0.000 000 000 000 000 000 008 540 8 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 000 000 008 540 8(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
0.000 000 000 000 000 000 008 540 8(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: 0.000 000 000 000 000 000 008 540 8.

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) 0.000 000 000 000 000 000 008 540 8 × 2 = 0 + 0.000 000 000 000 000 000 017 081 6;
  • 2) 0.000 000 000 000 000 000 017 081 6 × 2 = 0 + 0.000 000 000 000 000 000 034 163 2;
  • 3) 0.000 000 000 000 000 000 034 163 2 × 2 = 0 + 0.000 000 000 000 000 000 068 326 4;
  • 4) 0.000 000 000 000 000 000 068 326 4 × 2 = 0 + 0.000 000 000 000 000 000 136 652 8;
  • 5) 0.000 000 000 000 000 000 136 652 8 × 2 = 0 + 0.000 000 000 000 000 000 273 305 6;
  • 6) 0.000 000 000 000 000 000 273 305 6 × 2 = 0 + 0.000 000 000 000 000 000 546 611 2;
  • 7) 0.000 000 000 000 000 000 546 611 2 × 2 = 0 + 0.000 000 000 000 000 001 093 222 4;
  • 8) 0.000 000 000 000 000 001 093 222 4 × 2 = 0 + 0.000 000 000 000 000 002 186 444 8;
  • 9) 0.000 000 000 000 000 002 186 444 8 × 2 = 0 + 0.000 000 000 000 000 004 372 889 6;
  • 10) 0.000 000 000 000 000 004 372 889 6 × 2 = 0 + 0.000 000 000 000 000 008 745 779 2;
  • 11) 0.000 000 000 000 000 008 745 779 2 × 2 = 0 + 0.000 000 000 000 000 017 491 558 4;
  • 12) 0.000 000 000 000 000 017 491 558 4 × 2 = 0 + 0.000 000 000 000 000 034 983 116 8;
  • 13) 0.000 000 000 000 000 034 983 116 8 × 2 = 0 + 0.000 000 000 000 000 069 966 233 6;
  • 14) 0.000 000 000 000 000 069 966 233 6 × 2 = 0 + 0.000 000 000 000 000 139 932 467 2;
  • 15) 0.000 000 000 000 000 139 932 467 2 × 2 = 0 + 0.000 000 000 000 000 279 864 934 4;
  • 16) 0.000 000 000 000 000 279 864 934 4 × 2 = 0 + 0.000 000 000 000 000 559 729 868 8;
  • 17) 0.000 000 000 000 000 559 729 868 8 × 2 = 0 + 0.000 000 000 000 001 119 459 737 6;
  • 18) 0.000 000 000 000 001 119 459 737 6 × 2 = 0 + 0.000 000 000 000 002 238 919 475 2;
  • 19) 0.000 000 000 000 002 238 919 475 2 × 2 = 0 + 0.000 000 000 000 004 477 838 950 4;
  • 20) 0.000 000 000 000 004 477 838 950 4 × 2 = 0 + 0.000 000 000 000 008 955 677 900 8;
  • 21) 0.000 000 000 000 008 955 677 900 8 × 2 = 0 + 0.000 000 000 000 017 911 355 801 6;
  • 22) 0.000 000 000 000 017 911 355 801 6 × 2 = 0 + 0.000 000 000 000 035 822 711 603 2;
  • 23) 0.000 000 000 000 035 822 711 603 2 × 2 = 0 + 0.000 000 000 000 071 645 423 206 4;
  • 24) 0.000 000 000 000 071 645 423 206 4 × 2 = 0 + 0.000 000 000 000 143 290 846 412 8;
  • 25) 0.000 000 000 000 143 290 846 412 8 × 2 = 0 + 0.000 000 000 000 286 581 692 825 6;
  • 26) 0.000 000 000 000 286 581 692 825 6 × 2 = 0 + 0.000 000 000 000 573 163 385 651 2;
  • 27) 0.000 000 000 000 573 163 385 651 2 × 2 = 0 + 0.000 000 000 001 146 326 771 302 4;
  • 28) 0.000 000 000 001 146 326 771 302 4 × 2 = 0 + 0.000 000 000 002 292 653 542 604 8;
  • 29) 0.000 000 000 002 292 653 542 604 8 × 2 = 0 + 0.000 000 000 004 585 307 085 209 6;
  • 30) 0.000 000 000 004 585 307 085 209 6 × 2 = 0 + 0.000 000 000 009 170 614 170 419 2;
  • 31) 0.000 000 000 009 170 614 170 419 2 × 2 = 0 + 0.000 000 000 018 341 228 340 838 4;
  • 32) 0.000 000 000 018 341 228 340 838 4 × 2 = 0 + 0.000 000 000 036 682 456 681 676 8;
  • 33) 0.000 000 000 036 682 456 681 676 8 × 2 = 0 + 0.000 000 000 073 364 913 363 353 6;
  • 34) 0.000 000 000 073 364 913 363 353 6 × 2 = 0 + 0.000 000 000 146 729 826 726 707 2;
  • 35) 0.000 000 000 146 729 826 726 707 2 × 2 = 0 + 0.000 000 000 293 459 653 453 414 4;
  • 36) 0.000 000 000 293 459 653 453 414 4 × 2 = 0 + 0.000 000 000 586 919 306 906 828 8;
  • 37) 0.000 000 000 586 919 306 906 828 8 × 2 = 0 + 0.000 000 001 173 838 613 813 657 6;
  • 38) 0.000 000 001 173 838 613 813 657 6 × 2 = 0 + 0.000 000 002 347 677 227 627 315 2;
  • 39) 0.000 000 002 347 677 227 627 315 2 × 2 = 0 + 0.000 000 004 695 354 455 254 630 4;
  • 40) 0.000 000 004 695 354 455 254 630 4 × 2 = 0 + 0.000 000 009 390 708 910 509 260 8;
  • 41) 0.000 000 009 390 708 910 509 260 8 × 2 = 0 + 0.000 000 018 781 417 821 018 521 6;
  • 42) 0.000 000 018 781 417 821 018 521 6 × 2 = 0 + 0.000 000 037 562 835 642 037 043 2;
  • 43) 0.000 000 037 562 835 642 037 043 2 × 2 = 0 + 0.000 000 075 125 671 284 074 086 4;
  • 44) 0.000 000 075 125 671 284 074 086 4 × 2 = 0 + 0.000 000 150 251 342 568 148 172 8;
  • 45) 0.000 000 150 251 342 568 148 172 8 × 2 = 0 + 0.000 000 300 502 685 136 296 345 6;
  • 46) 0.000 000 300 502 685 136 296 345 6 × 2 = 0 + 0.000 000 601 005 370 272 592 691 2;
  • 47) 0.000 000 601 005 370 272 592 691 2 × 2 = 0 + 0.000 001 202 010 740 545 185 382 4;
  • 48) 0.000 001 202 010 740 545 185 382 4 × 2 = 0 + 0.000 002 404 021 481 090 370 764 8;
  • 49) 0.000 002 404 021 481 090 370 764 8 × 2 = 0 + 0.000 004 808 042 962 180 741 529 6;
  • 50) 0.000 004 808 042 962 180 741 529 6 × 2 = 0 + 0.000 009 616 085 924 361 483 059 2;
  • 51) 0.000 009 616 085 924 361 483 059 2 × 2 = 0 + 0.000 019 232 171 848 722 966 118 4;
  • 52) 0.000 019 232 171 848 722 966 118 4 × 2 = 0 + 0.000 038 464 343 697 445 932 236 8;
  • 53) 0.000 038 464 343 697 445 932 236 8 × 2 = 0 + 0.000 076 928 687 394 891 864 473 6;
  • 54) 0.000 076 928 687 394 891 864 473 6 × 2 = 0 + 0.000 153 857 374 789 783 728 947 2;
  • 55) 0.000 153 857 374 789 783 728 947 2 × 2 = 0 + 0.000 307 714 749 579 567 457 894 4;
  • 56) 0.000 307 714 749 579 567 457 894 4 × 2 = 0 + 0.000 615 429 499 159 134 915 788 8;
  • 57) 0.000 615 429 499 159 134 915 788 8 × 2 = 0 + 0.001 230 858 998 318 269 831 577 6;
  • 58) 0.001 230 858 998 318 269 831 577 6 × 2 = 0 + 0.002 461 717 996 636 539 663 155 2;
  • 59) 0.002 461 717 996 636 539 663 155 2 × 2 = 0 + 0.004 923 435 993 273 079 326 310 4;
  • 60) 0.004 923 435 993 273 079 326 310 4 × 2 = 0 + 0.009 846 871 986 546 158 652 620 8;
  • 61) 0.009 846 871 986 546 158 652 620 8 × 2 = 0 + 0.019 693 743 973 092 317 305 241 6;
  • 62) 0.019 693 743 973 092 317 305 241 6 × 2 = 0 + 0.039 387 487 946 184 634 610 483 2;
  • 63) 0.039 387 487 946 184 634 610 483 2 × 2 = 0 + 0.078 774 975 892 369 269 220 966 4;
  • 64) 0.078 774 975 892 369 269 220 966 4 × 2 = 0 + 0.157 549 951 784 738 538 441 932 8;
  • 65) 0.157 549 951 784 738 538 441 932 8 × 2 = 0 + 0.315 099 903 569 477 076 883 865 6;
  • 66) 0.315 099 903 569 477 076 883 865 6 × 2 = 0 + 0.630 199 807 138 954 153 767 731 2;
  • 67) 0.630 199 807 138 954 153 767 731 2 × 2 = 1 + 0.260 399 614 277 908 307 535 462 4;
  • 68) 0.260 399 614 277 908 307 535 462 4 × 2 = 0 + 0.520 799 228 555 816 615 070 924 8;
  • 69) 0.520 799 228 555 816 615 070 924 8 × 2 = 1 + 0.041 598 457 111 633 230 141 849 6;
  • 70) 0.041 598 457 111 633 230 141 849 6 × 2 = 0 + 0.083 196 914 223 266 460 283 699 2;
  • 71) 0.083 196 914 223 266 460 283 699 2 × 2 = 0 + 0.166 393 828 446 532 920 567 398 4;
  • 72) 0.166 393 828 446 532 920 567 398 4 × 2 = 0 + 0.332 787 656 893 065 841 134 796 8;
  • 73) 0.332 787 656 893 065 841 134 796 8 × 2 = 0 + 0.665 575 313 786 131 682 269 593 6;
  • 74) 0.665 575 313 786 131 682 269 593 6 × 2 = 1 + 0.331 150 627 572 263 364 539 187 2;
  • 75) 0.331 150 627 572 263 364 539 187 2 × 2 = 0 + 0.662 301 255 144 526 729 078 374 4;
  • 76) 0.662 301 255 144 526 729 078 374 4 × 2 = 1 + 0.324 602 510 289 053 458 156 748 8;
  • 77) 0.324 602 510 289 053 458 156 748 8 × 2 = 0 + 0.649 205 020 578 106 916 313 497 6;
  • 78) 0.649 205 020 578 106 916 313 497 6 × 2 = 1 + 0.298 410 041 156 213 832 626 995 2;
  • 79) 0.298 410 041 156 213 832 626 995 2 × 2 = 0 + 0.596 820 082 312 427 665 253 990 4;
  • 80) 0.596 820 082 312 427 665 253 990 4 × 2 = 1 + 0.193 640 164 624 855 330 507 980 8;
  • 81) 0.193 640 164 624 855 330 507 980 8 × 2 = 0 + 0.387 280 329 249 710 661 015 961 6;
  • 82) 0.387 280 329 249 710 661 015 961 6 × 2 = 0 + 0.774 560 658 499 421 322 031 923 2;
  • 83) 0.774 560 658 499 421 322 031 923 2 × 2 = 1 + 0.549 121 316 998 842 644 063 846 4;
  • 84) 0.549 121 316 998 842 644 063 846 4 × 2 = 1 + 0.098 242 633 997 685 288 127 692 8;
  • 85) 0.098 242 633 997 685 288 127 692 8 × 2 = 0 + 0.196 485 267 995 370 576 255 385 6;
  • 86) 0.196 485 267 995 370 576 255 385 6 × 2 = 0 + 0.392 970 535 990 741 152 510 771 2;
  • 87) 0.392 970 535 990 741 152 510 771 2 × 2 = 0 + 0.785 941 071 981 482 305 021 542 4;
  • 88) 0.785 941 071 981 482 305 021 542 4 × 2 = 1 + 0.571 882 143 962 964 610 043 084 8;
  • 89) 0.571 882 143 962 964 610 043 084 8 × 2 = 1 + 0.143 764 287 925 929 220 086 169 6;
  • 90) 0.143 764 287 925 929 220 086 169 6 × 2 = 0 + 0.287 528 575 851 858 440 172 339 2;
  • 91) 0.287 528 575 851 858 440 172 339 2 × 2 = 0 + 0.575 057 151 703 716 880 344 678 4;
  • 92) 0.575 057 151 703 716 880 344 678 4 × 2 = 1 + 0.150 114 303 407 433 760 689 356 8;
  • 93) 0.150 114 303 407 433 760 689 356 8 × 2 = 0 + 0.300 228 606 814 867 521 378 713 6;
  • 94) 0.300 228 606 814 867 521 378 713 6 × 2 = 0 + 0.600 457 213 629 735 042 757 427 2;
  • 95) 0.600 457 213 629 735 042 757 427 2 × 2 = 1 + 0.200 914 427 259 470 085 514 854 4;
  • 96) 0.200 914 427 259 470 085 514 854 4 × 2 = 0 + 0.401 828 854 518 940 171 029 708 8;
  • 97) 0.401 828 854 518 940 171 029 708 8 × 2 = 0 + 0.803 657 709 037 880 342 059 417 6;
  • 98) 0.803 657 709 037 880 342 059 417 6 × 2 = 1 + 0.607 315 418 075 760 684 118 835 2;
  • 99) 0.607 315 418 075 760 684 118 835 2 × 2 = 1 + 0.214 630 836 151 521 368 237 670 4;
  • 100) 0.214 630 836 151 521 368 237 670 4 × 2 = 0 + 0.429 261 672 303 042 736 475 340 8;
  • 101) 0.429 261 672 303 042 736 475 340 8 × 2 = 0 + 0.858 523 344 606 085 472 950 681 6;
  • 102) 0.858 523 344 606 085 472 950 681 6 × 2 = 1 + 0.717 046 689 212 170 945 901 363 2;
  • 103) 0.717 046 689 212 170 945 901 363 2 × 2 = 1 + 0.434 093 378 424 341 891 802 726 4;
  • 104) 0.434 093 378 424 341 891 802 726 4 × 2 = 0 + 0.868 186 756 848 683 783 605 452 8;
  • 105) 0.868 186 756 848 683 783 605 452 8 × 2 = 1 + 0.736 373 513 697 367 567 210 905 6;
  • 106) 0.736 373 513 697 367 567 210 905 6 × 2 = 1 + 0.472 747 027 394 735 134 421 811 2;
  • 107) 0.472 747 027 394 735 134 421 811 2 × 2 = 0 + 0.945 494 054 789 470 268 843 622 4;
  • 108) 0.945 494 054 789 470 268 843 622 4 × 2 = 1 + 0.890 988 109 578 940 537 687 244 8;
  • 109) 0.890 988 109 578 940 537 687 244 8 × 2 = 1 + 0.781 976 219 157 881 075 374 489 6;
  • 110) 0.781 976 219 157 881 075 374 489 6 × 2 = 1 + 0.563 952 438 315 762 150 748 979 2;
  • 111) 0.563 952 438 315 762 150 748 979 2 × 2 = 1 + 0.127 904 876 631 524 301 497 958 4;
  • 112) 0.127 904 876 631 524 301 497 958 4 × 2 = 0 + 0.255 809 753 263 048 602 995 916 8;
  • 113) 0.255 809 753 263 048 602 995 916 8 × 2 = 0 + 0.511 619 506 526 097 205 991 833 6;
  • 114) 0.511 619 506 526 097 205 991 833 6 × 2 = 1 + 0.023 239 013 052 194 411 983 667 2;
  • 115) 0.023 239 013 052 194 411 983 667 2 × 2 = 0 + 0.046 478 026 104 388 823 967 334 4;
  • 116) 0.046 478 026 104 388 823 967 334 4 × 2 = 0 + 0.092 956 052 208 777 647 934 668 8;
  • 117) 0.092 956 052 208 777 647 934 668 8 × 2 = 0 + 0.185 912 104 417 555 295 869 337 6;
  • 118) 0.185 912 104 417 555 295 869 337 6 × 2 = 0 + 0.371 824 208 835 110 591 738 675 2;
  • 119) 0.371 824 208 835 110 591 738 675 2 × 2 = 0 + 0.743 648 417 670 221 183 477 350 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:


0.000 000 000 000 000 000 008 540 8(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0101 0101 0011 0001 1001 0010 0110 0110 1101 1110 0100 000(2)

5. Positive number before normalization:

0.000 000 000 000 000 000 008 540 8(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0101 0101 0011 0001 1001 0010 0110 0110 1101 1110 0100 000(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:


0.000 000 000 000 000 000 008 540 8(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0101 0101 0011 0001 1001 0010 0110 0110 1101 1110 0100 000(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0101 0101 0011 0001 1001 0010 0110 0110 1101 1110 0100 000(2) × 20 =

1.0100 0010 1010 1001 1000 1100 1001 0011 0011 0110 1111 0010 0000(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 1010 1001 1000 1100 1001 0011 0011 0110 1111 0010 0000


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 1010 1001 1000 1100 1001 0011 0011 0110 1111 0010 0000 =

0100 0010 1010 1001 1000 1100 1001 0011 0011 0110 1111 0010 0000


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 1010 1001 1000 1100 1001 0011 0011 0110 1111 0010 0000


Decimal number 0.000 000 000 000 000 000 008 540 8 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1011 1100 - 0100 0010 1010 1001 1000 1100 1001 0011 0011 0110 1111 0010 0000

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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