0.000 000 000 000 000 000 000 048 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 000 000 000 048(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
0.000 000 000 000 000 000 000 048(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: 0.000 000 000 000 000 000 000 048.

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) 0.000 000 000 000 000 000 000 048 × 2 = 0 + 0.000 000 000 000 000 000 000 096;
  • 2) 0.000 000 000 000 000 000 000 096 × 2 = 0 + 0.000 000 000 000 000 000 000 192;
  • 3) 0.000 000 000 000 000 000 000 192 × 2 = 0 + 0.000 000 000 000 000 000 000 384;
  • 4) 0.000 000 000 000 000 000 000 384 × 2 = 0 + 0.000 000 000 000 000 000 000 768;
  • 5) 0.000 000 000 000 000 000 000 768 × 2 = 0 + 0.000 000 000 000 000 000 001 536;
  • 6) 0.000 000 000 000 000 000 001 536 × 2 = 0 + 0.000 000 000 000 000 000 003 072;
  • 7) 0.000 000 000 000 000 000 003 072 × 2 = 0 + 0.000 000 000 000 000 000 006 144;
  • 8) 0.000 000 000 000 000 000 006 144 × 2 = 0 + 0.000 000 000 000 000 000 012 288;
  • 9) 0.000 000 000 000 000 000 012 288 × 2 = 0 + 0.000 000 000 000 000 000 024 576;
  • 10) 0.000 000 000 000 000 000 024 576 × 2 = 0 + 0.000 000 000 000 000 000 049 152;
  • 11) 0.000 000 000 000 000 000 049 152 × 2 = 0 + 0.000 000 000 000 000 000 098 304;
  • 12) 0.000 000 000 000 000 000 098 304 × 2 = 0 + 0.000 000 000 000 000 000 196 608;
  • 13) 0.000 000 000 000 000 000 196 608 × 2 = 0 + 0.000 000 000 000 000 000 393 216;
  • 14) 0.000 000 000 000 000 000 393 216 × 2 = 0 + 0.000 000 000 000 000 000 786 432;
  • 15) 0.000 000 000 000 000 000 786 432 × 2 = 0 + 0.000 000 000 000 000 001 572 864;
  • 16) 0.000 000 000 000 000 001 572 864 × 2 = 0 + 0.000 000 000 000 000 003 145 728;
  • 17) 0.000 000 000 000 000 003 145 728 × 2 = 0 + 0.000 000 000 000 000 006 291 456;
  • 18) 0.000 000 000 000 000 006 291 456 × 2 = 0 + 0.000 000 000 000 000 012 582 912;
  • 19) 0.000 000 000 000 000 012 582 912 × 2 = 0 + 0.000 000 000 000 000 025 165 824;
  • 20) 0.000 000 000 000 000 025 165 824 × 2 = 0 + 0.000 000 000 000 000 050 331 648;
  • 21) 0.000 000 000 000 000 050 331 648 × 2 = 0 + 0.000 000 000 000 000 100 663 296;
  • 22) 0.000 000 000 000 000 100 663 296 × 2 = 0 + 0.000 000 000 000 000 201 326 592;
  • 23) 0.000 000 000 000 000 201 326 592 × 2 = 0 + 0.000 000 000 000 000 402 653 184;
  • 24) 0.000 000 000 000 000 402 653 184 × 2 = 0 + 0.000 000 000 000 000 805 306 368;
  • 25) 0.000 000 000 000 000 805 306 368 × 2 = 0 + 0.000 000 000 000 001 610 612 736;
  • 26) 0.000 000 000 000 001 610 612 736 × 2 = 0 + 0.000 000 000 000 003 221 225 472;
  • 27) 0.000 000 000 000 003 221 225 472 × 2 = 0 + 0.000 000 000 000 006 442 450 944;
  • 28) 0.000 000 000 000 006 442 450 944 × 2 = 0 + 0.000 000 000 000 012 884 901 888;
  • 29) 0.000 000 000 000 012 884 901 888 × 2 = 0 + 0.000 000 000 000 025 769 803 776;
  • 30) 0.000 000 000 000 025 769 803 776 × 2 = 0 + 0.000 000 000 000 051 539 607 552;
  • 31) 0.000 000 000 000 051 539 607 552 × 2 = 0 + 0.000 000 000 000 103 079 215 104;
  • 32) 0.000 000 000 000 103 079 215 104 × 2 = 0 + 0.000 000 000 000 206 158 430 208;
  • 33) 0.000 000 000 000 206 158 430 208 × 2 = 0 + 0.000 000 000 000 412 316 860 416;
  • 34) 0.000 000 000 000 412 316 860 416 × 2 = 0 + 0.000 000 000 000 824 633 720 832;
  • 35) 0.000 000 000 000 824 633 720 832 × 2 = 0 + 0.000 000 000 001 649 267 441 664;
  • 36) 0.000 000 000 001 649 267 441 664 × 2 = 0 + 0.000 000 000 003 298 534 883 328;
  • 37) 0.000 000 000 003 298 534 883 328 × 2 = 0 + 0.000 000 000 006 597 069 766 656;
  • 38) 0.000 000 000 006 597 069 766 656 × 2 = 0 + 0.000 000 000 013 194 139 533 312;
  • 39) 0.000 000 000 013 194 139 533 312 × 2 = 0 + 0.000 000 000 026 388 279 066 624;
  • 40) 0.000 000 000 026 388 279 066 624 × 2 = 0 + 0.000 000 000 052 776 558 133 248;
  • 41) 0.000 000 000 052 776 558 133 248 × 2 = 0 + 0.000 000 000 105 553 116 266 496;
  • 42) 0.000 000 000 105 553 116 266 496 × 2 = 0 + 0.000 000 000 211 106 232 532 992;
  • 43) 0.000 000 000 211 106 232 532 992 × 2 = 0 + 0.000 000 000 422 212 465 065 984;
  • 44) 0.000 000 000 422 212 465 065 984 × 2 = 0 + 0.000 000 000 844 424 930 131 968;
  • 45) 0.000 000 000 844 424 930 131 968 × 2 = 0 + 0.000 000 001 688 849 860 263 936;
  • 46) 0.000 000 001 688 849 860 263 936 × 2 = 0 + 0.000 000 003 377 699 720 527 872;
  • 47) 0.000 000 003 377 699 720 527 872 × 2 = 0 + 0.000 000 006 755 399 441 055 744;
  • 48) 0.000 000 006 755 399 441 055 744 × 2 = 0 + 0.000 000 013 510 798 882 111 488;
  • 49) 0.000 000 013 510 798 882 111 488 × 2 = 0 + 0.000 000 027 021 597 764 222 976;
  • 50) 0.000 000 027 021 597 764 222 976 × 2 = 0 + 0.000 000 054 043 195 528 445 952;
  • 51) 0.000 000 054 043 195 528 445 952 × 2 = 0 + 0.000 000 108 086 391 056 891 904;
  • 52) 0.000 000 108 086 391 056 891 904 × 2 = 0 + 0.000 000 216 172 782 113 783 808;
  • 53) 0.000 000 216 172 782 113 783 808 × 2 = 0 + 0.000 000 432 345 564 227 567 616;
  • 54) 0.000 000 432 345 564 227 567 616 × 2 = 0 + 0.000 000 864 691 128 455 135 232;
  • 55) 0.000 000 864 691 128 455 135 232 × 2 = 0 + 0.000 001 729 382 256 910 270 464;
  • 56) 0.000 001 729 382 256 910 270 464 × 2 = 0 + 0.000 003 458 764 513 820 540 928;
  • 57) 0.000 003 458 764 513 820 540 928 × 2 = 0 + 0.000 006 917 529 027 641 081 856;
  • 58) 0.000 006 917 529 027 641 081 856 × 2 = 0 + 0.000 013 835 058 055 282 163 712;
  • 59) 0.000 013 835 058 055 282 163 712 × 2 = 0 + 0.000 027 670 116 110 564 327 424;
  • 60) 0.000 027 670 116 110 564 327 424 × 2 = 0 + 0.000 055 340 232 221 128 654 848;
  • 61) 0.000 055 340 232 221 128 654 848 × 2 = 0 + 0.000 110 680 464 442 257 309 696;
  • 62) 0.000 110 680 464 442 257 309 696 × 2 = 0 + 0.000 221 360 928 884 514 619 392;
  • 63) 0.000 221 360 928 884 514 619 392 × 2 = 0 + 0.000 442 721 857 769 029 238 784;
  • 64) 0.000 442 721 857 769 029 238 784 × 2 = 0 + 0.000 885 443 715 538 058 477 568;
  • 65) 0.000 885 443 715 538 058 477 568 × 2 = 0 + 0.001 770 887 431 076 116 955 136;
  • 66) 0.001 770 887 431 076 116 955 136 × 2 = 0 + 0.003 541 774 862 152 233 910 272;
  • 67) 0.003 541 774 862 152 233 910 272 × 2 = 0 + 0.007 083 549 724 304 467 820 544;
  • 68) 0.007 083 549 724 304 467 820 544 × 2 = 0 + 0.014 167 099 448 608 935 641 088;
  • 69) 0.014 167 099 448 608 935 641 088 × 2 = 0 + 0.028 334 198 897 217 871 282 176;
  • 70) 0.028 334 198 897 217 871 282 176 × 2 = 0 + 0.056 668 397 794 435 742 564 352;
  • 71) 0.056 668 397 794 435 742 564 352 × 2 = 0 + 0.113 336 795 588 871 485 128 704;
  • 72) 0.113 336 795 588 871 485 128 704 × 2 = 0 + 0.226 673 591 177 742 970 257 408;
  • 73) 0.226 673 591 177 742 970 257 408 × 2 = 0 + 0.453 347 182 355 485 940 514 816;
  • 74) 0.453 347 182 355 485 940 514 816 × 2 = 0 + 0.906 694 364 710 971 881 029 632;
  • 75) 0.906 694 364 710 971 881 029 632 × 2 = 1 + 0.813 388 729 421 943 762 059 264;
  • 76) 0.813 388 729 421 943 762 059 264 × 2 = 1 + 0.626 777 458 843 887 524 118 528;
  • 77) 0.626 777 458 843 887 524 118 528 × 2 = 1 + 0.253 554 917 687 775 048 237 056;
  • 78) 0.253 554 917 687 775 048 237 056 × 2 = 0 + 0.507 109 835 375 550 096 474 112;
  • 79) 0.507 109 835 375 550 096 474 112 × 2 = 1 + 0.014 219 670 751 100 192 948 224;
  • 80) 0.014 219 670 751 100 192 948 224 × 2 = 0 + 0.028 439 341 502 200 385 896 448;
  • 81) 0.028 439 341 502 200 385 896 448 × 2 = 0 + 0.056 878 683 004 400 771 792 896;
  • 82) 0.056 878 683 004 400 771 792 896 × 2 = 0 + 0.113 757 366 008 801 543 585 792;
  • 83) 0.113 757 366 008 801 543 585 792 × 2 = 0 + 0.227 514 732 017 603 087 171 584;
  • 84) 0.227 514 732 017 603 087 171 584 × 2 = 0 + 0.455 029 464 035 206 174 343 168;
  • 85) 0.455 029 464 035 206 174 343 168 × 2 = 0 + 0.910 058 928 070 412 348 686 336;
  • 86) 0.910 058 928 070 412 348 686 336 × 2 = 1 + 0.820 117 856 140 824 697 372 672;
  • 87) 0.820 117 856 140 824 697 372 672 × 2 = 1 + 0.640 235 712 281 649 394 745 344;
  • 88) 0.640 235 712 281 649 394 745 344 × 2 = 1 + 0.280 471 424 563 298 789 490 688;
  • 89) 0.280 471 424 563 298 789 490 688 × 2 = 0 + 0.560 942 849 126 597 578 981 376;
  • 90) 0.560 942 849 126 597 578 981 376 × 2 = 1 + 0.121 885 698 253 195 157 962 752;
  • 91) 0.121 885 698 253 195 157 962 752 × 2 = 0 + 0.243 771 396 506 390 315 925 504;
  • 92) 0.243 771 396 506 390 315 925 504 × 2 = 0 + 0.487 542 793 012 780 631 851 008;
  • 93) 0.487 542 793 012 780 631 851 008 × 2 = 0 + 0.975 085 586 025 561 263 702 016;
  • 94) 0.975 085 586 025 561 263 702 016 × 2 = 1 + 0.950 171 172 051 122 527 404 032;
  • 95) 0.950 171 172 051 122 527 404 032 × 2 = 1 + 0.900 342 344 102 245 054 808 064;
  • 96) 0.900 342 344 102 245 054 808 064 × 2 = 1 + 0.800 684 688 204 490 109 616 128;
  • 97) 0.800 684 688 204 490 109 616 128 × 2 = 1 + 0.601 369 376 408 980 219 232 256;
  • 98) 0.601 369 376 408 980 219 232 256 × 2 = 1 + 0.202 738 752 817 960 438 464 512;
  • 99) 0.202 738 752 817 960 438 464 512 × 2 = 0 + 0.405 477 505 635 920 876 929 024;
  • 100) 0.405 477 505 635 920 876 929 024 × 2 = 0 + 0.810 955 011 271 841 753 858 048;
  • 101) 0.810 955 011 271 841 753 858 048 × 2 = 1 + 0.621 910 022 543 683 507 716 096;
  • 102) 0.621 910 022 543 683 507 716 096 × 2 = 1 + 0.243 820 045 087 367 015 432 192;
  • 103) 0.243 820 045 087 367 015 432 192 × 2 = 0 + 0.487 640 090 174 734 030 864 384;
  • 104) 0.487 640 090 174 734 030 864 384 × 2 = 0 + 0.975 280 180 349 468 061 728 768;
  • 105) 0.975 280 180 349 468 061 728 768 × 2 = 1 + 0.950 560 360 698 936 123 457 536;
  • 106) 0.950 560 360 698 936 123 457 536 × 2 = 1 + 0.901 120 721 397 872 246 915 072;
  • 107) 0.901 120 721 397 872 246 915 072 × 2 = 1 + 0.802 241 442 795 744 493 830 144;
  • 108) 0.802 241 442 795 744 493 830 144 × 2 = 1 + 0.604 482 885 591 488 987 660 288;
  • 109) 0.604 482 885 591 488 987 660 288 × 2 = 1 + 0.208 965 771 182 977 975 320 576;
  • 110) 0.208 965 771 182 977 975 320 576 × 2 = 0 + 0.417 931 542 365 955 950 641 152;
  • 111) 0.417 931 542 365 955 950 641 152 × 2 = 0 + 0.835 863 084 731 911 901 282 304;
  • 112) 0.835 863 084 731 911 901 282 304 × 2 = 1 + 0.671 726 169 463 823 802 564 608;
  • 113) 0.671 726 169 463 823 802 564 608 × 2 = 1 + 0.343 452 338 927 647 605 129 216;
  • 114) 0.343 452 338 927 647 605 129 216 × 2 = 0 + 0.686 904 677 855 295 210 258 432;
  • 115) 0.686 904 677 855 295 210 258 432 × 2 = 1 + 0.373 809 355 710 590 420 516 864;
  • 116) 0.373 809 355 710 590 420 516 864 × 2 = 0 + 0.747 618 711 421 180 841 033 728;
  • 117) 0.747 618 711 421 180 841 033 728 × 2 = 1 + 0.495 237 422 842 361 682 067 456;
  • 118) 0.495 237 422 842 361 682 067 456 × 2 = 0 + 0.990 474 845 684 723 364 134 912;
  • 119) 0.990 474 845 684 723 364 134 912 × 2 = 1 + 0.980 949 691 369 446 728 269 824;
  • 120) 0.980 949 691 369 446 728 269 824 × 2 = 1 + 0.961 899 382 738 893 456 539 648;
  • 121) 0.961 899 382 738 893 456 539 648 × 2 = 1 + 0.923 798 765 477 786 913 079 296;
  • 122) 0.923 798 765 477 786 913 079 296 × 2 = 1 + 0.847 597 530 955 573 826 158 592;
  • 123) 0.847 597 530 955 573 826 158 592 × 2 = 1 + 0.695 195 061 911 147 652 317 184;
  • 124) 0.695 195 061 911 147 652 317 184 × 2 = 1 + 0.390 390 123 822 295 304 634 368;
  • 125) 0.390 390 123 822 295 304 634 368 × 2 = 0 + 0.780 780 247 644 590 609 268 736;
  • 126) 0.780 780 247 644 590 609 268 736 × 2 = 1 + 0.561 560 495 289 181 218 537 472;
  • 127) 0.561 560 495 289 181 218 537 472 × 2 = 1 + 0.123 120 990 578 362 437 074 944;

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:


0.000 000 000 000 000 000 000 048(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 1010 0000 0111 0100 0111 1100 1100 1111 1001 1010 1011 1111 011(2)

5. Positive number before normalization:

0.000 000 000 000 000 000 000 048(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 1010 0000 0111 0100 0111 1100 1100 1111 1001 1010 1011 1111 011(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 75 positions to the right, so that only one non zero digit remains to the left of it:


0.000 000 000 000 000 000 000 048(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 1010 0000 0111 0100 0111 1100 1100 1111 1001 1010 1011 1111 011(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 1010 0000 0111 0100 0111 1100 1100 1111 1001 1010 1011 1111 011(2) × 20 =

1.1101 0000 0011 1010 0011 1110 0110 0111 1100 1101 0101 1111 1011(2) × 2-75


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): -75

Mantissa (not normalized):
1.1101 0000 0011 1010 0011 1110 0110 0111 1100 1101 0101 1111 1011


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

-75 + 2(11-1) - 1 =

(-75 + 1 023)(10) =

948(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;
  • 948 ÷ 2 = 474 + 0;
  • 474 ÷ 2 = 237 + 0;
  • 237 ÷ 2 = 118 + 1;
  • 118 ÷ 2 = 59 + 0;
  • 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) =

948(10) =

011 1011 0100(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. 1101 0000 0011 1010 0011 1110 0110 0111 1100 1101 0101 1111 1011 =

1101 0000 0011 1010 0011 1110 0110 0111 1100 1101 0101 1111 1011


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 0100

Mantissa (52 bits) =
1101 0000 0011 1010 0011 1110 0110 0111 1100 1101 0101 1111 1011


Decimal number 0.000 000 000 000 000 000 000 048 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1011 0100 - 1101 0000 0011 1010 0011 1110 0110 0111 1100 1101 0101 1111 1011

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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