0.000 000 000 000 000 000 008 537 35 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 000 000 008 537 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
0.000 000 000 000 000 000 008 537 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: 0.000 000 000 000 000 000 008 537 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) 0.000 000 000 000 000 000 008 537 35 × 2 = 0 + 0.000 000 000 000 000 000 017 074 7;
  • 2) 0.000 000 000 000 000 000 017 074 7 × 2 = 0 + 0.000 000 000 000 000 000 034 149 4;
  • 3) 0.000 000 000 000 000 000 034 149 4 × 2 = 0 + 0.000 000 000 000 000 000 068 298 8;
  • 4) 0.000 000 000 000 000 000 068 298 8 × 2 = 0 + 0.000 000 000 000 000 000 136 597 6;
  • 5) 0.000 000 000 000 000 000 136 597 6 × 2 = 0 + 0.000 000 000 000 000 000 273 195 2;
  • 6) 0.000 000 000 000 000 000 273 195 2 × 2 = 0 + 0.000 000 000 000 000 000 546 390 4;
  • 7) 0.000 000 000 000 000 000 546 390 4 × 2 = 0 + 0.000 000 000 000 000 001 092 780 8;
  • 8) 0.000 000 000 000 000 001 092 780 8 × 2 = 0 + 0.000 000 000 000 000 002 185 561 6;
  • 9) 0.000 000 000 000 000 002 185 561 6 × 2 = 0 + 0.000 000 000 000 000 004 371 123 2;
  • 10) 0.000 000 000 000 000 004 371 123 2 × 2 = 0 + 0.000 000 000 000 000 008 742 246 4;
  • 11) 0.000 000 000 000 000 008 742 246 4 × 2 = 0 + 0.000 000 000 000 000 017 484 492 8;
  • 12) 0.000 000 000 000 000 017 484 492 8 × 2 = 0 + 0.000 000 000 000 000 034 968 985 6;
  • 13) 0.000 000 000 000 000 034 968 985 6 × 2 = 0 + 0.000 000 000 000 000 069 937 971 2;
  • 14) 0.000 000 000 000 000 069 937 971 2 × 2 = 0 + 0.000 000 000 000 000 139 875 942 4;
  • 15) 0.000 000 000 000 000 139 875 942 4 × 2 = 0 + 0.000 000 000 000 000 279 751 884 8;
  • 16) 0.000 000 000 000 000 279 751 884 8 × 2 = 0 + 0.000 000 000 000 000 559 503 769 6;
  • 17) 0.000 000 000 000 000 559 503 769 6 × 2 = 0 + 0.000 000 000 000 001 119 007 539 2;
  • 18) 0.000 000 000 000 001 119 007 539 2 × 2 = 0 + 0.000 000 000 000 002 238 015 078 4;
  • 19) 0.000 000 000 000 002 238 015 078 4 × 2 = 0 + 0.000 000 000 000 004 476 030 156 8;
  • 20) 0.000 000 000 000 004 476 030 156 8 × 2 = 0 + 0.000 000 000 000 008 952 060 313 6;
  • 21) 0.000 000 000 000 008 952 060 313 6 × 2 = 0 + 0.000 000 000 000 017 904 120 627 2;
  • 22) 0.000 000 000 000 017 904 120 627 2 × 2 = 0 + 0.000 000 000 000 035 808 241 254 4;
  • 23) 0.000 000 000 000 035 808 241 254 4 × 2 = 0 + 0.000 000 000 000 071 616 482 508 8;
  • 24) 0.000 000 000 000 071 616 482 508 8 × 2 = 0 + 0.000 000 000 000 143 232 965 017 6;
  • 25) 0.000 000 000 000 143 232 965 017 6 × 2 = 0 + 0.000 000 000 000 286 465 930 035 2;
  • 26) 0.000 000 000 000 286 465 930 035 2 × 2 = 0 + 0.000 000 000 000 572 931 860 070 4;
  • 27) 0.000 000 000 000 572 931 860 070 4 × 2 = 0 + 0.000 000 000 001 145 863 720 140 8;
  • 28) 0.000 000 000 001 145 863 720 140 8 × 2 = 0 + 0.000 000 000 002 291 727 440 281 6;
  • 29) 0.000 000 000 002 291 727 440 281 6 × 2 = 0 + 0.000 000 000 004 583 454 880 563 2;
  • 30) 0.000 000 000 004 583 454 880 563 2 × 2 = 0 + 0.000 000 000 009 166 909 761 126 4;
  • 31) 0.000 000 000 009 166 909 761 126 4 × 2 = 0 + 0.000 000 000 018 333 819 522 252 8;
  • 32) 0.000 000 000 018 333 819 522 252 8 × 2 = 0 + 0.000 000 000 036 667 639 044 505 6;
  • 33) 0.000 000 000 036 667 639 044 505 6 × 2 = 0 + 0.000 000 000 073 335 278 089 011 2;
  • 34) 0.000 000 000 073 335 278 089 011 2 × 2 = 0 + 0.000 000 000 146 670 556 178 022 4;
  • 35) 0.000 000 000 146 670 556 178 022 4 × 2 = 0 + 0.000 000 000 293 341 112 356 044 8;
  • 36) 0.000 000 000 293 341 112 356 044 8 × 2 = 0 + 0.000 000 000 586 682 224 712 089 6;
  • 37) 0.000 000 000 586 682 224 712 089 6 × 2 = 0 + 0.000 000 001 173 364 449 424 179 2;
  • 38) 0.000 000 001 173 364 449 424 179 2 × 2 = 0 + 0.000 000 002 346 728 898 848 358 4;
  • 39) 0.000 000 002 346 728 898 848 358 4 × 2 = 0 + 0.000 000 004 693 457 797 696 716 8;
  • 40) 0.000 000 004 693 457 797 696 716 8 × 2 = 0 + 0.000 000 009 386 915 595 393 433 6;
  • 41) 0.000 000 009 386 915 595 393 433 6 × 2 = 0 + 0.000 000 018 773 831 190 786 867 2;
  • 42) 0.000 000 018 773 831 190 786 867 2 × 2 = 0 + 0.000 000 037 547 662 381 573 734 4;
  • 43) 0.000 000 037 547 662 381 573 734 4 × 2 = 0 + 0.000 000 075 095 324 763 147 468 8;
  • 44) 0.000 000 075 095 324 763 147 468 8 × 2 = 0 + 0.000 000 150 190 649 526 294 937 6;
  • 45) 0.000 000 150 190 649 526 294 937 6 × 2 = 0 + 0.000 000 300 381 299 052 589 875 2;
  • 46) 0.000 000 300 381 299 052 589 875 2 × 2 = 0 + 0.000 000 600 762 598 105 179 750 4;
  • 47) 0.000 000 600 762 598 105 179 750 4 × 2 = 0 + 0.000 001 201 525 196 210 359 500 8;
  • 48) 0.000 001 201 525 196 210 359 500 8 × 2 = 0 + 0.000 002 403 050 392 420 719 001 6;
  • 49) 0.000 002 403 050 392 420 719 001 6 × 2 = 0 + 0.000 004 806 100 784 841 438 003 2;
  • 50) 0.000 004 806 100 784 841 438 003 2 × 2 = 0 + 0.000 009 612 201 569 682 876 006 4;
  • 51) 0.000 009 612 201 569 682 876 006 4 × 2 = 0 + 0.000 019 224 403 139 365 752 012 8;
  • 52) 0.000 019 224 403 139 365 752 012 8 × 2 = 0 + 0.000 038 448 806 278 731 504 025 6;
  • 53) 0.000 038 448 806 278 731 504 025 6 × 2 = 0 + 0.000 076 897 612 557 463 008 051 2;
  • 54) 0.000 076 897 612 557 463 008 051 2 × 2 = 0 + 0.000 153 795 225 114 926 016 102 4;
  • 55) 0.000 153 795 225 114 926 016 102 4 × 2 = 0 + 0.000 307 590 450 229 852 032 204 8;
  • 56) 0.000 307 590 450 229 852 032 204 8 × 2 = 0 + 0.000 615 180 900 459 704 064 409 6;
  • 57) 0.000 615 180 900 459 704 064 409 6 × 2 = 0 + 0.001 230 361 800 919 408 128 819 2;
  • 58) 0.001 230 361 800 919 408 128 819 2 × 2 = 0 + 0.002 460 723 601 838 816 257 638 4;
  • 59) 0.002 460 723 601 838 816 257 638 4 × 2 = 0 + 0.004 921 447 203 677 632 515 276 8;
  • 60) 0.004 921 447 203 677 632 515 276 8 × 2 = 0 + 0.009 842 894 407 355 265 030 553 6;
  • 61) 0.009 842 894 407 355 265 030 553 6 × 2 = 0 + 0.019 685 788 814 710 530 061 107 2;
  • 62) 0.019 685 788 814 710 530 061 107 2 × 2 = 0 + 0.039 371 577 629 421 060 122 214 4;
  • 63) 0.039 371 577 629 421 060 122 214 4 × 2 = 0 + 0.078 743 155 258 842 120 244 428 8;
  • 64) 0.078 743 155 258 842 120 244 428 8 × 2 = 0 + 0.157 486 310 517 684 240 488 857 6;
  • 65) 0.157 486 310 517 684 240 488 857 6 × 2 = 0 + 0.314 972 621 035 368 480 977 715 2;
  • 66) 0.314 972 621 035 368 480 977 715 2 × 2 = 0 + 0.629 945 242 070 736 961 955 430 4;
  • 67) 0.629 945 242 070 736 961 955 430 4 × 2 = 1 + 0.259 890 484 141 473 923 910 860 8;
  • 68) 0.259 890 484 141 473 923 910 860 8 × 2 = 0 + 0.519 780 968 282 947 847 821 721 6;
  • 69) 0.519 780 968 282 947 847 821 721 6 × 2 = 1 + 0.039 561 936 565 895 695 643 443 2;
  • 70) 0.039 561 936 565 895 695 643 443 2 × 2 = 0 + 0.079 123 873 131 791 391 286 886 4;
  • 71) 0.079 123 873 131 791 391 286 886 4 × 2 = 0 + 0.158 247 746 263 582 782 573 772 8;
  • 72) 0.158 247 746 263 582 782 573 772 8 × 2 = 0 + 0.316 495 492 527 165 565 147 545 6;
  • 73) 0.316 495 492 527 165 565 147 545 6 × 2 = 0 + 0.632 990 985 054 331 130 295 091 2;
  • 74) 0.632 990 985 054 331 130 295 091 2 × 2 = 1 + 0.265 981 970 108 662 260 590 182 4;
  • 75) 0.265 981 970 108 662 260 590 182 4 × 2 = 0 + 0.531 963 940 217 324 521 180 364 8;
  • 76) 0.531 963 940 217 324 521 180 364 8 × 2 = 1 + 0.063 927 880 434 649 042 360 729 6;
  • 77) 0.063 927 880 434 649 042 360 729 6 × 2 = 0 + 0.127 855 760 869 298 084 721 459 2;
  • 78) 0.127 855 760 869 298 084 721 459 2 × 2 = 0 + 0.255 711 521 738 596 169 442 918 4;
  • 79) 0.255 711 521 738 596 169 442 918 4 × 2 = 0 + 0.511 423 043 477 192 338 885 836 8;
  • 80) 0.511 423 043 477 192 338 885 836 8 × 2 = 1 + 0.022 846 086 954 384 677 771 673 6;
  • 81) 0.022 846 086 954 384 677 771 673 6 × 2 = 0 + 0.045 692 173 908 769 355 543 347 2;
  • 82) 0.045 692 173 908 769 355 543 347 2 × 2 = 0 + 0.091 384 347 817 538 711 086 694 4;
  • 83) 0.091 384 347 817 538 711 086 694 4 × 2 = 0 + 0.182 768 695 635 077 422 173 388 8;
  • 84) 0.182 768 695 635 077 422 173 388 8 × 2 = 0 + 0.365 537 391 270 154 844 346 777 6;
  • 85) 0.365 537 391 270 154 844 346 777 6 × 2 = 0 + 0.731 074 782 540 309 688 693 555 2;
  • 86) 0.731 074 782 540 309 688 693 555 2 × 2 = 1 + 0.462 149 565 080 619 377 387 110 4;
  • 87) 0.462 149 565 080 619 377 387 110 4 × 2 = 0 + 0.924 299 130 161 238 754 774 220 8;
  • 88) 0.924 299 130 161 238 754 774 220 8 × 2 = 1 + 0.848 598 260 322 477 509 548 441 6;
  • 89) 0.848 598 260 322 477 509 548 441 6 × 2 = 1 + 0.697 196 520 644 955 019 096 883 2;
  • 90) 0.697 196 520 644 955 019 096 883 2 × 2 = 1 + 0.394 393 041 289 910 038 193 766 4;
  • 91) 0.394 393 041 289 910 038 193 766 4 × 2 = 0 + 0.788 786 082 579 820 076 387 532 8;
  • 92) 0.788 786 082 579 820 076 387 532 8 × 2 = 1 + 0.577 572 165 159 640 152 775 065 6;
  • 93) 0.577 572 165 159 640 152 775 065 6 × 2 = 1 + 0.155 144 330 319 280 305 550 131 2;
  • 94) 0.155 144 330 319 280 305 550 131 2 × 2 = 0 + 0.310 288 660 638 560 611 100 262 4;
  • 95) 0.310 288 660 638 560 611 100 262 4 × 2 = 0 + 0.620 577 321 277 121 222 200 524 8;
  • 96) 0.620 577 321 277 121 222 200 524 8 × 2 = 1 + 0.241 154 642 554 242 444 401 049 6;
  • 97) 0.241 154 642 554 242 444 401 049 6 × 2 = 0 + 0.482 309 285 108 484 888 802 099 2;
  • 98) 0.482 309 285 108 484 888 802 099 2 × 2 = 0 + 0.964 618 570 216 969 777 604 198 4;
  • 99) 0.964 618 570 216 969 777 604 198 4 × 2 = 1 + 0.929 237 140 433 939 555 208 396 8;
  • 100) 0.929 237 140 433 939 555 208 396 8 × 2 = 1 + 0.858 474 280 867 879 110 416 793 6;
  • 101) 0.858 474 280 867 879 110 416 793 6 × 2 = 1 + 0.716 948 561 735 758 220 833 587 2;
  • 102) 0.716 948 561 735 758 220 833 587 2 × 2 = 1 + 0.433 897 123 471 516 441 667 174 4;
  • 103) 0.433 897 123 471 516 441 667 174 4 × 2 = 0 + 0.867 794 246 943 032 883 334 348 8;
  • 104) 0.867 794 246 943 032 883 334 348 8 × 2 = 1 + 0.735 588 493 886 065 766 668 697 6;
  • 105) 0.735 588 493 886 065 766 668 697 6 × 2 = 1 + 0.471 176 987 772 131 533 337 395 2;
  • 106) 0.471 176 987 772 131 533 337 395 2 × 2 = 0 + 0.942 353 975 544 263 066 674 790 4;
  • 107) 0.942 353 975 544 263 066 674 790 4 × 2 = 1 + 0.884 707 951 088 526 133 349 580 8;
  • 108) 0.884 707 951 088 526 133 349 580 8 × 2 = 1 + 0.769 415 902 177 052 266 699 161 6;
  • 109) 0.769 415 902 177 052 266 699 161 6 × 2 = 1 + 0.538 831 804 354 104 533 398 323 2;
  • 110) 0.538 831 804 354 104 533 398 323 2 × 2 = 1 + 0.077 663 608 708 209 066 796 646 4;
  • 111) 0.077 663 608 708 209 066 796 646 4 × 2 = 0 + 0.155 327 217 416 418 133 593 292 8;
  • 112) 0.155 327 217 416 418 133 593 292 8 × 2 = 0 + 0.310 654 434 832 836 267 186 585 6;
  • 113) 0.310 654 434 832 836 267 186 585 6 × 2 = 0 + 0.621 308 869 665 672 534 373 171 2;
  • 114) 0.621 308 869 665 672 534 373 171 2 × 2 = 1 + 0.242 617 739 331 345 068 746 342 4;
  • 115) 0.242 617 739 331 345 068 746 342 4 × 2 = 0 + 0.485 235 478 662 690 137 492 684 8;
  • 116) 0.485 235 478 662 690 137 492 684 8 × 2 = 0 + 0.970 470 957 325 380 274 985 369 6;
  • 117) 0.970 470 957 325 380 274 985 369 6 × 2 = 1 + 0.940 941 914 650 760 549 970 739 2;
  • 118) 0.940 941 914 650 760 549 970 739 2 × 2 = 1 + 0.881 883 829 301 521 099 941 478 4;
  • 119) 0.881 883 829 301 521 099 941 478 4 × 2 = 1 + 0.763 767 658 603 042 199 882 956 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:


0.000 000 000 000 000 000 008 537 35(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0101 0001 0000 0101 1101 1001 0011 1101 1011 1100 0100 111(2)

5. Positive number before normalization:

0.000 000 000 000 000 000 008 537 35(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0101 0001 0000 0101 1101 1001 0011 1101 1011 1100 0100 111(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 35(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0101 0001 0000 0101 1101 1001 0011 1101 1011 1100 0100 111(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0101 0001 0000 0101 1101 1001 0011 1101 1011 1100 0100 111(2) × 20 =

1.0100 0010 1000 1000 0010 1110 1100 1001 1110 1101 1110 0010 0111(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 1000 0010 1110 1100 1001 1110 1101 1110 0010 0111


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 1000 0010 1110 1100 1001 1110 1101 1110 0010 0111 =

0100 0010 1000 1000 0010 1110 1100 1001 1110 1101 1110 0010 0111


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 1000 0010 1110 1100 1001 1110 1101 1110 0010 0111


Decimal number 0.000 000 000 000 000 000 008 537 35 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1011 1100 - 0100 0010 1000 1000 0010 1110 1100 1001 1110 1101 1110 0010 0111

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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