0.000 000 000 000 000 000 008 537 15 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 000 000 008 537 15(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
0.000 000 000 000 000 000 008 537 15(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: 0.000 000 000 000 000 000 008 537 15.

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) 0.000 000 000 000 000 000 008 537 15 × 2 = 0 + 0.000 000 000 000 000 000 017 074 3;
  • 2) 0.000 000 000 000 000 000 017 074 3 × 2 = 0 + 0.000 000 000 000 000 000 034 148 6;
  • 3) 0.000 000 000 000 000 000 034 148 6 × 2 = 0 + 0.000 000 000 000 000 000 068 297 2;
  • 4) 0.000 000 000 000 000 000 068 297 2 × 2 = 0 + 0.000 000 000 000 000 000 136 594 4;
  • 5) 0.000 000 000 000 000 000 136 594 4 × 2 = 0 + 0.000 000 000 000 000 000 273 188 8;
  • 6) 0.000 000 000 000 000 000 273 188 8 × 2 = 0 + 0.000 000 000 000 000 000 546 377 6;
  • 7) 0.000 000 000 000 000 000 546 377 6 × 2 = 0 + 0.000 000 000 000 000 001 092 755 2;
  • 8) 0.000 000 000 000 000 001 092 755 2 × 2 = 0 + 0.000 000 000 000 000 002 185 510 4;
  • 9) 0.000 000 000 000 000 002 185 510 4 × 2 = 0 + 0.000 000 000 000 000 004 371 020 8;
  • 10) 0.000 000 000 000 000 004 371 020 8 × 2 = 0 + 0.000 000 000 000 000 008 742 041 6;
  • 11) 0.000 000 000 000 000 008 742 041 6 × 2 = 0 + 0.000 000 000 000 000 017 484 083 2;
  • 12) 0.000 000 000 000 000 017 484 083 2 × 2 = 0 + 0.000 000 000 000 000 034 968 166 4;
  • 13) 0.000 000 000 000 000 034 968 166 4 × 2 = 0 + 0.000 000 000 000 000 069 936 332 8;
  • 14) 0.000 000 000 000 000 069 936 332 8 × 2 = 0 + 0.000 000 000 000 000 139 872 665 6;
  • 15) 0.000 000 000 000 000 139 872 665 6 × 2 = 0 + 0.000 000 000 000 000 279 745 331 2;
  • 16) 0.000 000 000 000 000 279 745 331 2 × 2 = 0 + 0.000 000 000 000 000 559 490 662 4;
  • 17) 0.000 000 000 000 000 559 490 662 4 × 2 = 0 + 0.000 000 000 000 001 118 981 324 8;
  • 18) 0.000 000 000 000 001 118 981 324 8 × 2 = 0 + 0.000 000 000 000 002 237 962 649 6;
  • 19) 0.000 000 000 000 002 237 962 649 6 × 2 = 0 + 0.000 000 000 000 004 475 925 299 2;
  • 20) 0.000 000 000 000 004 475 925 299 2 × 2 = 0 + 0.000 000 000 000 008 951 850 598 4;
  • 21) 0.000 000 000 000 008 951 850 598 4 × 2 = 0 + 0.000 000 000 000 017 903 701 196 8;
  • 22) 0.000 000 000 000 017 903 701 196 8 × 2 = 0 + 0.000 000 000 000 035 807 402 393 6;
  • 23) 0.000 000 000 000 035 807 402 393 6 × 2 = 0 + 0.000 000 000 000 071 614 804 787 2;
  • 24) 0.000 000 000 000 071 614 804 787 2 × 2 = 0 + 0.000 000 000 000 143 229 609 574 4;
  • 25) 0.000 000 000 000 143 229 609 574 4 × 2 = 0 + 0.000 000 000 000 286 459 219 148 8;
  • 26) 0.000 000 000 000 286 459 219 148 8 × 2 = 0 + 0.000 000 000 000 572 918 438 297 6;
  • 27) 0.000 000 000 000 572 918 438 297 6 × 2 = 0 + 0.000 000 000 001 145 836 876 595 2;
  • 28) 0.000 000 000 001 145 836 876 595 2 × 2 = 0 + 0.000 000 000 002 291 673 753 190 4;
  • 29) 0.000 000 000 002 291 673 753 190 4 × 2 = 0 + 0.000 000 000 004 583 347 506 380 8;
  • 30) 0.000 000 000 004 583 347 506 380 8 × 2 = 0 + 0.000 000 000 009 166 695 012 761 6;
  • 31) 0.000 000 000 009 166 695 012 761 6 × 2 = 0 + 0.000 000 000 018 333 390 025 523 2;
  • 32) 0.000 000 000 018 333 390 025 523 2 × 2 = 0 + 0.000 000 000 036 666 780 051 046 4;
  • 33) 0.000 000 000 036 666 780 051 046 4 × 2 = 0 + 0.000 000 000 073 333 560 102 092 8;
  • 34) 0.000 000 000 073 333 560 102 092 8 × 2 = 0 + 0.000 000 000 146 667 120 204 185 6;
  • 35) 0.000 000 000 146 667 120 204 185 6 × 2 = 0 + 0.000 000 000 293 334 240 408 371 2;
  • 36) 0.000 000 000 293 334 240 408 371 2 × 2 = 0 + 0.000 000 000 586 668 480 816 742 4;
  • 37) 0.000 000 000 586 668 480 816 742 4 × 2 = 0 + 0.000 000 001 173 336 961 633 484 8;
  • 38) 0.000 000 001 173 336 961 633 484 8 × 2 = 0 + 0.000 000 002 346 673 923 266 969 6;
  • 39) 0.000 000 002 346 673 923 266 969 6 × 2 = 0 + 0.000 000 004 693 347 846 533 939 2;
  • 40) 0.000 000 004 693 347 846 533 939 2 × 2 = 0 + 0.000 000 009 386 695 693 067 878 4;
  • 41) 0.000 000 009 386 695 693 067 878 4 × 2 = 0 + 0.000 000 018 773 391 386 135 756 8;
  • 42) 0.000 000 018 773 391 386 135 756 8 × 2 = 0 + 0.000 000 037 546 782 772 271 513 6;
  • 43) 0.000 000 037 546 782 772 271 513 6 × 2 = 0 + 0.000 000 075 093 565 544 543 027 2;
  • 44) 0.000 000 075 093 565 544 543 027 2 × 2 = 0 + 0.000 000 150 187 131 089 086 054 4;
  • 45) 0.000 000 150 187 131 089 086 054 4 × 2 = 0 + 0.000 000 300 374 262 178 172 108 8;
  • 46) 0.000 000 300 374 262 178 172 108 8 × 2 = 0 + 0.000 000 600 748 524 356 344 217 6;
  • 47) 0.000 000 600 748 524 356 344 217 6 × 2 = 0 + 0.000 001 201 497 048 712 688 435 2;
  • 48) 0.000 001 201 497 048 712 688 435 2 × 2 = 0 + 0.000 002 402 994 097 425 376 870 4;
  • 49) 0.000 002 402 994 097 425 376 870 4 × 2 = 0 + 0.000 004 805 988 194 850 753 740 8;
  • 50) 0.000 004 805 988 194 850 753 740 8 × 2 = 0 + 0.000 009 611 976 389 701 507 481 6;
  • 51) 0.000 009 611 976 389 701 507 481 6 × 2 = 0 + 0.000 019 223 952 779 403 014 963 2;
  • 52) 0.000 019 223 952 779 403 014 963 2 × 2 = 0 + 0.000 038 447 905 558 806 029 926 4;
  • 53) 0.000 038 447 905 558 806 029 926 4 × 2 = 0 + 0.000 076 895 811 117 612 059 852 8;
  • 54) 0.000 076 895 811 117 612 059 852 8 × 2 = 0 + 0.000 153 791 622 235 224 119 705 6;
  • 55) 0.000 153 791 622 235 224 119 705 6 × 2 = 0 + 0.000 307 583 244 470 448 239 411 2;
  • 56) 0.000 307 583 244 470 448 239 411 2 × 2 = 0 + 0.000 615 166 488 940 896 478 822 4;
  • 57) 0.000 615 166 488 940 896 478 822 4 × 2 = 0 + 0.001 230 332 977 881 792 957 644 8;
  • 58) 0.001 230 332 977 881 792 957 644 8 × 2 = 0 + 0.002 460 665 955 763 585 915 289 6;
  • 59) 0.002 460 665 955 763 585 915 289 6 × 2 = 0 + 0.004 921 331 911 527 171 830 579 2;
  • 60) 0.004 921 331 911 527 171 830 579 2 × 2 = 0 + 0.009 842 663 823 054 343 661 158 4;
  • 61) 0.009 842 663 823 054 343 661 158 4 × 2 = 0 + 0.019 685 327 646 108 687 322 316 8;
  • 62) 0.019 685 327 646 108 687 322 316 8 × 2 = 0 + 0.039 370 655 292 217 374 644 633 6;
  • 63) 0.039 370 655 292 217 374 644 633 6 × 2 = 0 + 0.078 741 310 584 434 749 289 267 2;
  • 64) 0.078 741 310 584 434 749 289 267 2 × 2 = 0 + 0.157 482 621 168 869 498 578 534 4;
  • 65) 0.157 482 621 168 869 498 578 534 4 × 2 = 0 + 0.314 965 242 337 738 997 157 068 8;
  • 66) 0.314 965 242 337 738 997 157 068 8 × 2 = 0 + 0.629 930 484 675 477 994 314 137 6;
  • 67) 0.629 930 484 675 477 994 314 137 6 × 2 = 1 + 0.259 860 969 350 955 988 628 275 2;
  • 68) 0.259 860 969 350 955 988 628 275 2 × 2 = 0 + 0.519 721 938 701 911 977 256 550 4;
  • 69) 0.519 721 938 701 911 977 256 550 4 × 2 = 1 + 0.039 443 877 403 823 954 513 100 8;
  • 70) 0.039 443 877 403 823 954 513 100 8 × 2 = 0 + 0.078 887 754 807 647 909 026 201 6;
  • 71) 0.078 887 754 807 647 909 026 201 6 × 2 = 0 + 0.157 775 509 615 295 818 052 403 2;
  • 72) 0.157 775 509 615 295 818 052 403 2 × 2 = 0 + 0.315 551 019 230 591 636 104 806 4;
  • 73) 0.315 551 019 230 591 636 104 806 4 × 2 = 0 + 0.631 102 038 461 183 272 209 612 8;
  • 74) 0.631 102 038 461 183 272 209 612 8 × 2 = 1 + 0.262 204 076 922 366 544 419 225 6;
  • 75) 0.262 204 076 922 366 544 419 225 6 × 2 = 0 + 0.524 408 153 844 733 088 838 451 2;
  • 76) 0.524 408 153 844 733 088 838 451 2 × 2 = 1 + 0.048 816 307 689 466 177 676 902 4;
  • 77) 0.048 816 307 689 466 177 676 902 4 × 2 = 0 + 0.097 632 615 378 932 355 353 804 8;
  • 78) 0.097 632 615 378 932 355 353 804 8 × 2 = 0 + 0.195 265 230 757 864 710 707 609 6;
  • 79) 0.195 265 230 757 864 710 707 609 6 × 2 = 0 + 0.390 530 461 515 729 421 415 219 2;
  • 80) 0.390 530 461 515 729 421 415 219 2 × 2 = 0 + 0.781 060 923 031 458 842 830 438 4;
  • 81) 0.781 060 923 031 458 842 830 438 4 × 2 = 1 + 0.562 121 846 062 917 685 660 876 8;
  • 82) 0.562 121 846 062 917 685 660 876 8 × 2 = 1 + 0.124 243 692 125 835 371 321 753 6;
  • 83) 0.124 243 692 125 835 371 321 753 6 × 2 = 0 + 0.248 487 384 251 670 742 643 507 2;
  • 84) 0.248 487 384 251 670 742 643 507 2 × 2 = 0 + 0.496 974 768 503 341 485 287 014 4;
  • 85) 0.496 974 768 503 341 485 287 014 4 × 2 = 0 + 0.993 949 537 006 682 970 574 028 8;
  • 86) 0.993 949 537 006 682 970 574 028 8 × 2 = 1 + 0.987 899 074 013 365 941 148 057 6;
  • 87) 0.987 899 074 013 365 941 148 057 6 × 2 = 1 + 0.975 798 148 026 731 882 296 115 2;
  • 88) 0.975 798 148 026 731 882 296 115 2 × 2 = 1 + 0.951 596 296 053 463 764 592 230 4;
  • 89) 0.951 596 296 053 463 764 592 230 4 × 2 = 1 + 0.903 192 592 106 927 529 184 460 8;
  • 90) 0.903 192 592 106 927 529 184 460 8 × 2 = 1 + 0.806 385 184 213 855 058 368 921 6;
  • 91) 0.806 385 184 213 855 058 368 921 6 × 2 = 1 + 0.612 770 368 427 710 116 737 843 2;
  • 92) 0.612 770 368 427 710 116 737 843 2 × 2 = 1 + 0.225 540 736 855 420 233 475 686 4;
  • 93) 0.225 540 736 855 420 233 475 686 4 × 2 = 0 + 0.451 081 473 710 840 466 951 372 8;
  • 94) 0.451 081 473 710 840 466 951 372 8 × 2 = 0 + 0.902 162 947 421 680 933 902 745 6;
  • 95) 0.902 162 947 421 680 933 902 745 6 × 2 = 1 + 0.804 325 894 843 361 867 805 491 2;
  • 96) 0.804 325 894 843 361 867 805 491 2 × 2 = 1 + 0.608 651 789 686 723 735 610 982 4;
  • 97) 0.608 651 789 686 723 735 610 982 4 × 2 = 1 + 0.217 303 579 373 447 471 221 964 8;
  • 98) 0.217 303 579 373 447 471 221 964 8 × 2 = 0 + 0.434 607 158 746 894 942 443 929 6;
  • 99) 0.434 607 158 746 894 942 443 929 6 × 2 = 0 + 0.869 214 317 493 789 884 887 859 2;
  • 100) 0.869 214 317 493 789 884 887 859 2 × 2 = 1 + 0.738 428 634 987 579 769 775 718 4;
  • 101) 0.738 428 634 987 579 769 775 718 4 × 2 = 1 + 0.476 857 269 975 159 539 551 436 8;
  • 102) 0.476 857 269 975 159 539 551 436 8 × 2 = 0 + 0.953 714 539 950 319 079 102 873 6;
  • 103) 0.953 714 539 950 319 079 102 873 6 × 2 = 1 + 0.907 429 079 900 638 158 205 747 2;
  • 104) 0.907 429 079 900 638 158 205 747 2 × 2 = 1 + 0.814 858 159 801 276 316 411 494 4;
  • 105) 0.814 858 159 801 276 316 411 494 4 × 2 = 1 + 0.629 716 319 602 552 632 822 988 8;
  • 106) 0.629 716 319 602 552 632 822 988 8 × 2 = 1 + 0.259 432 639 205 105 265 645 977 6;
  • 107) 0.259 432 639 205 105 265 645 977 6 × 2 = 0 + 0.518 865 278 410 210 531 291 955 2;
  • 108) 0.518 865 278 410 210 531 291 955 2 × 2 = 1 + 0.037 730 556 820 421 062 583 910 4;
  • 109) 0.037 730 556 820 421 062 583 910 4 × 2 = 0 + 0.075 461 113 640 842 125 167 820 8;
  • 110) 0.075 461 113 640 842 125 167 820 8 × 2 = 0 + 0.150 922 227 281 684 250 335 641 6;
  • 111) 0.150 922 227 281 684 250 335 641 6 × 2 = 0 + 0.301 844 454 563 368 500 671 283 2;
  • 112) 0.301 844 454 563 368 500 671 283 2 × 2 = 0 + 0.603 688 909 126 737 001 342 566 4;
  • 113) 0.603 688 909 126 737 001 342 566 4 × 2 = 1 + 0.207 377 818 253 474 002 685 132 8;
  • 114) 0.207 377 818 253 474 002 685 132 8 × 2 = 0 + 0.414 755 636 506 948 005 370 265 6;
  • 115) 0.414 755 636 506 948 005 370 265 6 × 2 = 0 + 0.829 511 273 013 896 010 740 531 2;
  • 116) 0.829 511 273 013 896 010 740 531 2 × 2 = 1 + 0.659 022 546 027 792 021 481 062 4;
  • 117) 0.659 022 546 027 792 021 481 062 4 × 2 = 1 + 0.318 045 092 055 584 042 962 124 8;
  • 118) 0.318 045 092 055 584 042 962 124 8 × 2 = 0 + 0.636 090 184 111 168 085 924 249 6;
  • 119) 0.636 090 184 111 168 085 924 249 6 × 2 = 1 + 0.272 180 368 222 336 171 848 499 2;

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:


0.000 000 000 000 000 000 008 537 15(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0101 0000 1100 0111 1111 0011 1001 1011 1101 0000 1001 101(2)

5. Positive number before normalization:

0.000 000 000 000 000 000 008 537 15(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0101 0000 1100 0111 1111 0011 1001 1011 1101 0000 1001 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:


0.000 000 000 000 000 000 008 537 15(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0101 0000 1100 0111 1111 0011 1001 1011 1101 0000 1001 101(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0101 0000 1100 0111 1111 0011 1001 1011 1101 0000 1001 101(2) × 20 =

1.0100 0010 1000 0110 0011 1111 1001 1100 1101 1110 1000 0100 1101(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 1000 0110 0011 1111 1001 1100 1101 1110 1000 0100 1101


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 1000 0110 0011 1111 1001 1100 1101 1110 1000 0100 1101 =

0100 0010 1000 0110 0011 1111 1001 1100 1101 1110 1000 0100 1101


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 1000 0110 0011 1111 1001 1100 1101 1110 1000 0100 1101


Decimal number 0.000 000 000 000 000 000 008 537 15 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1011 1100 - 0100 0010 1000 0110 0011 1111 1001 1100 1101 1110 1000 0100 1101

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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