-0.000 000 349 613 904 800 000 238 907 549 083 000 015 8 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.000 000 349 613 904 800 000 238 907 549 083 000 015 8(10) to 64 bit double precision IEEE 754 binary floating point representation standard (1 bit for sign, 11 bits for exponent, 52 bits for mantissa)

What are the steps to convert decimal number
-0.000 000 349 613 904 800 000 238 907 549 083 000 015 8(10) to 64 bit double precision IEEE 754 binary floating point representation (1 bit for sign, 11 bits for exponent, 52 bits for mantissa)

1. Start with the positive version of the number:

|-0.000 000 349 613 904 800 000 238 907 549 083 000 015 8| = 0.000 000 349 613 904 800 000 238 907 549 083 000 015 8


2. 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;

3. 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)


4. Convert to binary (base 2) the fractional part: 0.000 000 349 613 904 800 000 238 907 549 083 000 015 8.

Multiply it repeatedly by 2.

Keep track of each integer part of the results.

Stop when we get a fractional part that is equal to zero.


  • #) multiplying = integer + fractional part;
  • 1) 0.000 000 349 613 904 800 000 238 907 549 083 000 015 8 × 2 = 0 + 0.000 000 699 227 809 600 000 477 815 098 166 000 031 6;
  • 2) 0.000 000 699 227 809 600 000 477 815 098 166 000 031 6 × 2 = 0 + 0.000 001 398 455 619 200 000 955 630 196 332 000 063 2;
  • 3) 0.000 001 398 455 619 200 000 955 630 196 332 000 063 2 × 2 = 0 + 0.000 002 796 911 238 400 001 911 260 392 664 000 126 4;
  • 4) 0.000 002 796 911 238 400 001 911 260 392 664 000 126 4 × 2 = 0 + 0.000 005 593 822 476 800 003 822 520 785 328 000 252 8;
  • 5) 0.000 005 593 822 476 800 003 822 520 785 328 000 252 8 × 2 = 0 + 0.000 011 187 644 953 600 007 645 041 570 656 000 505 6;
  • 6) 0.000 011 187 644 953 600 007 645 041 570 656 000 505 6 × 2 = 0 + 0.000 022 375 289 907 200 015 290 083 141 312 001 011 2;
  • 7) 0.000 022 375 289 907 200 015 290 083 141 312 001 011 2 × 2 = 0 + 0.000 044 750 579 814 400 030 580 166 282 624 002 022 4;
  • 8) 0.000 044 750 579 814 400 030 580 166 282 624 002 022 4 × 2 = 0 + 0.000 089 501 159 628 800 061 160 332 565 248 004 044 8;
  • 9) 0.000 089 501 159 628 800 061 160 332 565 248 004 044 8 × 2 = 0 + 0.000 179 002 319 257 600 122 320 665 130 496 008 089 6;
  • 10) 0.000 179 002 319 257 600 122 320 665 130 496 008 089 6 × 2 = 0 + 0.000 358 004 638 515 200 244 641 330 260 992 016 179 2;
  • 11) 0.000 358 004 638 515 200 244 641 330 260 992 016 179 2 × 2 = 0 + 0.000 716 009 277 030 400 489 282 660 521 984 032 358 4;
  • 12) 0.000 716 009 277 030 400 489 282 660 521 984 032 358 4 × 2 = 0 + 0.001 432 018 554 060 800 978 565 321 043 968 064 716 8;
  • 13) 0.001 432 018 554 060 800 978 565 321 043 968 064 716 8 × 2 = 0 + 0.002 864 037 108 121 601 957 130 642 087 936 129 433 6;
  • 14) 0.002 864 037 108 121 601 957 130 642 087 936 129 433 6 × 2 = 0 + 0.005 728 074 216 243 203 914 261 284 175 872 258 867 2;
  • 15) 0.005 728 074 216 243 203 914 261 284 175 872 258 867 2 × 2 = 0 + 0.011 456 148 432 486 407 828 522 568 351 744 517 734 4;
  • 16) 0.011 456 148 432 486 407 828 522 568 351 744 517 734 4 × 2 = 0 + 0.022 912 296 864 972 815 657 045 136 703 489 035 468 8;
  • 17) 0.022 912 296 864 972 815 657 045 136 703 489 035 468 8 × 2 = 0 + 0.045 824 593 729 945 631 314 090 273 406 978 070 937 6;
  • 18) 0.045 824 593 729 945 631 314 090 273 406 978 070 937 6 × 2 = 0 + 0.091 649 187 459 891 262 628 180 546 813 956 141 875 2;
  • 19) 0.091 649 187 459 891 262 628 180 546 813 956 141 875 2 × 2 = 0 + 0.183 298 374 919 782 525 256 361 093 627 912 283 750 4;
  • 20) 0.183 298 374 919 782 525 256 361 093 627 912 283 750 4 × 2 = 0 + 0.366 596 749 839 565 050 512 722 187 255 824 567 500 8;
  • 21) 0.366 596 749 839 565 050 512 722 187 255 824 567 500 8 × 2 = 0 + 0.733 193 499 679 130 101 025 444 374 511 649 135 001 6;
  • 22) 0.733 193 499 679 130 101 025 444 374 511 649 135 001 6 × 2 = 1 + 0.466 386 999 358 260 202 050 888 749 023 298 270 003 2;
  • 23) 0.466 386 999 358 260 202 050 888 749 023 298 270 003 2 × 2 = 0 + 0.932 773 998 716 520 404 101 777 498 046 596 540 006 4;
  • 24) 0.932 773 998 716 520 404 101 777 498 046 596 540 006 4 × 2 = 1 + 0.865 547 997 433 040 808 203 554 996 093 193 080 012 8;
  • 25) 0.865 547 997 433 040 808 203 554 996 093 193 080 012 8 × 2 = 1 + 0.731 095 994 866 081 616 407 109 992 186 386 160 025 6;
  • 26) 0.731 095 994 866 081 616 407 109 992 186 386 160 025 6 × 2 = 1 + 0.462 191 989 732 163 232 814 219 984 372 772 320 051 2;
  • 27) 0.462 191 989 732 163 232 814 219 984 372 772 320 051 2 × 2 = 0 + 0.924 383 979 464 326 465 628 439 968 745 544 640 102 4;
  • 28) 0.924 383 979 464 326 465 628 439 968 745 544 640 102 4 × 2 = 1 + 0.848 767 958 928 652 931 256 879 937 491 089 280 204 8;
  • 29) 0.848 767 958 928 652 931 256 879 937 491 089 280 204 8 × 2 = 1 + 0.697 535 917 857 305 862 513 759 874 982 178 560 409 6;
  • 30) 0.697 535 917 857 305 862 513 759 874 982 178 560 409 6 × 2 = 1 + 0.395 071 835 714 611 725 027 519 749 964 357 120 819 2;
  • 31) 0.395 071 835 714 611 725 027 519 749 964 357 120 819 2 × 2 = 0 + 0.790 143 671 429 223 450 055 039 499 928 714 241 638 4;
  • 32) 0.790 143 671 429 223 450 055 039 499 928 714 241 638 4 × 2 = 1 + 0.580 287 342 858 446 900 110 078 999 857 428 483 276 8;
  • 33) 0.580 287 342 858 446 900 110 078 999 857 428 483 276 8 × 2 = 1 + 0.160 574 685 716 893 800 220 157 999 714 856 966 553 6;
  • 34) 0.160 574 685 716 893 800 220 157 999 714 856 966 553 6 × 2 = 0 + 0.321 149 371 433 787 600 440 315 999 429 713 933 107 2;
  • 35) 0.321 149 371 433 787 600 440 315 999 429 713 933 107 2 × 2 = 0 + 0.642 298 742 867 575 200 880 631 998 859 427 866 214 4;
  • 36) 0.642 298 742 867 575 200 880 631 998 859 427 866 214 4 × 2 = 1 + 0.284 597 485 735 150 401 761 263 997 718 855 732 428 8;
  • 37) 0.284 597 485 735 150 401 761 263 997 718 855 732 428 8 × 2 = 0 + 0.569 194 971 470 300 803 522 527 995 437 711 464 857 6;
  • 38) 0.569 194 971 470 300 803 522 527 995 437 711 464 857 6 × 2 = 1 + 0.138 389 942 940 601 607 045 055 990 875 422 929 715 2;
  • 39) 0.138 389 942 940 601 607 045 055 990 875 422 929 715 2 × 2 = 0 + 0.276 779 885 881 203 214 090 111 981 750 845 859 430 4;
  • 40) 0.276 779 885 881 203 214 090 111 981 750 845 859 430 4 × 2 = 0 + 0.553 559 771 762 406 428 180 223 963 501 691 718 860 8;
  • 41) 0.553 559 771 762 406 428 180 223 963 501 691 718 860 8 × 2 = 1 + 0.107 119 543 524 812 856 360 447 927 003 383 437 721 6;
  • 42) 0.107 119 543 524 812 856 360 447 927 003 383 437 721 6 × 2 = 0 + 0.214 239 087 049 625 712 720 895 854 006 766 875 443 2;
  • 43) 0.214 239 087 049 625 712 720 895 854 006 766 875 443 2 × 2 = 0 + 0.428 478 174 099 251 425 441 791 708 013 533 750 886 4;
  • 44) 0.428 478 174 099 251 425 441 791 708 013 533 750 886 4 × 2 = 0 + 0.856 956 348 198 502 850 883 583 416 027 067 501 772 8;
  • 45) 0.856 956 348 198 502 850 883 583 416 027 067 501 772 8 × 2 = 1 + 0.713 912 696 397 005 701 767 166 832 054 135 003 545 6;
  • 46) 0.713 912 696 397 005 701 767 166 832 054 135 003 545 6 × 2 = 1 + 0.427 825 392 794 011 403 534 333 664 108 270 007 091 2;
  • 47) 0.427 825 392 794 011 403 534 333 664 108 270 007 091 2 × 2 = 0 + 0.855 650 785 588 022 807 068 667 328 216 540 014 182 4;
  • 48) 0.855 650 785 588 022 807 068 667 328 216 540 014 182 4 × 2 = 1 + 0.711 301 571 176 045 614 137 334 656 433 080 028 364 8;
  • 49) 0.711 301 571 176 045 614 137 334 656 433 080 028 364 8 × 2 = 1 + 0.422 603 142 352 091 228 274 669 312 866 160 056 729 6;
  • 50) 0.422 603 142 352 091 228 274 669 312 866 160 056 729 6 × 2 = 0 + 0.845 206 284 704 182 456 549 338 625 732 320 113 459 2;
  • 51) 0.845 206 284 704 182 456 549 338 625 732 320 113 459 2 × 2 = 1 + 0.690 412 569 408 364 913 098 677 251 464 640 226 918 4;
  • 52) 0.690 412 569 408 364 913 098 677 251 464 640 226 918 4 × 2 = 1 + 0.380 825 138 816 729 826 197 354 502 929 280 453 836 8;
  • 53) 0.380 825 138 816 729 826 197 354 502 929 280 453 836 8 × 2 = 0 + 0.761 650 277 633 459 652 394 709 005 858 560 907 673 6;
  • 54) 0.761 650 277 633 459 652 394 709 005 858 560 907 673 6 × 2 = 1 + 0.523 300 555 266 919 304 789 418 011 717 121 815 347 2;
  • 55) 0.523 300 555 266 919 304 789 418 011 717 121 815 347 2 × 2 = 1 + 0.046 601 110 533 838 609 578 836 023 434 243 630 694 4;
  • 56) 0.046 601 110 533 838 609 578 836 023 434 243 630 694 4 × 2 = 0 + 0.093 202 221 067 677 219 157 672 046 868 487 261 388 8;
  • 57) 0.093 202 221 067 677 219 157 672 046 868 487 261 388 8 × 2 = 0 + 0.186 404 442 135 354 438 315 344 093 736 974 522 777 6;
  • 58) 0.186 404 442 135 354 438 315 344 093 736 974 522 777 6 × 2 = 0 + 0.372 808 884 270 708 876 630 688 187 473 949 045 555 2;
  • 59) 0.372 808 884 270 708 876 630 688 187 473 949 045 555 2 × 2 = 0 + 0.745 617 768 541 417 753 261 376 374 947 898 091 110 4;
  • 60) 0.745 617 768 541 417 753 261 376 374 947 898 091 110 4 × 2 = 1 + 0.491 235 537 082 835 506 522 752 749 895 796 182 220 8;
  • 61) 0.491 235 537 082 835 506 522 752 749 895 796 182 220 8 × 2 = 0 + 0.982 471 074 165 671 013 045 505 499 791 592 364 441 6;
  • 62) 0.982 471 074 165 671 013 045 505 499 791 592 364 441 6 × 2 = 1 + 0.964 942 148 331 342 026 091 010 999 583 184 728 883 2;
  • 63) 0.964 942 148 331 342 026 091 010 999 583 184 728 883 2 × 2 = 1 + 0.929 884 296 662 684 052 182 021 999 166 369 457 766 4;
  • 64) 0.929 884 296 662 684 052 182 021 999 166 369 457 766 4 × 2 = 1 + 0.859 768 593 325 368 104 364 043 998 332 738 915 532 8;
  • 65) 0.859 768 593 325 368 104 364 043 998 332 738 915 532 8 × 2 = 1 + 0.719 537 186 650 736 208 728 087 996 665 477 831 065 6;
  • 66) 0.719 537 186 650 736 208 728 087 996 665 477 831 065 6 × 2 = 1 + 0.439 074 373 301 472 417 456 175 993 330 955 662 131 2;
  • 67) 0.439 074 373 301 472 417 456 175 993 330 955 662 131 2 × 2 = 0 + 0.878 148 746 602 944 834 912 351 986 661 911 324 262 4;
  • 68) 0.878 148 746 602 944 834 912 351 986 661 911 324 262 4 × 2 = 1 + 0.756 297 493 205 889 669 824 703 973 323 822 648 524 8;
  • 69) 0.756 297 493 205 889 669 824 703 973 323 822 648 524 8 × 2 = 1 + 0.512 594 986 411 779 339 649 407 946 647 645 297 049 6;
  • 70) 0.512 594 986 411 779 339 649 407 946 647 645 297 049 6 × 2 = 1 + 0.025 189 972 823 558 679 298 815 893 295 290 594 099 2;
  • 71) 0.025 189 972 823 558 679 298 815 893 295 290 594 099 2 × 2 = 0 + 0.050 379 945 647 117 358 597 631 786 590 581 188 198 4;
  • 72) 0.050 379 945 647 117 358 597 631 786 590 581 188 198 4 × 2 = 0 + 0.100 759 891 294 234 717 195 263 573 181 162 376 396 8;
  • 73) 0.100 759 891 294 234 717 195 263 573 181 162 376 396 8 × 2 = 0 + 0.201 519 782 588 469 434 390 527 146 362 324 752 793 6;
  • 74) 0.201 519 782 588 469 434 390 527 146 362 324 752 793 6 × 2 = 0 + 0.403 039 565 176 938 868 781 054 292 724 649 505 587 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).


5. 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:


0.000 000 349 613 904 800 000 238 907 549 083 000 015 8(10) =

0.0000 0000 0000 0000 0000 0101 1101 1101 1001 0100 1000 1101 1011 0110 0001 0111 1101 1100 00(2)

6. Positive number before normalization:

0.000 000 349 613 904 800 000 238 907 549 083 000 015 8(10) =

0.0000 0000 0000 0000 0000 0101 1101 1101 1001 0100 1000 1101 1011 0110 0001 0111 1101 1100 00(2)

7. Normalize the binary representation of the number.

Shift the decimal mark 22 positions to the right, so that only one non zero digit remains to the left of it:


0.000 000 349 613 904 800 000 238 907 549 083 000 015 8(10) =

0.0000 0000 0000 0000 0000 0101 1101 1101 1001 0100 1000 1101 1011 0110 0001 0111 1101 1100 00(2) =

0.0000 0000 0000 0000 0000 0101 1101 1101 1001 0100 1000 1101 1011 0110 0001 0111 1101 1100 00(2) × 20 =

1.0111 0111 0110 0101 0010 0011 0110 1101 1000 0101 1111 0111 0000(2) × 2-22


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): -22

Mantissa (not normalized):
1.0111 0111 0110 0101 0010 0011 0110 1101 1000 0101 1111 0111 0000


9. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

-22 + 2(11-1) - 1 =

(-22 + 1 023)(10) =

1 001(10)


10. 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;
  • 1 001 ÷ 2 = 500 + 1;
  • 500 ÷ 2 = 250 + 0;
  • 250 ÷ 2 = 125 + 0;
  • 125 ÷ 2 = 62 + 1;
  • 62 ÷ 2 = 31 + 0;
  • 31 ÷ 2 = 15 + 1;
  • 15 ÷ 2 = 7 + 1;
  • 7 ÷ 2 = 3 + 1;
  • 3 ÷ 2 = 1 + 1;
  • 1 ÷ 2 = 0 + 1;

11. Construct the base 2 representation of the adjusted exponent.

Take all the remainders starting from the bottom of the list constructed above.


Exponent (adjusted) =

1001(10) =

011 1110 1001(2)


12. 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. 0111 0111 0110 0101 0010 0011 0110 1101 1000 0101 1111 0111 0000 =

0111 0111 0110 0101 0010 0011 0110 1101 1000 0101 1111 0111 0000


13. The three elements that make up the number's 64 bit double precision IEEE 754 binary floating point representation:

Sign (1 bit) =
1 (a negative number)

Exponent (11 bits) =
011 1110 1001

Mantissa (52 bits) =
0111 0111 0110 0101 0010 0011 0110 1101 1000 0101 1111 0111 0000


Decimal number -0.000 000 349 613 904 800 000 238 907 549 083 000 015 8 converted to 64 bit double precision IEEE 754 binary floating point representation:

1 - 011 1110 1001 - 0111 0111 0110 0101 0010 0011 0110 1101 1000 0101 1111 0111 0000

» Convert decimal -0.000 000 349 613 904 800 000 238 907 549 083 000 013 to 64 bit double precision IEEE 754 binary floating point representation standard

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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