1.161 834 032 785 773 095 167 300 531 840 609 646 036 411 516 475 047 407 229 817 563 318 642 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 1.161 834 032 785 773 095 167 300 531 840 609 646 036 411 516 475 047 407 229 817 563 318 642(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
1.161 834 032 785 773 095 167 300 531 840 609 646 036 411 516 475 047 407 229 817 563 318 642(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: 1.
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;
  • 1 ÷ 2 = 0 + 1;

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.

1(10) =

1(2)


3. Convert to binary (base 2) the fractional part: 0.161 834 032 785 773 095 167 300 531 840 609 646 036 411 516 475 047 407 229 817 563 318 642.

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) 0.161 834 032 785 773 095 167 300 531 840 609 646 036 411 516 475 047 407 229 817 563 318 642 × 2 = 0 + 0.323 668 065 571 546 190 334 601 063 681 219 292 072 823 032 950 094 814 459 635 126 637 284;
  • 2) 0.323 668 065 571 546 190 334 601 063 681 219 292 072 823 032 950 094 814 459 635 126 637 284 × 2 = 0 + 0.647 336 131 143 092 380 669 202 127 362 438 584 145 646 065 900 189 628 919 270 253 274 568;
  • 3) 0.647 336 131 143 092 380 669 202 127 362 438 584 145 646 065 900 189 628 919 270 253 274 568 × 2 = 1 + 0.294 672 262 286 184 761 338 404 254 724 877 168 291 292 131 800 379 257 838 540 506 549 136;
  • 4) 0.294 672 262 286 184 761 338 404 254 724 877 168 291 292 131 800 379 257 838 540 506 549 136 × 2 = 0 + 0.589 344 524 572 369 522 676 808 509 449 754 336 582 584 263 600 758 515 677 081 013 098 272;
  • 5) 0.589 344 524 572 369 522 676 808 509 449 754 336 582 584 263 600 758 515 677 081 013 098 272 × 2 = 1 + 0.178 689 049 144 739 045 353 617 018 899 508 673 165 168 527 201 517 031 354 162 026 196 544;
  • 6) 0.178 689 049 144 739 045 353 617 018 899 508 673 165 168 527 201 517 031 354 162 026 196 544 × 2 = 0 + 0.357 378 098 289 478 090 707 234 037 799 017 346 330 337 054 403 034 062 708 324 052 393 088;
  • 7) 0.357 378 098 289 478 090 707 234 037 799 017 346 330 337 054 403 034 062 708 324 052 393 088 × 2 = 0 + 0.714 756 196 578 956 181 414 468 075 598 034 692 660 674 108 806 068 125 416 648 104 786 176;
  • 8) 0.714 756 196 578 956 181 414 468 075 598 034 692 660 674 108 806 068 125 416 648 104 786 176 × 2 = 1 + 0.429 512 393 157 912 362 828 936 151 196 069 385 321 348 217 612 136 250 833 296 209 572 352;
  • 9) 0.429 512 393 157 912 362 828 936 151 196 069 385 321 348 217 612 136 250 833 296 209 572 352 × 2 = 0 + 0.859 024 786 315 824 725 657 872 302 392 138 770 642 696 435 224 272 501 666 592 419 144 704;
  • 10) 0.859 024 786 315 824 725 657 872 302 392 138 770 642 696 435 224 272 501 666 592 419 144 704 × 2 = 1 + 0.718 049 572 631 649 451 315 744 604 784 277 541 285 392 870 448 545 003 333 184 838 289 408;
  • 11) 0.718 049 572 631 649 451 315 744 604 784 277 541 285 392 870 448 545 003 333 184 838 289 408 × 2 = 1 + 0.436 099 145 263 298 902 631 489 209 568 555 082 570 785 740 897 090 006 666 369 676 578 816;
  • 12) 0.436 099 145 263 298 902 631 489 209 568 555 082 570 785 740 897 090 006 666 369 676 578 816 × 2 = 0 + 0.872 198 290 526 597 805 262 978 419 137 110 165 141 571 481 794 180 013 332 739 353 157 632;
  • 13) 0.872 198 290 526 597 805 262 978 419 137 110 165 141 571 481 794 180 013 332 739 353 157 632 × 2 = 1 + 0.744 396 581 053 195 610 525 956 838 274 220 330 283 142 963 588 360 026 665 478 706 315 264;
  • 14) 0.744 396 581 053 195 610 525 956 838 274 220 330 283 142 963 588 360 026 665 478 706 315 264 × 2 = 1 + 0.488 793 162 106 391 221 051 913 676 548 440 660 566 285 927 176 720 053 330 957 412 630 528;
  • 15) 0.488 793 162 106 391 221 051 913 676 548 440 660 566 285 927 176 720 053 330 957 412 630 528 × 2 = 0 + 0.977 586 324 212 782 442 103 827 353 096 881 321 132 571 854 353 440 106 661 914 825 261 056;
  • 16) 0.977 586 324 212 782 442 103 827 353 096 881 321 132 571 854 353 440 106 661 914 825 261 056 × 2 = 1 + 0.955 172 648 425 564 884 207 654 706 193 762 642 265 143 708 706 880 213 323 829 650 522 112;
  • 17) 0.955 172 648 425 564 884 207 654 706 193 762 642 265 143 708 706 880 213 323 829 650 522 112 × 2 = 1 + 0.910 345 296 851 129 768 415 309 412 387 525 284 530 287 417 413 760 426 647 659 301 044 224;
  • 18) 0.910 345 296 851 129 768 415 309 412 387 525 284 530 287 417 413 760 426 647 659 301 044 224 × 2 = 1 + 0.820 690 593 702 259 536 830 618 824 775 050 569 060 574 834 827 520 853 295 318 602 088 448;
  • 19) 0.820 690 593 702 259 536 830 618 824 775 050 569 060 574 834 827 520 853 295 318 602 088 448 × 2 = 1 + 0.641 381 187 404 519 073 661 237 649 550 101 138 121 149 669 655 041 706 590 637 204 176 896;
  • 20) 0.641 381 187 404 519 073 661 237 649 550 101 138 121 149 669 655 041 706 590 637 204 176 896 × 2 = 1 + 0.282 762 374 809 038 147 322 475 299 100 202 276 242 299 339 310 083 413 181 274 408 353 792;
  • 21) 0.282 762 374 809 038 147 322 475 299 100 202 276 242 299 339 310 083 413 181 274 408 353 792 × 2 = 0 + 0.565 524 749 618 076 294 644 950 598 200 404 552 484 598 678 620 166 826 362 548 816 707 584;
  • 22) 0.565 524 749 618 076 294 644 950 598 200 404 552 484 598 678 620 166 826 362 548 816 707 584 × 2 = 1 + 0.131 049 499 236 152 589 289 901 196 400 809 104 969 197 357 240 333 652 725 097 633 415 168;
  • 23) 0.131 049 499 236 152 589 289 901 196 400 809 104 969 197 357 240 333 652 725 097 633 415 168 × 2 = 0 + 0.262 098 998 472 305 178 579 802 392 801 618 209 938 394 714 480 667 305 450 195 266 830 336;
  • 24) 0.262 098 998 472 305 178 579 802 392 801 618 209 938 394 714 480 667 305 450 195 266 830 336 × 2 = 0 + 0.524 197 996 944 610 357 159 604 785 603 236 419 876 789 428 961 334 610 900 390 533 660 672;
  • 25) 0.524 197 996 944 610 357 159 604 785 603 236 419 876 789 428 961 334 610 900 390 533 660 672 × 2 = 1 + 0.048 395 993 889 220 714 319 209 571 206 472 839 753 578 857 922 669 221 800 781 067 321 344;
  • 26) 0.048 395 993 889 220 714 319 209 571 206 472 839 753 578 857 922 669 221 800 781 067 321 344 × 2 = 0 + 0.096 791 987 778 441 428 638 419 142 412 945 679 507 157 715 845 338 443 601 562 134 642 688;
  • 27) 0.096 791 987 778 441 428 638 419 142 412 945 679 507 157 715 845 338 443 601 562 134 642 688 × 2 = 0 + 0.193 583 975 556 882 857 276 838 284 825 891 359 014 315 431 690 676 887 203 124 269 285 376;
  • 28) 0.193 583 975 556 882 857 276 838 284 825 891 359 014 315 431 690 676 887 203 124 269 285 376 × 2 = 0 + 0.387 167 951 113 765 714 553 676 569 651 782 718 028 630 863 381 353 774 406 248 538 570 752;
  • 29) 0.387 167 951 113 765 714 553 676 569 651 782 718 028 630 863 381 353 774 406 248 538 570 752 × 2 = 0 + 0.774 335 902 227 531 429 107 353 139 303 565 436 057 261 726 762 707 548 812 497 077 141 504;
  • 30) 0.774 335 902 227 531 429 107 353 139 303 565 436 057 261 726 762 707 548 812 497 077 141 504 × 2 = 1 + 0.548 671 804 455 062 858 214 706 278 607 130 872 114 523 453 525 415 097 624 994 154 283 008;
  • 31) 0.548 671 804 455 062 858 214 706 278 607 130 872 114 523 453 525 415 097 624 994 154 283 008 × 2 = 1 + 0.097 343 608 910 125 716 429 412 557 214 261 744 229 046 907 050 830 195 249 988 308 566 016;
  • 32) 0.097 343 608 910 125 716 429 412 557 214 261 744 229 046 907 050 830 195 249 988 308 566 016 × 2 = 0 + 0.194 687 217 820 251 432 858 825 114 428 523 488 458 093 814 101 660 390 499 976 617 132 032;
  • 33) 0.194 687 217 820 251 432 858 825 114 428 523 488 458 093 814 101 660 390 499 976 617 132 032 × 2 = 0 + 0.389 374 435 640 502 865 717 650 228 857 046 976 916 187 628 203 320 780 999 953 234 264 064;
  • 34) 0.389 374 435 640 502 865 717 650 228 857 046 976 916 187 628 203 320 780 999 953 234 264 064 × 2 = 0 + 0.778 748 871 281 005 731 435 300 457 714 093 953 832 375 256 406 641 561 999 906 468 528 128;
  • 35) 0.778 748 871 281 005 731 435 300 457 714 093 953 832 375 256 406 641 561 999 906 468 528 128 × 2 = 1 + 0.557 497 742 562 011 462 870 600 915 428 187 907 664 750 512 813 283 123 999 812 937 056 256;
  • 36) 0.557 497 742 562 011 462 870 600 915 428 187 907 664 750 512 813 283 123 999 812 937 056 256 × 2 = 1 + 0.114 995 485 124 022 925 741 201 830 856 375 815 329 501 025 626 566 247 999 625 874 112 512;
  • 37) 0.114 995 485 124 022 925 741 201 830 856 375 815 329 501 025 626 566 247 999 625 874 112 512 × 2 = 0 + 0.229 990 970 248 045 851 482 403 661 712 751 630 659 002 051 253 132 495 999 251 748 225 024;
  • 38) 0.229 990 970 248 045 851 482 403 661 712 751 630 659 002 051 253 132 495 999 251 748 225 024 × 2 = 0 + 0.459 981 940 496 091 702 964 807 323 425 503 261 318 004 102 506 264 991 998 503 496 450 048;
  • 39) 0.459 981 940 496 091 702 964 807 323 425 503 261 318 004 102 506 264 991 998 503 496 450 048 × 2 = 0 + 0.919 963 880 992 183 405 929 614 646 851 006 522 636 008 205 012 529 983 997 006 992 900 096;
  • 40) 0.919 963 880 992 183 405 929 614 646 851 006 522 636 008 205 012 529 983 997 006 992 900 096 × 2 = 1 + 0.839 927 761 984 366 811 859 229 293 702 013 045 272 016 410 025 059 967 994 013 985 800 192;
  • 41) 0.839 927 761 984 366 811 859 229 293 702 013 045 272 016 410 025 059 967 994 013 985 800 192 × 2 = 1 + 0.679 855 523 968 733 623 718 458 587 404 026 090 544 032 820 050 119 935 988 027 971 600 384;
  • 42) 0.679 855 523 968 733 623 718 458 587 404 026 090 544 032 820 050 119 935 988 027 971 600 384 × 2 = 1 + 0.359 711 047 937 467 247 436 917 174 808 052 181 088 065 640 100 239 871 976 055 943 200 768;
  • 43) 0.359 711 047 937 467 247 436 917 174 808 052 181 088 065 640 100 239 871 976 055 943 200 768 × 2 = 0 + 0.719 422 095 874 934 494 873 834 349 616 104 362 176 131 280 200 479 743 952 111 886 401 536;
  • 44) 0.719 422 095 874 934 494 873 834 349 616 104 362 176 131 280 200 479 743 952 111 886 401 536 × 2 = 1 + 0.438 844 191 749 868 989 747 668 699 232 208 724 352 262 560 400 959 487 904 223 772 803 072;
  • 45) 0.438 844 191 749 868 989 747 668 699 232 208 724 352 262 560 400 959 487 904 223 772 803 072 × 2 = 0 + 0.877 688 383 499 737 979 495 337 398 464 417 448 704 525 120 801 918 975 808 447 545 606 144;
  • 46) 0.877 688 383 499 737 979 495 337 398 464 417 448 704 525 120 801 918 975 808 447 545 606 144 × 2 = 1 + 0.755 376 766 999 475 958 990 674 796 928 834 897 409 050 241 603 837 951 616 895 091 212 288;
  • 47) 0.755 376 766 999 475 958 990 674 796 928 834 897 409 050 241 603 837 951 616 895 091 212 288 × 2 = 1 + 0.510 753 533 998 951 917 981 349 593 857 669 794 818 100 483 207 675 903 233 790 182 424 576;
  • 48) 0.510 753 533 998 951 917 981 349 593 857 669 794 818 100 483 207 675 903 233 790 182 424 576 × 2 = 1 + 0.021 507 067 997 903 835 962 699 187 715 339 589 636 200 966 415 351 806 467 580 364 849 152;
  • 49) 0.021 507 067 997 903 835 962 699 187 715 339 589 636 200 966 415 351 806 467 580 364 849 152 × 2 = 0 + 0.043 014 135 995 807 671 925 398 375 430 679 179 272 401 932 830 703 612 935 160 729 698 304;
  • 50) 0.043 014 135 995 807 671 925 398 375 430 679 179 272 401 932 830 703 612 935 160 729 698 304 × 2 = 0 + 0.086 028 271 991 615 343 850 796 750 861 358 358 544 803 865 661 407 225 870 321 459 396 608;
  • 51) 0.086 028 271 991 615 343 850 796 750 861 358 358 544 803 865 661 407 225 870 321 459 396 608 × 2 = 0 + 0.172 056 543 983 230 687 701 593 501 722 716 717 089 607 731 322 814 451 740 642 918 793 216;
  • 52) 0.172 056 543 983 230 687 701 593 501 722 716 717 089 607 731 322 814 451 740 642 918 793 216 × 2 = 0 + 0.344 113 087 966 461 375 403 187 003 445 433 434 179 215 462 645 628 903 481 285 837 586 432;
  • 53) 0.344 113 087 966 461 375 403 187 003 445 433 434 179 215 462 645 628 903 481 285 837 586 432 × 2 = 0 + 0.688 226 175 932 922 750 806 374 006 890 866 868 358 430 925 291 257 806 962 571 675 172 864;

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:


0.161 834 032 785 773 095 167 300 531 840 609 646 036 411 516 475 047 407 229 817 563 318 642(10) =

0.0010 1001 0110 1101 1111 0100 1000 0110 0011 0001 1101 0111 0000 0(2)

5. Positive number before normalization:

1.161 834 032 785 773 095 167 300 531 840 609 646 036 411 516 475 047 407 229 817 563 318 642(10) =

1.0010 1001 0110 1101 1111 0100 1000 0110 0011 0001 1101 0111 0000 0(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 0 positions to the left, so that only one non zero digit remains to the left of it:


1.161 834 032 785 773 095 167 300 531 840 609 646 036 411 516 475 047 407 229 817 563 318 642(10) =

1.0010 1001 0110 1101 1111 0100 1000 0110 0011 0001 1101 0111 0000 0(2) =

1.0010 1001 0110 1101 1111 0100 1000 0110 0011 0001 1101 0111 0000 0(2) × 20


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): 0

Mantissa (not normalized):
1.0010 1001 0110 1101 1111 0100 1000 0110 0011 0001 1101 0111 0000 0


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

Exponent (unadjusted) + 2(11-1) - 1 =

0 + 2(11-1) - 1 =

(0 + 1 023)(10) =

1 023(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;
  • 1 023 ÷ 2 = 511 + 1;
  • 511 ÷ 2 = 255 + 1;
  • 255 ÷ 2 = 127 + 1;
  • 127 ÷ 2 = 63 + 1;
  • 63 ÷ 2 = 31 + 1;
  • 31 ÷ 2 = 15 + 1;
  • 15 ÷ 2 = 7 + 1;
  • 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) =

1023(10) =

011 1111 1111(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, by removing the excess bits, from the right (if any of the excess bits is set on 1, we are losing precision...).


Mantissa (normalized) =

1. 0010 1001 0110 1101 1111 0100 1000 0110 0011 0001 1101 0111 0000 0 =

0010 1001 0110 1101 1111 0100 1000 0110 0011 0001 1101 0111 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 1111 1111

Mantissa (52 bits) =
0010 1001 0110 1101 1111 0100 1000 0110 0011 0001 1101 0111 0000


Decimal number 1.161 834 032 785 773 095 167 300 531 840 609 646 036 411 516 475 047 407 229 817 563 318 642 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1111 1111 - 0010 1001 0110 1101 1111 0100 1000 0110 0011 0001 1101 0111 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