 707 2;
  • 38) 0.000 000 001 173 227 010 470 707 2 × 2 = 0 + 0.000 000 002 346 454 020 941 414 4;
  • 39) 0.000 000 002 346 454 020 941 414 4 × 2 = 0 + 0.000 000 004 692 908 041 882 828 8;
  • 40) 0.000 000 004 692 908 041 882 828 8 × 2 = 0 + 0.000 000 009 385 816 083 765 657 6;
  • 41) 0.000 000 009 385 816 083 765 657 6 × 2 = 0 + 0.000 000 018 771 632 167 531 315 2;
  • 42) 0.000 000 018 771 632 167 531 315 2 × 2 = 0 + 0.000 000 037 543 264 335 062 630 4;
  • 43) 0.000 000 037 543 264 335 062 630 4 × 2 = 0 + 0.000 000 075 086 528 670 125 260 8;
  • 44) 0.000 000 075 086 528 670 125 260 8 × 2 = 0 + 0.000 000 150 173 057 340 250 521 6;
  • 45) 0.000 000 150 173 057 340 250 521 6 × 2 = 0 + 0.000 000 300 346 114 680 501 043 2;
  • 46) 0.000 000 300 346 114 680 501 043 2 × 2 = 0 + 0.000 000 600 692 229 361 002 086 4;
  • 47) 0.000 000 600 692 229 361 002 086 4 × 2 = 0 + 0.000 001 201 384 458 722 004 172 8;
  • 48) 0.000 001 201 384 458 722 004 172 8 × 2 = 0 + 0.000 002 402 768 917 444 008 345 6;
  • 49) 0.000 002 402 768 917 444 008 345 6 × 2 = 0 + 0.000 004 805 537 834 888 016 691 2;
  • 50) 0.000 004 805 537 834 888 016 691 2 × 2 = 0 + 0.000 009 611 075 669 776 033 382 4;
  • 51) 0.000 009 611 075 669 776 033 382 4 × 2 = 0 + 0.000 019 222 151 339 552 066 764 8;
  • 52) 0.000 019 222 151 339 552 066 764 8 × 2 = 0 + 0.000 038 444 302 679 104 133 529 6;
  • 53) 0.000 038 444 302 679 104 133 529 6 × 2 = 0 + 0.000 076 888 605 358 208 267 059 2;
  • 54) 0.000 076 888 605 358 208 267 059 2 × 2 = 0 + 0.000 153 777 210 716 416 534 118 4;
  • 55) 0.000 153 777 210 716 416 534 118 4 × 2 = 0 + 0.000 307 554 421 432 833 068 236 8;
  • 56) 0.000 307 554 421 432 833 068 236 8 × 2 = 0 + 0.000 615 108 842 865 666 136 473 6;
  • 57) 0.000 615 108 842 865 666 136 473 6 × 2 = 0 + 0.001 230 217 685 731 332 272 947 2;
  • 58) 0.001 230 217 685 731 332 272 947 2 × 2 = 0 + 0.002 460 435 371 462 664 545 894 4;
  • 59) 0.002 460 435 371 462 664 545 894 4 × 2 = 0 + 0.004 920 870 742 925 329 091 788 8;
  • 60) 0.004 920 870 742 925 329 091 788 8 × 2 = 0 + 0.009 841 741 485 850 658 183 577 6;
  • 61) 0.009 841 741 485 850 658 183 577 6 × 2 = 0 + 0.019 683 482 971 701 316 367 155 2;
  • 62) 0.019 683 482 971 701 316 367 155 2 × 2 = 0 + 0.039 366 965 943 402 632 734 310 4;
  • 63) 0.039 366 965 943 402 632 734 310 4 × 2 = 0 + 0.078 733 931 886 805 265 468 620 8;
  • 64) 0.078 733 931 886 805 265 468 620 8 × 2 = 0 + 0.157 467 863 773 610 530 937 241 6;
  • 65) 0.157 467 863 773 610 530 937 241 6 × 2 = 0 + 0.314 935 727 547 221 061 874 483 2;
  • 66) 0.314 935 727 547 221 061 874 483 2 × 2 = 0 + 0.629 871 455 094 442 123 748 966 4;
  • 67) 0.629 871 455 094 442 123 748 966 4 × 2 = 1 + 0.259 742 910 188 884 247 497 932 8;
  • 68) 0.259 742 910 188 884 247 497 932 8 × 2 = 0 + 0.519 485 820 377 768 494 995 865 6;
  • 69) 0.519 485 820 377 768 494 995 865 6 × 2 = 1 + 0.038 971 640 755 536 989 991 731 2;
  • 70) 0.038 971 640 755 536 989 991 731 2 × 2 = 0 + 0.077 943 281 511 073 979 983 462 4;
  • 71) 0.077 943 281 511 073 979 983 462 4 × 2 = 0 + 0.155 886 563 022 147 959 966 924 8;
  • 72) 0.155 886 563 022 147 959 966 924 8 × 2 = 0 + 0.311 773 126 044 295 919 933 849 6;
  • 73) 0.311 773 126 044 295 919 933 849 6 × 2 = 0 + 0.623 546 252 088 591 839 867 699 2;
  • 74) 0.623 546 252 088 591 839 867 699 2 × 2 = 1 + 0.247 092 504 177 183 679 735 398 4;
  • 75) 0.247 092 504 177 183 679 735 398 4 × 2 = 0 + 0.494 185 008 354 367 359 470 796 8;
  • 76) 0.494 185 008 354 367 359 470 796 8 × 2 = 0 + 0.988 370 016 708 734 718 941 593 6;
  • 77) 0.988 370 016 708 734 718 941 593 6 × 2 = 1 + 0.976 740 033 417 469 437 883 187 2;
  • 78) 0.976 740 033 417 469 437 883 187 2 × 2 = 1 + 0.953 480 066 834 938 875 766 374 4;
  • 79) 0.953 480 066 834 938 875 766 374 4 × 2 = 1 + 0.906 960 133 669 877 751 532 748 8;
  • 80) 0.906 960 133 669 877 751 532 748 8 × 2 = 1 + 0.813 920 267 339 755 503 065 497 6;
  • 81) 0.813 920 267 339 755 503 065 497 6 × 2 = 1 + 0.627 840 534 679 511 006 130 995 2;
  • 82) 0.627 840 534 679 511 006 130 995 2 × 2 = 1 + 0.255 681 069 359 022 012 261 990 4;
  • 83) 0.255 681 069 359 022 012 261 990 4 × 2 = 0 + 0.511 362 138 718 044 024 523 980 8;
  • 84) 0.511 362 138 718 044 024 523 980 8 × 2 = 1 + 0.022 724 277 436 088 049 047 961 6;
  • 85) 0.022 724 277 436 088 049 047 961 6 × 2 = 0 + 0.045 448 554 872 176 098 095 923 2;
  • 86) 0.045 448 554 872 176 098 095 923 2 × 2 = 0 + 0.090 897 109 744 352 196 191 846 4;
  • 87) 0.090 897 109 744 352 196 191 846 4 × 2 = 0 + 0.181 794 219 488 704 392 383 692 8;
  • 88) 0.181 794 219 488 704 392 383 692 8 × 2 = 0 + 0.363 588 438 977 408 784 767 385 6;
  • 89) 0.363 588 438 977 408 784 767 385 6 × 2 = 0 + 0.727 176 877 954 817 569 534 771 2;
  • 90) 0.727 176 877 954 817 569 534 771 2 × 2 = 1 + 0.454 353 755 909 635 139 069 542 4;
  • 91) 0.454 353 755 909 635 139 069 542 4 × 2 = 0 + 0.908 707 511 819 270 278 139 084 8;
  • 92) 0.908 707 511 819 270 278 139 084 8 × 2 = 1 + 0.817 415 023 638 540 556 278 169 6;
  • 93) 0.817 415 023 638 540 556 278 169 6 × 2 = 1 + 0.634 830 047 277 081 112 556 339 2;
  • 94) 0.634 830 047 277 081 112 556 339 2 × 2 = 1 + 0.269 660 094 554 162 225 112 678 4;
  • 95) 0.269 660 094 554 162 225 112 678 4 × 2 = 0 + 0.539 320 189 108 324 450 225 356 8;
  • 96) 0.539 320 189 108 324 450 225 356 8 × 2 = 1 + 0.078 640 378 216 648 900 450 713 6;
  • 97) 0.078 640 378 216 648 900 450 713 6 × 2 = 0 + 0.157 280 756 433 297 800 901 427 2;
  • 98) 0.157 280 756 433 297 800 901 427 2 × 2 = 0 + 0.314 561 512 866 595 601 802 854 4;
  • 99) 0.314 561 512 866 595 601 802 854 4 × 2 = 0 + 0.629 123 025 733 191 203 605 708 8;
  • 100) 0.629 123 025 733 191 203 605 708 8 × 2 = 1 + 0.258 246 051 466 382 407 211 417 6;
  • 101) 0.258 246 051 466 382 407 211 417 6 × 2 = 0 + 0.516 492 102 932 764 814 422 835 2;
  • 102) 0.516 492 102 932 764 814 422 835 2 × 2 = 1 + 0.032 984 205 865 529 628 845 670 4;
  • 103) 0.032 984 205 865 529 628 845 670 4 × 2 = 0 + 0.065 968 411 731 059 257 691 340 8;
  • 104) 0.065 968 411 731 059 257 691 340 8 × 2 = 0 + 0.131 936 823 462 118 515 382 681 6;
  • 105) 0.131 936 823 462 118 515 382 681 6 × 2 = 0 + 0.263 873 646 924 237 030 765 363 2;
  • 106) 0.263 873 646 924 237 030 765 363 2 × 2 = 0 + 0.527 747 293 848 474 061 530 726 4;
  • 107) 0.527 747 293 848 474 061 530 726 4 × 2 = 1 + 0.055 494 587 696 948 123 061 452 8;
  • 108) 0.055 494 587 696 948 123 061 452 8 × 2 = 0 + 0.110 989 175 393 896 246 122 905 6;
  • 109) 0.110 989 175 393 896 246 122 905 6 × 2 = 0 + 0.221 978 350 787 792 492 245 811 2;
  • 110) 0.221 978 350 787 792 492 245 811 2 × 2 = 0 + 0.443 956 701 575 584 984 491 622 4;
  • 111) 0.443 956 701 575 584 984 491 622 4 × 2 = 0 + 0.887 913 403 151 169 968 983 244 8;
  • 112) 0.887 913 403 151 169 968 983 244 8 × 2 = 1 + 0.775 826 806 302 339 937 966 489 6;
  • 113) 0.775 826 806 302 339 937 966 489 6 × 2 = 1 + 0.551 653 612 604 679 875 932 979 2;
  • 114) 0.551 653 612 604 679 875 932 979 2 × 2 = 1 + 0.103 307 225 209 359 751 865 958 4;
  • 115) 0.103 307 225 209 359 751 865 958 4 × 2 = 0 + 0.206 614 450 418 719 503 731 916 8;
  • 116) 0.206 614 450 418 719 503 731 916 8 × 2 = 0 + 0.413 228 900 837 439 007 463 833 6;
  • 117) 0.413 228 900 837 439 007 463 833 6 × 2 = 0 + 0.826 457 801 674 878 014 927 667 2;
  • 118) 0.826 457 801 674 878 014 927 667 2 × 2 = 1 + 0.652 915 603 349 756 029 855 334 4;
  • 119) 0.652 915 603 349 756 029 855 334 4 × 2 = 1 + 0.305 831 206 699 512 059 710 668 8;

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 35(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1111 1101 0000 0101 1101 0001 0100 0010 0001 1100 011(2)

5. Positive number before normalization:

0.000 000 000 000 000 000 008 536 35(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1111 1101 0000 0101 1101 0001 0100 0010 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:


0.000 000 000 000 000 000 008 536 35(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1111 1101 0000 0101 1101 0001 0100 0010 0001 1100 011(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1111 1101 0000 0101 1101 0001 0100 0010 0001 1100 011(2) × 20 =

1.0100 0010 0111 1110 1000 0010 1110 1000 1010 0001 0000 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 0111 1110 1000 0010 1110 1000 1010 0001 0000 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 0111 1110 1000 0010 1110 1000 1010 0001 0000 1110 0011 =

0100 0010 0111 1110 1000 0010 1110 1000 1010 0001 0000 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 0111 1110 1000 0010 1110 1000 1010 0001 0000 1110 0011


Decimal number 0.000 000 000 000 000 000 008 536 35 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1011 1100 - 0100 0010 0111 1110 1000 0010 1110 1000 1010 0001 0000 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