1.618 033 988 749 894 848 204 586 834 365 638 117 720 309 179 805 762 861 82 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 1.618 033 988 749 894 848 204 586 834 365 638 117 720 309 179 805 762 861 82(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
1.618 033 988 749 894 848 204 586 834 365 638 117 720 309 179 805 762 861 82(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: 0.618 033 988 749 894 848 204 586 834 365 638 117 720 309 179 805 762 861 82.

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) 0.618 033 988 749 894 848 204 586 834 365 638 117 720 309 179 805 762 861 82 × 2 = 1 + 0.236 067 977 499 789 696 409 173 668 731 276 235 440 618 359 611 525 723 64;
  • 2) 0.236 067 977 499 789 696 409 173 668 731 276 235 440 618 359 611 525 723 64 × 2 = 0 + 0.472 135 954 999 579 392 818 347 337 462 552 470 881 236 719 223 051 447 28;
  • 3) 0.472 135 954 999 579 392 818 347 337 462 552 470 881 236 719 223 051 447 28 × 2 = 0 + 0.944 271 909 999 158 785 636 694 674 925 104 941 762 473 438 446 102 894 56;
  • 4) 0.944 271 909 999 158 785 636 694 674 925 104 941 762 473 438 446 102 894 56 × 2 = 1 + 0.888 543 819 998 317 571 273 389 349 850 209 883 524 946 876 892 205 789 12;
  • 5) 0.888 543 819 998 317 571 273 389 349 850 209 883 524 946 876 892 205 789 12 × 2 = 1 + 0.777 087 639 996 635 142 546 778 699 700 419 767 049 893 753 784 411 578 24;
  • 6) 0.777 087 639 996 635 142 546 778 699 700 419 767 049 893 753 784 411 578 24 × 2 = 1 + 0.554 175 279 993 270 285 093 557 399 400 839 534 099 787 507 568 823 156 48;
  • 7) 0.554 175 279 993 270 285 093 557 399 400 839 534 099 787 507 568 823 156 48 × 2 = 1 + 0.108 350 559 986 540 570 187 114 798 801 679 068 199 575 015 137 646 312 96;
  • 8) 0.108 350 559 986 540 570 187 114 798 801 679 068 199 575 015 137 646 312 96 × 2 = 0 + 0.216 701 119 973 081 140 374 229 597 603 358 136 399 150 030 275 292 625 92;
  • 9) 0.216 701 119 973 081 140 374 229 597 603 358 136 399 150 030 275 292 625 92 × 2 = 0 + 0.433 402 239 946 162 280 748 459 195 206 716 272 798 300 060 550 585 251 84;
  • 10) 0.433 402 239 946 162 280 748 459 195 206 716 272 798 300 060 550 585 251 84 × 2 = 0 + 0.866 804 479 892 324 561 496 918 390 413 432 545 596 600 121 101 170 503 68;
  • 11) 0.866 804 479 892 324 561 496 918 390 413 432 545 596 600 121 101 170 503 68 × 2 = 1 + 0.733 608 959 784 649 122 993 836 780 826 865 091 193 200 242 202 341 007 36;
  • 12) 0.733 608 959 784 649 122 993 836 780 826 865 091 193 200 242 202 341 007 36 × 2 = 1 + 0.467 217 919 569 298 245 987 673 561 653 730 182 386 400 484 404 682 014 72;
  • 13) 0.467 217 919 569 298 245 987 673 561 653 730 182 386 400 484 404 682 014 72 × 2 = 0 + 0.934 435 839 138 596 491 975 347 123 307 460 364 772 800 968 809 364 029 44;
  • 14) 0.934 435 839 138 596 491 975 347 123 307 460 364 772 800 968 809 364 029 44 × 2 = 1 + 0.868 871 678 277 192 983 950 694 246 614 920 729 545 601 937 618 728 058 88;
  • 15) 0.868 871 678 277 192 983 950 694 246 614 920 729 545 601 937 618 728 058 88 × 2 = 1 + 0.737 743 356 554 385 967 901 388 493 229 841 459 091 203 875 237 456 117 76;
  • 16) 0.737 743 356 554 385 967 901 388 493 229 841 459 091 203 875 237 456 117 76 × 2 = 1 + 0.475 486 713 108 771 935 802 776 986 459 682 918 182 407 750 474 912 235 52;
  • 17) 0.475 486 713 108 771 935 802 776 986 459 682 918 182 407 750 474 912 235 52 × 2 = 0 + 0.950 973 426 217 543 871 605 553 972 919 365 836 364 815 500 949 824 471 04;
  • 18) 0.950 973 426 217 543 871 605 553 972 919 365 836 364 815 500 949 824 471 04 × 2 = 1 + 0.901 946 852 435 087 743 211 107 945 838 731 672 729 631 001 899 648 942 08;
  • 19) 0.901 946 852 435 087 743 211 107 945 838 731 672 729 631 001 899 648 942 08 × 2 = 1 + 0.803 893 704 870 175 486 422 215 891 677 463 345 459 262 003 799 297 884 16;
  • 20) 0.803 893 704 870 175 486 422 215 891 677 463 345 459 262 003 799 297 884 16 × 2 = 1 + 0.607 787 409 740 350 972 844 431 783 354 926 690 918 524 007 598 595 768 32;
  • 21) 0.607 787 409 740 350 972 844 431 783 354 926 690 918 524 007 598 595 768 32 × 2 = 1 + 0.215 574 819 480 701 945 688 863 566 709 853 381 837 048 015 197 191 536 64;
  • 22) 0.215 574 819 480 701 945 688 863 566 709 853 381 837 048 015 197 191 536 64 × 2 = 0 + 0.431 149 638 961 403 891 377 727 133 419 706 763 674 096 030 394 383 073 28;
  • 23) 0.431 149 638 961 403 891 377 727 133 419 706 763 674 096 030 394 383 073 28 × 2 = 0 + 0.862 299 277 922 807 782 755 454 266 839 413 527 348 192 060 788 766 146 56;
  • 24) 0.862 299 277 922 807 782 755 454 266 839 413 527 348 192 060 788 766 146 56 × 2 = 1 + 0.724 598 555 845 615 565 510 908 533 678 827 054 696 384 121 577 532 293 12;
  • 25) 0.724 598 555 845 615 565 510 908 533 678 827 054 696 384 121 577 532 293 12 × 2 = 1 + 0.449 197 111 691 231 131 021 817 067 357 654 109 392 768 243 155 064 586 24;
  • 26) 0.449 197 111 691 231 131 021 817 067 357 654 109 392 768 243 155 064 586 24 × 2 = 0 + 0.898 394 223 382 462 262 043 634 134 715 308 218 785 536 486 310 129 172 48;
  • 27) 0.898 394 223 382 462 262 043 634 134 715 308 218 785 536 486 310 129 172 48 × 2 = 1 + 0.796 788 446 764 924 524 087 268 269 430 616 437 571 072 972 620 258 344 96;
  • 28) 0.796 788 446 764 924 524 087 268 269 430 616 437 571 072 972 620 258 344 96 × 2 = 1 + 0.593 576 893 529 849 048 174 536 538 861 232 875 142 145 945 240 516 689 92;
  • 29) 0.593 576 893 529 849 048 174 536 538 861 232 875 142 145 945 240 516 689 92 × 2 = 1 + 0.187 153 787 059 698 096 349 073 077 722 465 750 284 291 890 481 033 379 84;
  • 30) 0.187 153 787 059 698 096 349 073 077 722 465 750 284 291 890 481 033 379 84 × 2 = 0 + 0.374 307 574 119 396 192 698 146 155 444 931 500 568 583 780 962 066 759 68;
  • 31) 0.374 307 574 119 396 192 698 146 155 444 931 500 568 583 780 962 066 759 68 × 2 = 0 + 0.748 615 148 238 792 385 396 292 310 889 863 001 137 167 561 924 133 519 36;
  • 32) 0.748 615 148 238 792 385 396 292 310 889 863 001 137 167 561 924 133 519 36 × 2 = 1 + 0.497 230 296 477 584 770 792 584 621 779 726 002 274 335 123 848 267 038 72;
  • 33) 0.497 230 296 477 584 770 792 584 621 779 726 002 274 335 123 848 267 038 72 × 2 = 0 + 0.994 460 592 955 169 541 585 169 243 559 452 004 548 670 247 696 534 077 44;
  • 34) 0.994 460 592 955 169 541 585 169 243 559 452 004 548 670 247 696 534 077 44 × 2 = 1 + 0.988 921 185 910 339 083 170 338 487 118 904 009 097 340 495 393 068 154 88;
  • 35) 0.988 921 185 910 339 083 170 338 487 118 904 009 097 340 495 393 068 154 88 × 2 = 1 + 0.977 842 371 820 678 166 340 676 974 237 808 018 194 680 990 786 136 309 76;
  • 36) 0.977 842 371 820 678 166 340 676 974 237 808 018 194 680 990 786 136 309 76 × 2 = 1 + 0.955 684 743 641 356 332 681 353 948 475 616 036 389 361 981 572 272 619 52;
  • 37) 0.955 684 743 641 356 332 681 353 948 475 616 036 389 361 981 572 272 619 52 × 2 = 1 + 0.911 369 487 282 712 665 362 707 896 951 232 072 778 723 963 144 545 239 04;
  • 38) 0.911 369 487 282 712 665 362 707 896 951 232 072 778 723 963 144 545 239 04 × 2 = 1 + 0.822 738 974 565 425 330 725 415 793 902 464 145 557 447 926 289 090 478 08;
  • 39) 0.822 738 974 565 425 330 725 415 793 902 464 145 557 447 926 289 090 478 08 × 2 = 1 + 0.645 477 949 130 850 661 450 831 587 804 928 291 114 895 852 578 180 956 16;
  • 40) 0.645 477 949 130 850 661 450 831 587 804 928 291 114 895 852 578 180 956 16 × 2 = 1 + 0.290 955 898 261 701 322 901 663 175 609 856 582 229 791 705 156 361 912 32;
  • 41) 0.290 955 898 261 701 322 901 663 175 609 856 582 229 791 705 156 361 912 32 × 2 = 0 + 0.581 911 796 523 402 645 803 326 351 219 713 164 459 583 410 312 723 824 64;
  • 42) 0.581 911 796 523 402 645 803 326 351 219 713 164 459 583 410 312 723 824 64 × 2 = 1 + 0.163 823 593 046 805 291 606 652 702 439 426 328 919 166 820 625 447 649 28;
  • 43) 0.163 823 593 046 805 291 606 652 702 439 426 328 919 166 820 625 447 649 28 × 2 = 0 + 0.327 647 186 093 610 583 213 305 404 878 852 657 838 333 641 250 895 298 56;
  • 44) 0.327 647 186 093 610 583 213 305 404 878 852 657 838 333 641 250 895 298 56 × 2 = 0 + 0.655 294 372 187 221 166 426 610 809 757 705 315 676 667 282 501 790 597 12;
  • 45) 0.655 294 372 187 221 166 426 610 809 757 705 315 676 667 282 501 790 597 12 × 2 = 1 + 0.310 588 744 374 442 332 853 221 619 515 410 631 353 334 565 003 581 194 24;
  • 46) 0.310 588 744 374 442 332 853 221 619 515 410 631 353 334 565 003 581 194 24 × 2 = 0 + 0.621 177 488 748 884 665 706 443 239 030 821 262 706 669 130 007 162 388 48;
  • 47) 0.621 177 488 748 884 665 706 443 239 030 821 262 706 669 130 007 162 388 48 × 2 = 1 + 0.242 354 977 497 769 331 412 886 478 061 642 525 413 338 260 014 324 776 96;
  • 48) 0.242 354 977 497 769 331 412 886 478 061 642 525 413 338 260 014 324 776 96 × 2 = 0 + 0.484 709 954 995 538 662 825 772 956 123 285 050 826 676 520 028 649 553 92;
  • 49) 0.484 709 954 995 538 662 825 772 956 123 285 050 826 676 520 028 649 553 92 × 2 = 0 + 0.969 419 909 991 077 325 651 545 912 246 570 101 653 353 040 057 299 107 84;
  • 50) 0.969 419 909 991 077 325 651 545 912 246 570 101 653 353 040 057 299 107 84 × 2 = 1 + 0.938 839 819 982 154 651 303 091 824 493 140 203 306 706 080 114 598 215 68;
  • 51) 0.938 839 819 982 154 651 303 091 824 493 140 203 306 706 080 114 598 215 68 × 2 = 1 + 0.877 679 639 964 309 302 606 183 648 986 280 406 613 412 160 229 196 431 36;
  • 52) 0.877 679 639 964 309 302 606 183 648 986 280 406 613 412 160 229 196 431 36 × 2 = 1 + 0.755 359 279 928 618 605 212 367 297 972 560 813 226 824 320 458 392 862 72;
  • 53) 0.755 359 279 928 618 605 212 367 297 972 560 813 226 824 320 458 392 862 72 × 2 = 1 + 0.510 718 559 857 237 210 424 734 595 945 121 626 453 648 640 916 785 725 44;

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:


0.618 033 988 749 894 848 204 586 834 365 638 117 720 309 179 805 762 861 82(10) =

0.1001 1110 0011 0111 0111 1001 1011 1001 0111 1111 0100 1010 0111 1(2)

5. Positive number before normalization:

1.618 033 988 749 894 848 204 586 834 365 638 117 720 309 179 805 762 861 82(10) =

1.1001 1110 0011 0111 0111 1001 1011 1001 0111 1111 0100 1010 0111 1(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:


1.618 033 988 749 894 848 204 586 834 365 638 117 720 309 179 805 762 861 82(10) =

1.1001 1110 0011 0111 0111 1001 1011 1001 0111 1111 0100 1010 0111 1(2) =

1.1001 1110 0011 0111 0111 1001 1011 1001 0111 1111 0100 1010 0111 1(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):
1.1001 1110 0011 0111 0111 1001 1011 1001 0111 1111 0100 1010 0111 1


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. 1001 1110 0011 0111 0111 1001 1011 1001 0111 1111 0100 1010 0111 1 =

1001 1110 0011 0111 0111 1001 1011 1001 0111 1111 0100 1010 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 1111 1111

Mantissa (52 bits) =
1001 1110 0011 0111 0111 1001 1011 1001 0111 1111 0100 1010 0111


Decimal number 1.618 033 988 749 894 848 204 586 834 365 638 117 720 309 179 805 762 861 82 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1111 1111 - 1001 1110 0011 0111 0111 1001 1011 1001 0111 1111 0100 1010 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