-0.000 000 000 000 000 222 044 604 925 052 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.000 000 000 000 000 222 044 604 925 052(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
-0.000 000 000 000 000 222 044 604 925 052(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:

|-0.000 000 000 000 000 222 044 604 925 052| = 0.000 000 000 000 000 222 044 604 925 052


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: 0.000 000 000 000 000 222 044 604 925 052.

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) 0.000 000 000 000 000 222 044 604 925 052 × 2 = 0 + 0.000 000 000 000 000 444 089 209 850 104;
  • 2) 0.000 000 000 000 000 444 089 209 850 104 × 2 = 0 + 0.000 000 000 000 000 888 178 419 700 208;
  • 3) 0.000 000 000 000 000 888 178 419 700 208 × 2 = 0 + 0.000 000 000 000 001 776 356 839 400 416;
  • 4) 0.000 000 000 000 001 776 356 839 400 416 × 2 = 0 + 0.000 000 000 000 003 552 713 678 800 832;
  • 5) 0.000 000 000 000 003 552 713 678 800 832 × 2 = 0 + 0.000 000 000 000 007 105 427 357 601 664;
  • 6) 0.000 000 000 000 007 105 427 357 601 664 × 2 = 0 + 0.000 000 000 000 014 210 854 715 203 328;
  • 7) 0.000 000 000 000 014 210 854 715 203 328 × 2 = 0 + 0.000 000 000 000 028 421 709 430 406 656;
  • 8) 0.000 000 000 000 028 421 709 430 406 656 × 2 = 0 + 0.000 000 000 000 056 843 418 860 813 312;
  • 9) 0.000 000 000 000 056 843 418 860 813 312 × 2 = 0 + 0.000 000 000 000 113 686 837 721 626 624;
  • 10) 0.000 000 000 000 113 686 837 721 626 624 × 2 = 0 + 0.000 000 000 000 227 373 675 443 253 248;
  • 11) 0.000 000 000 000 227 373 675 443 253 248 × 2 = 0 + 0.000 000 000 000 454 747 350 886 506 496;
  • 12) 0.000 000 000 000 454 747 350 886 506 496 × 2 = 0 + 0.000 000 000 000 909 494 701 773 012 992;
  • 13) 0.000 000 000 000 909 494 701 773 012 992 × 2 = 0 + 0.000 000 000 001 818 989 403 546 025 984;
  • 14) 0.000 000 000 001 818 989 403 546 025 984 × 2 = 0 + 0.000 000 000 003 637 978 807 092 051 968;
  • 15) 0.000 000 000 003 637 978 807 092 051 968 × 2 = 0 + 0.000 000 000 007 275 957 614 184 103 936;
  • 16) 0.000 000 000 007 275 957 614 184 103 936 × 2 = 0 + 0.000 000 000 014 551 915 228 368 207 872;
  • 17) 0.000 000 000 014 551 915 228 368 207 872 × 2 = 0 + 0.000 000 000 029 103 830 456 736 415 744;
  • 18) 0.000 000 000 029 103 830 456 736 415 744 × 2 = 0 + 0.000 000 000 058 207 660 913 472 831 488;
  • 19) 0.000 000 000 058 207 660 913 472 831 488 × 2 = 0 + 0.000 000 000 116 415 321 826 945 662 976;
  • 20) 0.000 000 000 116 415 321 826 945 662 976 × 2 = 0 + 0.000 000 000 232 830 643 653 891 325 952;
  • 21) 0.000 000 000 232 830 643 653 891 325 952 × 2 = 0 + 0.000 000 000 465 661 287 307 782 651 904;
  • 22) 0.000 000 000 465 661 287 307 782 651 904 × 2 = 0 + 0.000 000 000 931 322 574 615 565 303 808;
  • 23) 0.000 000 000 931 322 574 615 565 303 808 × 2 = 0 + 0.000 000 001 862 645 149 231 130 607 616;
  • 24) 0.000 000 001 862 645 149 231 130 607 616 × 2 = 0 + 0.000 000 003 725 290 298 462 261 215 232;
  • 25) 0.000 000 003 725 290 298 462 261 215 232 × 2 = 0 + 0.000 000 007 450 580 596 924 522 430 464;
  • 26) 0.000 000 007 450 580 596 924 522 430 464 × 2 = 0 + 0.000 000 014 901 161 193 849 044 860 928;
  • 27) 0.000 000 014 901 161 193 849 044 860 928 × 2 = 0 + 0.000 000 029 802 322 387 698 089 721 856;
  • 28) 0.000 000 029 802 322 387 698 089 721 856 × 2 = 0 + 0.000 000 059 604 644 775 396 179 443 712;
  • 29) 0.000 000 059 604 644 775 396 179 443 712 × 2 = 0 + 0.000 000 119 209 289 550 792 358 887 424;
  • 30) 0.000 000 119 209 289 550 792 358 887 424 × 2 = 0 + 0.000 000 238 418 579 101 584 717 774 848;
  • 31) 0.000 000 238 418 579 101 584 717 774 848 × 2 = 0 + 0.000 000 476 837 158 203 169 435 549 696;
  • 32) 0.000 000 476 837 158 203 169 435 549 696 × 2 = 0 + 0.000 000 953 674 316 406 338 871 099 392;
  • 33) 0.000 000 953 674 316 406 338 871 099 392 × 2 = 0 + 0.000 001 907 348 632 812 677 742 198 784;
  • 34) 0.000 001 907 348 632 812 677 742 198 784 × 2 = 0 + 0.000 003 814 697 265 625 355 484 397 568;
  • 35) 0.000 003 814 697 265 625 355 484 397 568 × 2 = 0 + 0.000 007 629 394 531 250 710 968 795 136;
  • 36) 0.000 007 629 394 531 250 710 968 795 136 × 2 = 0 + 0.000 015 258 789 062 501 421 937 590 272;
  • 37) 0.000 015 258 789 062 501 421 937 590 272 × 2 = 0 + 0.000 030 517 578 125 002 843 875 180 544;
  • 38) 0.000 030 517 578 125 002 843 875 180 544 × 2 = 0 + 0.000 061 035 156 250 005 687 750 361 088;
  • 39) 0.000 061 035 156 250 005 687 750 361 088 × 2 = 0 + 0.000 122 070 312 500 011 375 500 722 176;
  • 40) 0.000 122 070 312 500 011 375 500 722 176 × 2 = 0 + 0.000 244 140 625 000 022 751 001 444 352;
  • 41) 0.000 244 140 625 000 022 751 001 444 352 × 2 = 0 + 0.000 488 281 250 000 045 502 002 888 704;
  • 42) 0.000 488 281 250 000 045 502 002 888 704 × 2 = 0 + 0.000 976 562 500 000 091 004 005 777 408;
  • 43) 0.000 976 562 500 000 091 004 005 777 408 × 2 = 0 + 0.001 953 125 000 000 182 008 011 554 816;
  • 44) 0.001 953 125 000 000 182 008 011 554 816 × 2 = 0 + 0.003 906 250 000 000 364 016 023 109 632;
  • 45) 0.003 906 250 000 000 364 016 023 109 632 × 2 = 0 + 0.007 812 500 000 000 728 032 046 219 264;
  • 46) 0.007 812 500 000 000 728 032 046 219 264 × 2 = 0 + 0.015 625 000 000 001 456 064 092 438 528;
  • 47) 0.015 625 000 000 001 456 064 092 438 528 × 2 = 0 + 0.031 250 000 000 002 912 128 184 877 056;
  • 48) 0.031 250 000 000 002 912 128 184 877 056 × 2 = 0 + 0.062 500 000 000 005 824 256 369 754 112;
  • 49) 0.062 500 000 000 005 824 256 369 754 112 × 2 = 0 + 0.125 000 000 000 011 648 512 739 508 224;
  • 50) 0.125 000 000 000 011 648 512 739 508 224 × 2 = 0 + 0.250 000 000 000 023 297 025 479 016 448;
  • 51) 0.250 000 000 000 023 297 025 479 016 448 × 2 = 0 + 0.500 000 000 000 046 594 050 958 032 896;
  • 52) 0.500 000 000 000 046 594 050 958 032 896 × 2 = 1 + 0.000 000 000 000 093 188 101 916 065 792;
  • 53) 0.000 000 000 000 093 188 101 916 065 792 × 2 = 0 + 0.000 000 000 000 186 376 203 832 131 584;
  • 54) 0.000 000 000 000 186 376 203 832 131 584 × 2 = 0 + 0.000 000 000 000 372 752 407 664 263 168;
  • 55) 0.000 000 000 000 372 752 407 664 263 168 × 2 = 0 + 0.000 000 000 000 745 504 815 328 526 336;
  • 56) 0.000 000 000 000 745 504 815 328 526 336 × 2 = 0 + 0.000 000 000 001 491 009 630 657 052 672;
  • 57) 0.000 000 000 001 491 009 630 657 052 672 × 2 = 0 + 0.000 000 000 002 982 019 261 314 105 344;
  • 58) 0.000 000 000 002 982 019 261 314 105 344 × 2 = 0 + 0.000 000 000 005 964 038 522 628 210 688;
  • 59) 0.000 000 000 005 964 038 522 628 210 688 × 2 = 0 + 0.000 000 000 011 928 077 045 256 421 376;
  • 60) 0.000 000 000 011 928 077 045 256 421 376 × 2 = 0 + 0.000 000 000 023 856 154 090 512 842 752;
  • 61) 0.000 000 000 023 856 154 090 512 842 752 × 2 = 0 + 0.000 000 000 047 712 308 181 025 685 504;
  • 62) 0.000 000 000 047 712 308 181 025 685 504 × 2 = 0 + 0.000 000 000 095 424 616 362 051 371 008;
  • 63) 0.000 000 000 095 424 616 362 051 371 008 × 2 = 0 + 0.000 000 000 190 849 232 724 102 742 016;
  • 64) 0.000 000 000 190 849 232 724 102 742 016 × 2 = 0 + 0.000 000 000 381 698 465 448 205 484 032;
  • 65) 0.000 000 000 381 698 465 448 205 484 032 × 2 = 0 + 0.000 000 000 763 396 930 896 410 968 064;
  • 66) 0.000 000 000 763 396 930 896 410 968 064 × 2 = 0 + 0.000 000 001 526 793 861 792 821 936 128;
  • 67) 0.000 000 001 526 793 861 792 821 936 128 × 2 = 0 + 0.000 000 003 053 587 723 585 643 872 256;
  • 68) 0.000 000 003 053 587 723 585 643 872 256 × 2 = 0 + 0.000 000 006 107 175 447 171 287 744 512;
  • 69) 0.000 000 006 107 175 447 171 287 744 512 × 2 = 0 + 0.000 000 012 214 350 894 342 575 489 024;
  • 70) 0.000 000 012 214 350 894 342 575 489 024 × 2 = 0 + 0.000 000 024 428 701 788 685 150 978 048;
  • 71) 0.000 000 024 428 701 788 685 150 978 048 × 2 = 0 + 0.000 000 048 857 403 577 370 301 956 096;
  • 72) 0.000 000 048 857 403 577 370 301 956 096 × 2 = 0 + 0.000 000 097 714 807 154 740 603 912 192;
  • 73) 0.000 000 097 714 807 154 740 603 912 192 × 2 = 0 + 0.000 000 195 429 614 309 481 207 824 384;
  • 74) 0.000 000 195 429 614 309 481 207 824 384 × 2 = 0 + 0.000 000 390 859 228 618 962 415 648 768;
  • 75) 0.000 000 390 859 228 618 962 415 648 768 × 2 = 0 + 0.000 000 781 718 457 237 924 831 297 536;
  • 76) 0.000 000 781 718 457 237 924 831 297 536 × 2 = 0 + 0.000 001 563 436 914 475 849 662 595 072;
  • 77) 0.000 001 563 436 914 475 849 662 595 072 × 2 = 0 + 0.000 003 126 873 828 951 699 325 190 144;
  • 78) 0.000 003 126 873 828 951 699 325 190 144 × 2 = 0 + 0.000 006 253 747 657 903 398 650 380 288;
  • 79) 0.000 006 253 747 657 903 398 650 380 288 × 2 = 0 + 0.000 012 507 495 315 806 797 300 760 576;
  • 80) 0.000 012 507 495 315 806 797 300 760 576 × 2 = 0 + 0.000 025 014 990 631 613 594 601 521 152;
  • 81) 0.000 025 014 990 631 613 594 601 521 152 × 2 = 0 + 0.000 050 029 981 263 227 189 203 042 304;
  • 82) 0.000 050 029 981 263 227 189 203 042 304 × 2 = 0 + 0.000 100 059 962 526 454 378 406 084 608;
  • 83) 0.000 100 059 962 526 454 378 406 084 608 × 2 = 0 + 0.000 200 119 925 052 908 756 812 169 216;
  • 84) 0.000 200 119 925 052 908 756 812 169 216 × 2 = 0 + 0.000 400 239 850 105 817 513 624 338 432;
  • 85) 0.000 400 239 850 105 817 513 624 338 432 × 2 = 0 + 0.000 800 479 700 211 635 027 248 676 864;
  • 86) 0.000 800 479 700 211 635 027 248 676 864 × 2 = 0 + 0.001 600 959 400 423 270 054 497 353 728;
  • 87) 0.001 600 959 400 423 270 054 497 353 728 × 2 = 0 + 0.003 201 918 800 846 540 108 994 707 456;
  • 88) 0.003 201 918 800 846 540 108 994 707 456 × 2 = 0 + 0.006 403 837 601 693 080 217 989 414 912;
  • 89) 0.006 403 837 601 693 080 217 989 414 912 × 2 = 0 + 0.012 807 675 203 386 160 435 978 829 824;
  • 90) 0.012 807 675 203 386 160 435 978 829 824 × 2 = 0 + 0.025 615 350 406 772 320 871 957 659 648;
  • 91) 0.025 615 350 406 772 320 871 957 659 648 × 2 = 0 + 0.051 230 700 813 544 641 743 915 319 296;
  • 92) 0.051 230 700 813 544 641 743 915 319 296 × 2 = 0 + 0.102 461 401 627 089 283 487 830 638 592;
  • 93) 0.102 461 401 627 089 283 487 830 638 592 × 2 = 0 + 0.204 922 803 254 178 566 975 661 277 184;
  • 94) 0.204 922 803 254 178 566 975 661 277 184 × 2 = 0 + 0.409 845 606 508 357 133 951 322 554 368;
  • 95) 0.409 845 606 508 357 133 951 322 554 368 × 2 = 0 + 0.819 691 213 016 714 267 902 645 108 736;
  • 96) 0.819 691 213 016 714 267 902 645 108 736 × 2 = 1 + 0.639 382 426 033 428 535 805 290 217 472;
  • 97) 0.639 382 426 033 428 535 805 290 217 472 × 2 = 1 + 0.278 764 852 066 857 071 610 580 434 944;
  • 98) 0.278 764 852 066 857 071 610 580 434 944 × 2 = 0 + 0.557 529 704 133 714 143 221 160 869 888;
  • 99) 0.557 529 704 133 714 143 221 160 869 888 × 2 = 1 + 0.115 059 408 267 428 286 442 321 739 776;
  • 100) 0.115 059 408 267 428 286 442 321 739 776 × 2 = 0 + 0.230 118 816 534 856 572 884 643 479 552;
  • 101) 0.230 118 816 534 856 572 884 643 479 552 × 2 = 0 + 0.460 237 633 069 713 145 769 286 959 104;
  • 102) 0.460 237 633 069 713 145 769 286 959 104 × 2 = 0 + 0.920 475 266 139 426 291 538 573 918 208;
  • 103) 0.920 475 266 139 426 291 538 573 918 208 × 2 = 1 + 0.840 950 532 278 852 583 077 147 836 416;
  • 104) 0.840 950 532 278 852 583 077 147 836 416 × 2 = 1 + 0.681 901 064 557 705 166 154 295 672 832;

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:


0.000 000 000 000 000 222 044 604 925 052(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0001 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0001 1010 0011(2)

6. Positive number before normalization:

0.000 000 000 000 000 222 044 604 925 052(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0001 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0001 1010 0011(2)

7. Normalize the binary representation of the number.

Shift the decimal mark 52 positions to the right, so that only one non zero digit remains to the left of it:


0.000 000 000 000 000 222 044 604 925 052(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0001 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0001 1010 0011(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0001 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0001 1010 0011(2) × 20 =

1.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0001 1010 0011(2) × 2-52


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): -52

Mantissa (not normalized):
1.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0001 1010 0011


9. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

-52 + 2(11-1) - 1 =

(-52 + 1 023)(10) =

971(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;
  • 971 ÷ 2 = 485 + 1;
  • 485 ÷ 2 = 242 + 1;
  • 242 ÷ 2 = 121 + 0;
  • 121 ÷ 2 = 60 + 1;
  • 60 ÷ 2 = 30 + 0;
  • 30 ÷ 2 = 15 + 0;
  • 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) =

971(10) =

011 1100 1011(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. 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0001 1010 0011 =

0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0001 1010 0011


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 1100 1011

Mantissa (52 bits) =
0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0001 1010 0011


Decimal number -0.000 000 000 000 000 222 044 604 925 052 converted to 64 bit double precision IEEE 754 binary floating point representation:

1 - 011 1100 1011 - 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0001 1010 0011

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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