0.000 000 000 000 000 000 008 538 7 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 000 000 008 538 7(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
0.000 000 000 000 000 000 008 538 7(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: 0.000 000 000 000 000 000 008 538 7.

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) 0.000 000 000 000 000 000 008 538 7 × 2 = 0 + 0.000 000 000 000 000 000 017 077 4;
  • 2) 0.000 000 000 000 000 000 017 077 4 × 2 = 0 + 0.000 000 000 000 000 000 034 154 8;
  • 3) 0.000 000 000 000 000 000 034 154 8 × 2 = 0 + 0.000 000 000 000 000 000 068 309 6;
  • 4) 0.000 000 000 000 000 000 068 309 6 × 2 = 0 + 0.000 000 000 000 000 000 136 619 2;
  • 5) 0.000 000 000 000 000 000 136 619 2 × 2 = 0 + 0.000 000 000 000 000 000 273 238 4;
  • 6) 0.000 000 000 000 000 000 273 238 4 × 2 = 0 + 0.000 000 000 000 000 000 546 476 8;
  • 7) 0.000 000 000 000 000 000 546 476 8 × 2 = 0 + 0.000 000 000 000 000 001 092 953 6;
  • 8) 0.000 000 000 000 000 001 092 953 6 × 2 = 0 + 0.000 000 000 000 000 002 185 907 2;
  • 9) 0.000 000 000 000 000 002 185 907 2 × 2 = 0 + 0.000 000 000 000 000 004 371 814 4;
  • 10) 0.000 000 000 000 000 004 371 814 4 × 2 = 0 + 0.000 000 000 000 000 008 743 628 8;
  • 11) 0.000 000 000 000 000 008 743 628 8 × 2 = 0 + 0.000 000 000 000 000 017 487 257 6;
  • 12) 0.000 000 000 000 000 017 487 257 6 × 2 = 0 + 0.000 000 000 000 000 034 974 515 2;
  • 13) 0.000 000 000 000 000 034 974 515 2 × 2 = 0 + 0.000 000 000 000 000 069 949 030 4;
  • 14) 0.000 000 000 000 000 069 949 030 4 × 2 = 0 + 0.000 000 000 000 000 139 898 060 8;
  • 15) 0.000 000 000 000 000 139 898 060 8 × 2 = 0 + 0.000 000 000 000 000 279 796 121 6;
  • 16) 0.000 000 000 000 000 279 796 121 6 × 2 = 0 + 0.000 000 000 000 000 559 592 243 2;
  • 17) 0.000 000 000 000 000 559 592 243 2 × 2 = 0 + 0.000 000 000 000 001 119 184 486 4;
  • 18) 0.000 000 000 000 001 119 184 486 4 × 2 = 0 + 0.000 000 000 000 002 238 368 972 8;
  • 19) 0.000 000 000 000 002 238 368 972 8 × 2 = 0 + 0.000 000 000 000 004 476 737 945 6;
  • 20) 0.000 000 000 000 004 476 737 945 6 × 2 = 0 + 0.000 000 000 000 008 953 475 891 2;
  • 21) 0.000 000 000 000 008 953 475 891 2 × 2 = 0 + 0.000 000 000 000 017 906 951 782 4;
  • 22) 0.000 000 000 000 017 906 951 782 4 × 2 = 0 + 0.000 000 000 000 035 813 903 564 8;
  • 23) 0.000 000 000 000 035 813 903 564 8 × 2 = 0 + 0.000 000 000 000 071 627 807 129 6;
  • 24) 0.000 000 000 000 071 627 807 129 6 × 2 = 0 + 0.000 000 000 000 143 255 614 259 2;
  • 25) 0.000 000 000 000 143 255 614 259 2 × 2 = 0 + 0.000 000 000 000 286 511 228 518 4;
  • 26) 0.000 000 000 000 286 511 228 518 4 × 2 = 0 + 0.000 000 000 000 573 022 457 036 8;
  • 27) 0.000 000 000 000 573 022 457 036 8 × 2 = 0 + 0.000 000 000 001 146 044 914 073 6;
  • 28) 0.000 000 000 001 146 044 914 073 6 × 2 = 0 + 0.000 000 000 002 292 089 828 147 2;
  • 29) 0.000 000 000 002 292 089 828 147 2 × 2 = 0 + 0.000 000 000 004 584 179 656 294 4;
  • 30) 0.000 000 000 004 584 179 656 294 4 × 2 = 0 + 0.000 000 000 009 168 359 312 588 8;
  • 31) 0.000 000 000 009 168 359 312 588 8 × 2 = 0 + 0.000 000 000 018 336 718 625 177 6;
  • 32) 0.000 000 000 018 336 718 625 177 6 × 2 = 0 + 0.000 000 000 036 673 437 250 355 2;
  • 33) 0.000 000 000 036 673 437 250 355 2 × 2 = 0 + 0.000 000 000 073 346 874 500 710 4;
  • 34) 0.000 000 000 073 346 874 500 710 4 × 2 = 0 + 0.000 000 000 146 693 749 001 420 8;
  • 35) 0.000 000 000 146 693 749 001 420 8 × 2 = 0 + 0.000 000 000 293 387 498 002 841 6;
  • 36) 0.000 000 000 293 387 498 002 841 6 × 2 = 0 + 0.000 000 000 586 774 996 005 683 2;
  • 37) 0.000 000 000 586 774 996 005 683 2 × 2 = 0 + 0.000 000 001 173 549 992 011 366 4;
  • 38) 0.000 000 001 173 549 992 011 366 4 × 2 = 0 + 0.000 000 002 347 099 984 022 732 8;
  • 39) 0.000 000 002 347 099 984 022 732 8 × 2 = 0 + 0.000 000 004 694 199 968 045 465 6;
  • 40) 0.000 000 004 694 199 968 045 465 6 × 2 = 0 + 0.000 000 009 388 399 936 090 931 2;
  • 41) 0.000 000 009 388 399 936 090 931 2 × 2 = 0 + 0.000 000 018 776 799 872 181 862 4;
  • 42) 0.000 000 018 776 799 872 181 862 4 × 2 = 0 + 0.000 000 037 553 599 744 363 724 8;
  • 43) 0.000 000 037 553 599 744 363 724 8 × 2 = 0 + 0.000 000 075 107 199 488 727 449 6;
  • 44) 0.000 000 075 107 199 488 727 449 6 × 2 = 0 + 0.000 000 150 214 398 977 454 899 2;
  • 45) 0.000 000 150 214 398 977 454 899 2 × 2 = 0 + 0.000 000 300 428 797 954 909 798 4;
  • 46) 0.000 000 300 428 797 954 909 798 4 × 2 = 0 + 0.000 000 600 857 595 909 819 596 8;
  • 47) 0.000 000 600 857 595 909 819 596 8 × 2 = 0 + 0.000 001 201 715 191 819 639 193 6;
  • 48) 0.000 001 201 715 191 819 639 193 6 × 2 = 0 + 0.000 002 403 430 383 639 278 387 2;
  • 49) 0.000 002 403 430 383 639 278 387 2 × 2 = 0 + 0.000 004 806 860 767 278 556 774 4;
  • 50) 0.000 004 806 860 767 278 556 774 4 × 2 = 0 + 0.000 009 613 721 534 557 113 548 8;
  • 51) 0.000 009 613 721 534 557 113 548 8 × 2 = 0 + 0.000 019 227 443 069 114 227 097 6;
  • 52) 0.000 019 227 443 069 114 227 097 6 × 2 = 0 + 0.000 038 454 886 138 228 454 195 2;
  • 53) 0.000 038 454 886 138 228 454 195 2 × 2 = 0 + 0.000 076 909 772 276 456 908 390 4;
  • 54) 0.000 076 909 772 276 456 908 390 4 × 2 = 0 + 0.000 153 819 544 552 913 816 780 8;
  • 55) 0.000 153 819 544 552 913 816 780 8 × 2 = 0 + 0.000 307 639 089 105 827 633 561 6;
  • 56) 0.000 307 639 089 105 827 633 561 6 × 2 = 0 + 0.000 615 278 178 211 655 267 123 2;
  • 57) 0.000 615 278 178 211 655 267 123 2 × 2 = 0 + 0.001 230 556 356 423 310 534 246 4;
  • 58) 0.001 230 556 356 423 310 534 246 4 × 2 = 0 + 0.002 461 112 712 846 621 068 492 8;
  • 59) 0.002 461 112 712 846 621 068 492 8 × 2 = 0 + 0.004 922 225 425 693 242 136 985 6;
  • 60) 0.004 922 225 425 693 242 136 985 6 × 2 = 0 + 0.009 844 450 851 386 484 273 971 2;
  • 61) 0.009 844 450 851 386 484 273 971 2 × 2 = 0 + 0.019 688 901 702 772 968 547 942 4;
  • 62) 0.019 688 901 702 772 968 547 942 4 × 2 = 0 + 0.039 377 803 405 545 937 095 884 8;
  • 63) 0.039 377 803 405 545 937 095 884 8 × 2 = 0 + 0.078 755 606 811 091 874 191 769 6;
  • 64) 0.078 755 606 811 091 874 191 769 6 × 2 = 0 + 0.157 511 213 622 183 748 383 539 2;
  • 65) 0.157 511 213 622 183 748 383 539 2 × 2 = 0 + 0.315 022 427 244 367 496 767 078 4;
  • 66) 0.315 022 427 244 367 496 767 078 4 × 2 = 0 + 0.630 044 854 488 734 993 534 156 8;
  • 67) 0.630 044 854 488 734 993 534 156 8 × 2 = 1 + 0.260 089 708 977 469 987 068 313 6;
  • 68) 0.260 089 708 977 469 987 068 313 6 × 2 = 0 + 0.520 179 417 954 939 974 136 627 2;
  • 69) 0.520 179 417 954 939 974 136 627 2 × 2 = 1 + 0.040 358 835 909 879 948 273 254 4;
  • 70) 0.040 358 835 909 879 948 273 254 4 × 2 = 0 + 0.080 717 671 819 759 896 546 508 8;
  • 71) 0.080 717 671 819 759 896 546 508 8 × 2 = 0 + 0.161 435 343 639 519 793 093 017 6;
  • 72) 0.161 435 343 639 519 793 093 017 6 × 2 = 0 + 0.322 870 687 279 039 586 186 035 2;
  • 73) 0.322 870 687 279 039 586 186 035 2 × 2 = 0 + 0.645 741 374 558 079 172 372 070 4;
  • 74) 0.645 741 374 558 079 172 372 070 4 × 2 = 1 + 0.291 482 749 116 158 344 744 140 8;
  • 75) 0.291 482 749 116 158 344 744 140 8 × 2 = 0 + 0.582 965 498 232 316 689 488 281 6;
  • 76) 0.582 965 498 232 316 689 488 281 6 × 2 = 1 + 0.165 930 996 464 633 378 976 563 2;
  • 77) 0.165 930 996 464 633 378 976 563 2 × 2 = 0 + 0.331 861 992 929 266 757 953 126 4;
  • 78) 0.331 861 992 929 266 757 953 126 4 × 2 = 0 + 0.663 723 985 858 533 515 906 252 8;
  • 79) 0.663 723 985 858 533 515 906 252 8 × 2 = 1 + 0.327 447 971 717 067 031 812 505 6;
  • 80) 0.327 447 971 717 067 031 812 505 6 × 2 = 0 + 0.654 895 943 434 134 063 625 011 2;
  • 81) 0.654 895 943 434 134 063 625 011 2 × 2 = 1 + 0.309 791 886 868 268 127 250 022 4;
  • 82) 0.309 791 886 868 268 127 250 022 4 × 2 = 0 + 0.619 583 773 736 536 254 500 044 8;
  • 83) 0.619 583 773 736 536 254 500 044 8 × 2 = 1 + 0.239 167 547 473 072 509 000 089 6;
  • 84) 0.239 167 547 473 072 509 000 089 6 × 2 = 0 + 0.478 335 094 946 145 018 000 179 2;
  • 85) 0.478 335 094 946 145 018 000 179 2 × 2 = 0 + 0.956 670 189 892 290 036 000 358 4;
  • 86) 0.956 670 189 892 290 036 000 358 4 × 2 = 1 + 0.913 340 379 784 580 072 000 716 8;
  • 87) 0.913 340 379 784 580 072 000 716 8 × 2 = 1 + 0.826 680 759 569 160 144 001 433 6;
  • 88) 0.826 680 759 569 160 144 001 433 6 × 2 = 1 + 0.653 361 519 138 320 288 002 867 2;
  • 89) 0.653 361 519 138 320 288 002 867 2 × 2 = 1 + 0.306 723 038 276 640 576 005 734 4;
  • 90) 0.306 723 038 276 640 576 005 734 4 × 2 = 0 + 0.613 446 076 553 281 152 011 468 8;
  • 91) 0.613 446 076 553 281 152 011 468 8 × 2 = 1 + 0.226 892 153 106 562 304 022 937 6;
  • 92) 0.226 892 153 106 562 304 022 937 6 × 2 = 0 + 0.453 784 306 213 124 608 045 875 2;
  • 93) 0.453 784 306 213 124 608 045 875 2 × 2 = 0 + 0.907 568 612 426 249 216 091 750 4;
  • 94) 0.907 568 612 426 249 216 091 750 4 × 2 = 1 + 0.815 137 224 852 498 432 183 500 8;
  • 95) 0.815 137 224 852 498 432 183 500 8 × 2 = 1 + 0.630 274 449 704 996 864 367 001 6;
  • 96) 0.630 274 449 704 996 864 367 001 6 × 2 = 1 + 0.260 548 899 409 993 728 734 003 2;
  • 97) 0.260 548 899 409 993 728 734 003 2 × 2 = 0 + 0.521 097 798 819 987 457 468 006 4;
  • 98) 0.521 097 798 819 987 457 468 006 4 × 2 = 1 + 0.042 195 597 639 974 914 936 012 8;
  • 99) 0.042 195 597 639 974 914 936 012 8 × 2 = 0 + 0.084 391 195 279 949 829 872 025 6;
  • 100) 0.084 391 195 279 949 829 872 025 6 × 2 = 0 + 0.168 782 390 559 899 659 744 051 2;
  • 101) 0.168 782 390 559 899 659 744 051 2 × 2 = 0 + 0.337 564 781 119 799 319 488 102 4;
  • 102) 0.337 564 781 119 799 319 488 102 4 × 2 = 0 + 0.675 129 562 239 598 638 976 204 8;
  • 103) 0.675 129 562 239 598 638 976 204 8 × 2 = 1 + 0.350 259 124 479 197 277 952 409 6;
  • 104) 0.350 259 124 479 197 277 952 409 6 × 2 = 0 + 0.700 518 248 958 394 555 904 819 2;
  • 105) 0.700 518 248 958 394 555 904 819 2 × 2 = 1 + 0.401 036 497 916 789 111 809 638 4;
  • 106) 0.401 036 497 916 789 111 809 638 4 × 2 = 0 + 0.802 072 995 833 578 223 619 276 8;
  • 107) 0.802 072 995 833 578 223 619 276 8 × 2 = 1 + 0.604 145 991 667 156 447 238 553 6;
  • 108) 0.604 145 991 667 156 447 238 553 6 × 2 = 1 + 0.208 291 983 334 312 894 477 107 2;
  • 109) 0.208 291 983 334 312 894 477 107 2 × 2 = 0 + 0.416 583 966 668 625 788 954 214 4;
  • 110) 0.416 583 966 668 625 788 954 214 4 × 2 = 0 + 0.833 167 933 337 251 577 908 428 8;
  • 111) 0.833 167 933 337 251 577 908 428 8 × 2 = 1 + 0.666 335 866 674 503 155 816 857 6;
  • 112) 0.666 335 866 674 503 155 816 857 6 × 2 = 1 + 0.332 671 733 349 006 311 633 715 2;
  • 113) 0.332 671 733 349 006 311 633 715 2 × 2 = 0 + 0.665 343 466 698 012 623 267 430 4;
  • 114) 0.665 343 466 698 012 623 267 430 4 × 2 = 1 + 0.330 686 933 396 025 246 534 860 8;
  • 115) 0.330 686 933 396 025 246 534 860 8 × 2 = 0 + 0.661 373 866 792 050 493 069 721 6;
  • 116) 0.661 373 866 792 050 493 069 721 6 × 2 = 1 + 0.322 747 733 584 100 986 139 443 2;
  • 117) 0.322 747 733 584 100 986 139 443 2 × 2 = 0 + 0.645 495 467 168 201 972 278 886 4;
  • 118) 0.645 495 467 168 201 972 278 886 4 × 2 = 1 + 0.290 990 934 336 403 944 557 772 8;
  • 119) 0.290 990 934 336 403 944 557 772 8 × 2 = 0 + 0.581 981 868 672 807 889 115 545 6;

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:


0.000 000 000 000 000 000 008 538 7(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0101 0010 1010 0111 1010 0111 0100 0010 1011 0011 0101 010(2)

5. Positive number before normalization:

0.000 000 000 000 000 000 008 538 7(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0101 0010 1010 0111 1010 0111 0100 0010 1011 0011 0101 010(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:


0.000 000 000 000 000 000 008 538 7(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0101 0010 1010 0111 1010 0111 0100 0010 1011 0011 0101 010(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0101 0010 1010 0111 1010 0111 0100 0010 1011 0011 0101 010(2) × 20 =

1.0100 0010 1001 0101 0011 1101 0011 1010 0001 0101 1001 1010 1010(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 1001 0101 0011 1101 0011 1010 0001 0101 1001 1010 1010


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 1001 0101 0011 1101 0011 1010 0001 0101 1001 1010 1010 =

0100 0010 1001 0101 0011 1101 0011 1010 0001 0101 1001 1010 1010


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 1001 0101 0011 1101 0011 1010 0001 0101 1001 1010 1010


Decimal number 0.000 000 000 000 000 000 008 538 7 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1011 1100 - 0100 0010 1001 0101 0011 1101 0011 1010 0001 0101 1001 1010 1010

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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