0.000 000 000 000 000 000 008 589 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 000 000 008 589(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
0.000 000 000 000 000 000 008 589(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: 0.000 000 000 000 000 000 008 589.

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) 0.000 000 000 000 000 000 008 589 × 2 = 0 + 0.000 000 000 000 000 000 017 178;
  • 2) 0.000 000 000 000 000 000 017 178 × 2 = 0 + 0.000 000 000 000 000 000 034 356;
  • 3) 0.000 000 000 000 000 000 034 356 × 2 = 0 + 0.000 000 000 000 000 000 068 712;
  • 4) 0.000 000 000 000 000 000 068 712 × 2 = 0 + 0.000 000 000 000 000 000 137 424;
  • 5) 0.000 000 000 000 000 000 137 424 × 2 = 0 + 0.000 000 000 000 000 000 274 848;
  • 6) 0.000 000 000 000 000 000 274 848 × 2 = 0 + 0.000 000 000 000 000 000 549 696;
  • 7) 0.000 000 000 000 000 000 549 696 × 2 = 0 + 0.000 000 000 000 000 001 099 392;
  • 8) 0.000 000 000 000 000 001 099 392 × 2 = 0 + 0.000 000 000 000 000 002 198 784;
  • 9) 0.000 000 000 000 000 002 198 784 × 2 = 0 + 0.000 000 000 000 000 004 397 568;
  • 10) 0.000 000 000 000 000 004 397 568 × 2 = 0 + 0.000 000 000 000 000 008 795 136;
  • 11) 0.000 000 000 000 000 008 795 136 × 2 = 0 + 0.000 000 000 000 000 017 590 272;
  • 12) 0.000 000 000 000 000 017 590 272 × 2 = 0 + 0.000 000 000 000 000 035 180 544;
  • 13) 0.000 000 000 000 000 035 180 544 × 2 = 0 + 0.000 000 000 000 000 070 361 088;
  • 14) 0.000 000 000 000 000 070 361 088 × 2 = 0 + 0.000 000 000 000 000 140 722 176;
  • 15) 0.000 000 000 000 000 140 722 176 × 2 = 0 + 0.000 000 000 000 000 281 444 352;
  • 16) 0.000 000 000 000 000 281 444 352 × 2 = 0 + 0.000 000 000 000 000 562 888 704;
  • 17) 0.000 000 000 000 000 562 888 704 × 2 = 0 + 0.000 000 000 000 001 125 777 408;
  • 18) 0.000 000 000 000 001 125 777 408 × 2 = 0 + 0.000 000 000 000 002 251 554 816;
  • 19) 0.000 000 000 000 002 251 554 816 × 2 = 0 + 0.000 000 000 000 004 503 109 632;
  • 20) 0.000 000 000 000 004 503 109 632 × 2 = 0 + 0.000 000 000 000 009 006 219 264;
  • 21) 0.000 000 000 000 009 006 219 264 × 2 = 0 + 0.000 000 000 000 018 012 438 528;
  • 22) 0.000 000 000 000 018 012 438 528 × 2 = 0 + 0.000 000 000 000 036 024 877 056;
  • 23) 0.000 000 000 000 036 024 877 056 × 2 = 0 + 0.000 000 000 000 072 049 754 112;
  • 24) 0.000 000 000 000 072 049 754 112 × 2 = 0 + 0.000 000 000 000 144 099 508 224;
  • 25) 0.000 000 000 000 144 099 508 224 × 2 = 0 + 0.000 000 000 000 288 199 016 448;
  • 26) 0.000 000 000 000 288 199 016 448 × 2 = 0 + 0.000 000 000 000 576 398 032 896;
  • 27) 0.000 000 000 000 576 398 032 896 × 2 = 0 + 0.000 000 000 001 152 796 065 792;
  • 28) 0.000 000 000 001 152 796 065 792 × 2 = 0 + 0.000 000 000 002 305 592 131 584;
  • 29) 0.000 000 000 002 305 592 131 584 × 2 = 0 + 0.000 000 000 004 611 184 263 168;
  • 30) 0.000 000 000 004 611 184 263 168 × 2 = 0 + 0.000 000 000 009 222 368 526 336;
  • 31) 0.000 000 000 009 222 368 526 336 × 2 = 0 + 0.000 000 000 018 444 737 052 672;
  • 32) 0.000 000 000 018 444 737 052 672 × 2 = 0 + 0.000 000 000 036 889 474 105 344;
  • 33) 0.000 000 000 036 889 474 105 344 × 2 = 0 + 0.000 000 000 073 778 948 210 688;
  • 34) 0.000 000 000 073 778 948 210 688 × 2 = 0 + 0.000 000 000 147 557 896 421 376;
  • 35) 0.000 000 000 147 557 896 421 376 × 2 = 0 + 0.000 000 000 295 115 792 842 752;
  • 36) 0.000 000 000 295 115 792 842 752 × 2 = 0 + 0.000 000 000 590 231 585 685 504;
  • 37) 0.000 000 000 590 231 585 685 504 × 2 = 0 + 0.000 000 001 180 463 171 371 008;
  • 38) 0.000 000 001 180 463 171 371 008 × 2 = 0 + 0.000 000 002 360 926 342 742 016;
  • 39) 0.000 000 002 360 926 342 742 016 × 2 = 0 + 0.000 000 004 721 852 685 484 032;
  • 40) 0.000 000 004 721 852 685 484 032 × 2 = 0 + 0.000 000 009 443 705 370 968 064;
  • 41) 0.000 000 009 443 705 370 968 064 × 2 = 0 + 0.000 000 018 887 410 741 936 128;
  • 42) 0.000 000 018 887 410 741 936 128 × 2 = 0 + 0.000 000 037 774 821 483 872 256;
  • 43) 0.000 000 037 774 821 483 872 256 × 2 = 0 + 0.000 000 075 549 642 967 744 512;
  • 44) 0.000 000 075 549 642 967 744 512 × 2 = 0 + 0.000 000 151 099 285 935 489 024;
  • 45) 0.000 000 151 099 285 935 489 024 × 2 = 0 + 0.000 000 302 198 571 870 978 048;
  • 46) 0.000 000 302 198 571 870 978 048 × 2 = 0 + 0.000 000 604 397 143 741 956 096;
  • 47) 0.000 000 604 397 143 741 956 096 × 2 = 0 + 0.000 001 208 794 287 483 912 192;
  • 48) 0.000 001 208 794 287 483 912 192 × 2 = 0 + 0.000 002 417 588 574 967 824 384;
  • 49) 0.000 002 417 588 574 967 824 384 × 2 = 0 + 0.000 004 835 177 149 935 648 768;
  • 50) 0.000 004 835 177 149 935 648 768 × 2 = 0 + 0.000 009 670 354 299 871 297 536;
  • 51) 0.000 009 670 354 299 871 297 536 × 2 = 0 + 0.000 019 340 708 599 742 595 072;
  • 52) 0.000 019 340 708 599 742 595 072 × 2 = 0 + 0.000 038 681 417 199 485 190 144;
  • 53) 0.000 038 681 417 199 485 190 144 × 2 = 0 + 0.000 077 362 834 398 970 380 288;
  • 54) 0.000 077 362 834 398 970 380 288 × 2 = 0 + 0.000 154 725 668 797 940 760 576;
  • 55) 0.000 154 725 668 797 940 760 576 × 2 = 0 + 0.000 309 451 337 595 881 521 152;
  • 56) 0.000 309 451 337 595 881 521 152 × 2 = 0 + 0.000 618 902 675 191 763 042 304;
  • 57) 0.000 618 902 675 191 763 042 304 × 2 = 0 + 0.001 237 805 350 383 526 084 608;
  • 58) 0.001 237 805 350 383 526 084 608 × 2 = 0 + 0.002 475 610 700 767 052 169 216;
  • 59) 0.002 475 610 700 767 052 169 216 × 2 = 0 + 0.004 951 221 401 534 104 338 432;
  • 60) 0.004 951 221 401 534 104 338 432 × 2 = 0 + 0.009 902 442 803 068 208 676 864;
  • 61) 0.009 902 442 803 068 208 676 864 × 2 = 0 + 0.019 804 885 606 136 417 353 728;
  • 62) 0.019 804 885 606 136 417 353 728 × 2 = 0 + 0.039 609 771 212 272 834 707 456;
  • 63) 0.039 609 771 212 272 834 707 456 × 2 = 0 + 0.079 219 542 424 545 669 414 912;
  • 64) 0.079 219 542 424 545 669 414 912 × 2 = 0 + 0.158 439 084 849 091 338 829 824;
  • 65) 0.158 439 084 849 091 338 829 824 × 2 = 0 + 0.316 878 169 698 182 677 659 648;
  • 66) 0.316 878 169 698 182 677 659 648 × 2 = 0 + 0.633 756 339 396 365 355 319 296;
  • 67) 0.633 756 339 396 365 355 319 296 × 2 = 1 + 0.267 512 678 792 730 710 638 592;
  • 68) 0.267 512 678 792 730 710 638 592 × 2 = 0 + 0.535 025 357 585 461 421 277 184;
  • 69) 0.535 025 357 585 461 421 277 184 × 2 = 1 + 0.070 050 715 170 922 842 554 368;
  • 70) 0.070 050 715 170 922 842 554 368 × 2 = 0 + 0.140 101 430 341 845 685 108 736;
  • 71) 0.140 101 430 341 845 685 108 736 × 2 = 0 + 0.280 202 860 683 691 370 217 472;
  • 72) 0.280 202 860 683 691 370 217 472 × 2 = 0 + 0.560 405 721 367 382 740 434 944;
  • 73) 0.560 405 721 367 382 740 434 944 × 2 = 1 + 0.120 811 442 734 765 480 869 888;
  • 74) 0.120 811 442 734 765 480 869 888 × 2 = 0 + 0.241 622 885 469 530 961 739 776;
  • 75) 0.241 622 885 469 530 961 739 776 × 2 = 0 + 0.483 245 770 939 061 923 479 552;
  • 76) 0.483 245 770 939 061 923 479 552 × 2 = 0 + 0.966 491 541 878 123 846 959 104;
  • 77) 0.966 491 541 878 123 846 959 104 × 2 = 1 + 0.932 983 083 756 247 693 918 208;
  • 78) 0.932 983 083 756 247 693 918 208 × 2 = 1 + 0.865 966 167 512 495 387 836 416;
  • 79) 0.865 966 167 512 495 387 836 416 × 2 = 1 + 0.731 932 335 024 990 775 672 832;
  • 80) 0.731 932 335 024 990 775 672 832 × 2 = 1 + 0.463 864 670 049 981 551 345 664;
  • 81) 0.463 864 670 049 981 551 345 664 × 2 = 0 + 0.927 729 340 099 963 102 691 328;
  • 82) 0.927 729 340 099 963 102 691 328 × 2 = 1 + 0.855 458 680 199 926 205 382 656;
  • 83) 0.855 458 680 199 926 205 382 656 × 2 = 1 + 0.710 917 360 399 852 410 765 312;
  • 84) 0.710 917 360 399 852 410 765 312 × 2 = 1 + 0.421 834 720 799 704 821 530 624;
  • 85) 0.421 834 720 799 704 821 530 624 × 2 = 0 + 0.843 669 441 599 409 643 061 248;
  • 86) 0.843 669 441 599 409 643 061 248 × 2 = 1 + 0.687 338 883 198 819 286 122 496;
  • 87) 0.687 338 883 198 819 286 122 496 × 2 = 1 + 0.374 677 766 397 638 572 244 992;
  • 88) 0.374 677 766 397 638 572 244 992 × 2 = 0 + 0.749 355 532 795 277 144 489 984;
  • 89) 0.749 355 532 795 277 144 489 984 × 2 = 1 + 0.498 711 065 590 554 288 979 968;
  • 90) 0.498 711 065 590 554 288 979 968 × 2 = 0 + 0.997 422 131 181 108 577 959 936;
  • 91) 0.997 422 131 181 108 577 959 936 × 2 = 1 + 0.994 844 262 362 217 155 919 872;
  • 92) 0.994 844 262 362 217 155 919 872 × 2 = 1 + 0.989 688 524 724 434 311 839 744;
  • 93) 0.989 688 524 724 434 311 839 744 × 2 = 1 + 0.979 377 049 448 868 623 679 488;
  • 94) 0.979 377 049 448 868 623 679 488 × 2 = 1 + 0.958 754 098 897 737 247 358 976;
  • 95) 0.958 754 098 897 737 247 358 976 × 2 = 1 + 0.917 508 197 795 474 494 717 952;
  • 96) 0.917 508 197 795 474 494 717 952 × 2 = 1 + 0.835 016 395 590 948 989 435 904;
  • 97) 0.835 016 395 590 948 989 435 904 × 2 = 1 + 0.670 032 791 181 897 978 871 808;
  • 98) 0.670 032 791 181 897 978 871 808 × 2 = 1 + 0.340 065 582 363 795 957 743 616;
  • 99) 0.340 065 582 363 795 957 743 616 × 2 = 0 + 0.680 131 164 727 591 915 487 232;
  • 100) 0.680 131 164 727 591 915 487 232 × 2 = 1 + 0.360 262 329 455 183 830 974 464;
  • 101) 0.360 262 329 455 183 830 974 464 × 2 = 0 + 0.720 524 658 910 367 661 948 928;
  • 102) 0.720 524 658 910 367 661 948 928 × 2 = 1 + 0.441 049 317 820 735 323 897 856;
  • 103) 0.441 049 317 820 735 323 897 856 × 2 = 0 + 0.882 098 635 641 470 647 795 712;
  • 104) 0.882 098 635 641 470 647 795 712 × 2 = 1 + 0.764 197 271 282 941 295 591 424;
  • 105) 0.764 197 271 282 941 295 591 424 × 2 = 1 + 0.528 394 542 565 882 591 182 848;
  • 106) 0.528 394 542 565 882 591 182 848 × 2 = 1 + 0.056 789 085 131 765 182 365 696;
  • 107) 0.056 789 085 131 765 182 365 696 × 2 = 0 + 0.113 578 170 263 530 364 731 392;
  • 108) 0.113 578 170 263 530 364 731 392 × 2 = 0 + 0.227 156 340 527 060 729 462 784;
  • 109) 0.227 156 340 527 060 729 462 784 × 2 = 0 + 0.454 312 681 054 121 458 925 568;
  • 110) 0.454 312 681 054 121 458 925 568 × 2 = 0 + 0.908 625 362 108 242 917 851 136;
  • 111) 0.908 625 362 108 242 917 851 136 × 2 = 1 + 0.817 250 724 216 485 835 702 272;
  • 112) 0.817 250 724 216 485 835 702 272 × 2 = 1 + 0.634 501 448 432 971 671 404 544;
  • 113) 0.634 501 448 432 971 671 404 544 × 2 = 1 + 0.269 002 896 865 943 342 809 088;
  • 114) 0.269 002 896 865 943 342 809 088 × 2 = 0 + 0.538 005 793 731 886 685 618 176;
  • 115) 0.538 005 793 731 886 685 618 176 × 2 = 1 + 0.076 011 587 463 773 371 236 352;
  • 116) 0.076 011 587 463 773 371 236 352 × 2 = 0 + 0.152 023 174 927 546 742 472 704;
  • 117) 0.152 023 174 927 546 742 472 704 × 2 = 0 + 0.304 046 349 855 093 484 945 408;
  • 118) 0.304 046 349 855 093 484 945 408 × 2 = 0 + 0.608 092 699 710 186 969 890 816;
  • 119) 0.608 092 699 710 186 969 890 816 × 2 = 1 + 0.216 185 399 420 373 939 781 632;

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:


0.000 000 000 000 000 000 008 589(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 1000 1111 0111 0110 1011 1111 1101 0101 1100 0011 1010 001(2)

5. Positive number before normalization:

0.000 000 000 000 000 000 008 589(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 1000 1111 0111 0110 1011 1111 1101 0101 1100 0011 1010 001(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 67 positions to the right, so that only one non zero digit remains to the left of it:


0.000 000 000 000 000 000 008 589(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 1000 1111 0111 0110 1011 1111 1101 0101 1100 0011 1010 001(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 1000 1111 0111 0110 1011 1111 1101 0101 1100 0011 1010 001(2) × 20 =

1.0100 0100 0111 1011 1011 0101 1111 1110 1010 1110 0001 1101 0001(2) × 2-67


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): -67

Mantissa (not normalized):
1.0100 0100 0111 1011 1011 0101 1111 1110 1010 1110 0001 1101 0001


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

-67 + 2(11-1) - 1 =

(-67 + 1 023)(10) =

956(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;
  • 956 ÷ 2 = 478 + 0;
  • 478 ÷ 2 = 239 + 0;
  • 239 ÷ 2 = 119 + 1;
  • 119 ÷ 2 = 59 + 1;
  • 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) =

956(10) =

011 1011 1100(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. 0100 0100 0111 1011 1011 0101 1111 1110 1010 1110 0001 1101 0001 =

0100 0100 0111 1011 1011 0101 1111 1110 1010 1110 0001 1101 0001


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 1100

Mantissa (52 bits) =
0100 0100 0111 1011 1011 0101 1111 1110 1010 1110 0001 1101 0001


Decimal number 0.000 000 000 000 000 000 008 589 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1011 1100 - 0100 0100 0111 1011 1011 0101 1111 1110 1010 1110 0001 1101 0001

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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