0.000 000 000 000 000 000 008 530 9 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 000 000 008 530 9(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
0.000 000 000 000 000 000 008 530 9(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: 0.000 000 000 000 000 000 008 530 9.

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) 0.000 000 000 000 000 000 008 530 9 × 2 = 0 + 0.000 000 000 000 000 000 017 061 8;
  • 2) 0.000 000 000 000 000 000 017 061 8 × 2 = 0 + 0.000 000 000 000 000 000 034 123 6;
  • 3) 0.000 000 000 000 000 000 034 123 6 × 2 = 0 + 0.000 000 000 000 000 000 068 247 2;
  • 4) 0.000 000 000 000 000 000 068 247 2 × 2 = 0 + 0.000 000 000 000 000 000 136 494 4;
  • 5) 0.000 000 000 000 000 000 136 494 4 × 2 = 0 + 0.000 000 000 000 000 000 272 988 8;
  • 6) 0.000 000 000 000 000 000 272 988 8 × 2 = 0 + 0.000 000 000 000 000 000 545 977 6;
  • 7) 0.000 000 000 000 000 000 545 977 6 × 2 = 0 + 0.000 000 000 000 000 001 091 955 2;
  • 8) 0.000 000 000 000 000 001 091 955 2 × 2 = 0 + 0.000 000 000 000 000 002 183 910 4;
  • 9) 0.000 000 000 000 000 002 183 910 4 × 2 = 0 + 0.000 000 000 000 000 004 367 820 8;
  • 10) 0.000 000 000 000 000 004 367 820 8 × 2 = 0 + 0.000 000 000 000 000 008 735 641 6;
  • 11) 0.000 000 000 000 000 008 735 641 6 × 2 = 0 + 0.000 000 000 000 000 017 471 283 2;
  • 12) 0.000 000 000 000 000 017 471 283 2 × 2 = 0 + 0.000 000 000 000 000 034 942 566 4;
  • 13) 0.000 000 000 000 000 034 942 566 4 × 2 = 0 + 0.000 000 000 000 000 069 885 132 8;
  • 14) 0.000 000 000 000 000 069 885 132 8 × 2 = 0 + 0.000 000 000 000 000 139 770 265 6;
  • 15) 0.000 000 000 000 000 139 770 265 6 × 2 = 0 + 0.000 000 000 000 000 279 540 531 2;
  • 16) 0.000 000 000 000 000 279 540 531 2 × 2 = 0 + 0.000 000 000 000 000 559 081 062 4;
  • 17) 0.000 000 000 000 000 559 081 062 4 × 2 = 0 + 0.000 000 000 000 001 118 162 124 8;
  • 18) 0.000 000 000 000 001 118 162 124 8 × 2 = 0 + 0.000 000 000 000 002 236 324 249 6;
  • 19) 0.000 000 000 000 002 236 324 249 6 × 2 = 0 + 0.000 000 000 000 004 472 648 499 2;
  • 20) 0.000 000 000 000 004 472 648 499 2 × 2 = 0 + 0.000 000 000 000 008 945 296 998 4;
  • 21) 0.000 000 000 000 008 945 296 998 4 × 2 = 0 + 0.000 000 000 000 017 890 593 996 8;
  • 22) 0.000 000 000 000 017 890 593 996 8 × 2 = 0 + 0.000 000 000 000 035 781 187 993 6;
  • 23) 0.000 000 000 000 035 781 187 993 6 × 2 = 0 + 0.000 000 000 000 071 562 375 987 2;
  • 24) 0.000 000 000 000 071 562 375 987 2 × 2 = 0 + 0.000 000 000 000 143 124 751 974 4;
  • 25) 0.000 000 000 000 143 124 751 974 4 × 2 = 0 + 0.000 000 000 000 286 249 503 948 8;
  • 26) 0.000 000 000 000 286 249 503 948 8 × 2 = 0 + 0.000 000 000 000 572 499 007 897 6;
  • 27) 0.000 000 000 000 572 499 007 897 6 × 2 = 0 + 0.000 000 000 001 144 998 015 795 2;
  • 28) 0.000 000 000 001 144 998 015 795 2 × 2 = 0 + 0.000 000 000 002 289 996 031 590 4;
  • 29) 0.000 000 000 002 289 996 031 590 4 × 2 = 0 + 0.000 000 000 004 579 992 063 180 8;
  • 30) 0.000 000 000 004 579 992 063 180 8 × 2 = 0 + 0.000 000 000 009 159 984 126 361 6;
  • 31) 0.000 000 000 009 159 984 126 361 6 × 2 = 0 + 0.000 000 000 018 319 968 252 723 2;
  • 32) 0.000 000 000 018 319 968 252 723 2 × 2 = 0 + 0.000 000 000 036 639 936 505 446 4;
  • 33) 0.000 000 000 036 639 936 505 446 4 × 2 = 0 + 0.000 000 000 073 279 873 010 892 8;
  • 34) 0.000 000 000 073 279 873 010 892 8 × 2 = 0 + 0.000 000 000 146 559 746 021 785 6;
  • 35) 0.000 000 000 146 559 746 021 785 6 × 2 = 0 + 0.000 000 000 293 119 492 043 571 2;
  • 36) 0.000 000 000 293 119 492 043 571 2 × 2 = 0 + 0.000 000 000 586 238 984 087 142 4;
  • 37) 0.000 000 000 586 238 984 087 142 4 × 2 = 0 + 0.000 000 001 172 477 968 174 284 8;
  • 38) 0.000 000 001 172 477 968 174 284 8 × 2 = 0 + 0.000 000 002 344 955 936 348 569 6;
  • 39) 0.000 000 002 344 955 936 348 569 6 × 2 = 0 + 0.000 000 004 689 911 872 697 139 2;
  • 40) 0.000 000 004 689 911 872 697 139 2 × 2 = 0 + 0.000 000 009 379 823 745 394 278 4;
  • 41) 0.000 000 009 379 823 745 394 278 4 × 2 = 0 + 0.000 000 018 759 647 490 788 556 8;
  • 42) 0.000 000 018 759 647 490 788 556 8 × 2 = 0 + 0.000 000 037 519 294 981 577 113 6;
  • 43) 0.000 000 037 519 294 981 577 113 6 × 2 = 0 + 0.000 000 075 038 589 963 154 227 2;
  • 44) 0.000 000 075 038 589 963 154 227 2 × 2 = 0 + 0.000 000 150 077 179 926 308 454 4;
  • 45) 0.000 000 150 077 179 926 308 454 4 × 2 = 0 + 0.000 000 300 154 359 852 616 908 8;
  • 46) 0.000 000 300 154 359 852 616 908 8 × 2 = 0 + 0.000 000 600 308 719 705 233 817 6;
  • 47) 0.000 000 600 308 719 705 233 817 6 × 2 = 0 + 0.000 001 200 617 439 410 467 635 2;
  • 48) 0.000 001 200 617 439 410 467 635 2 × 2 = 0 + 0.000 002 401 234 878 820 935 270 4;
  • 49) 0.000 002 401 234 878 820 935 270 4 × 2 = 0 + 0.000 004 802 469 757 641 870 540 8;
  • 50) 0.000 004 802 469 757 641 870 540 8 × 2 = 0 + 0.000 009 604 939 515 283 741 081 6;
  • 51) 0.000 009 604 939 515 283 741 081 6 × 2 = 0 + 0.000 019 209 879 030 567 482 163 2;
  • 52) 0.000 019 209 879 030 567 482 163 2 × 2 = 0 + 0.000 038 419 758 061 134 964 326 4;
  • 53) 0.000 038 419 758 061 134 964 326 4 × 2 = 0 + 0.000 076 839 516 122 269 928 652 8;
  • 54) 0.000 076 839 516 122 269 928 652 8 × 2 = 0 + 0.000 153 679 032 244 539 857 305 6;
  • 55) 0.000 153 679 032 244 539 857 305 6 × 2 = 0 + 0.000 307 358 064 489 079 714 611 2;
  • 56) 0.000 307 358 064 489 079 714 611 2 × 2 = 0 + 0.000 614 716 128 978 159 429 222 4;
  • 57) 0.000 614 716 128 978 159 429 222 4 × 2 = 0 + 0.001 229 432 257 956 318 858 444 8;
  • 58) 0.001 229 432 257 956 318 858 444 8 × 2 = 0 + 0.002 458 864 515 912 637 716 889 6;
  • 59) 0.002 458 864 515 912 637 716 889 6 × 2 = 0 + 0.004 917 729 031 825 275 433 779 2;
  • 60) 0.004 917 729 031 825 275 433 779 2 × 2 = 0 + 0.009 835 458 063 650 550 867 558 4;
  • 61) 0.009 835 458 063 650 550 867 558 4 × 2 = 0 + 0.019 670 916 127 301 101 735 116 8;
  • 62) 0.019 670 916 127 301 101 735 116 8 × 2 = 0 + 0.039 341 832 254 602 203 470 233 6;
  • 63) 0.039 341 832 254 602 203 470 233 6 × 2 = 0 + 0.078 683 664 509 204 406 940 467 2;
  • 64) 0.078 683 664 509 204 406 940 467 2 × 2 = 0 + 0.157 367 329 018 408 813 880 934 4;
  • 65) 0.157 367 329 018 408 813 880 934 4 × 2 = 0 + 0.314 734 658 036 817 627 761 868 8;
  • 66) 0.314 734 658 036 817 627 761 868 8 × 2 = 0 + 0.629 469 316 073 635 255 523 737 6;
  • 67) 0.629 469 316 073 635 255 523 737 6 × 2 = 1 + 0.258 938 632 147 270 511 047 475 2;
  • 68) 0.258 938 632 147 270 511 047 475 2 × 2 = 0 + 0.517 877 264 294 541 022 094 950 4;
  • 69) 0.517 877 264 294 541 022 094 950 4 × 2 = 1 + 0.035 754 528 589 082 044 189 900 8;
  • 70) 0.035 754 528 589 082 044 189 900 8 × 2 = 0 + 0.071 509 057 178 164 088 379 801 6;
  • 71) 0.071 509 057 178 164 088 379 801 6 × 2 = 0 + 0.143 018 114 356 328 176 759 603 2;
  • 72) 0.143 018 114 356 328 176 759 603 2 × 2 = 0 + 0.286 036 228 712 656 353 519 206 4;
  • 73) 0.286 036 228 712 656 353 519 206 4 × 2 = 0 + 0.572 072 457 425 312 707 038 412 8;
  • 74) 0.572 072 457 425 312 707 038 412 8 × 2 = 1 + 0.144 144 914 850 625 414 076 825 6;
  • 75) 0.144 144 914 850 625 414 076 825 6 × 2 = 0 + 0.288 289 829 701 250 828 153 651 2;
  • 76) 0.288 289 829 701 250 828 153 651 2 × 2 = 0 + 0.576 579 659 402 501 656 307 302 4;
  • 77) 0.576 579 659 402 501 656 307 302 4 × 2 = 1 + 0.153 159 318 805 003 312 614 604 8;
  • 78) 0.153 159 318 805 003 312 614 604 8 × 2 = 0 + 0.306 318 637 610 006 625 229 209 6;
  • 79) 0.306 318 637 610 006 625 229 209 6 × 2 = 0 + 0.612 637 275 220 013 250 458 419 2;
  • 80) 0.612 637 275 220 013 250 458 419 2 × 2 = 1 + 0.225 274 550 440 026 500 916 838 4;
  • 81) 0.225 274 550 440 026 500 916 838 4 × 2 = 0 + 0.450 549 100 880 053 001 833 676 8;
  • 82) 0.450 549 100 880 053 001 833 676 8 × 2 = 0 + 0.901 098 201 760 106 003 667 353 6;
  • 83) 0.901 098 201 760 106 003 667 353 6 × 2 = 1 + 0.802 196 403 520 212 007 334 707 2;
  • 84) 0.802 196 403 520 212 007 334 707 2 × 2 = 1 + 0.604 392 807 040 424 014 669 414 4;
  • 85) 0.604 392 807 040 424 014 669 414 4 × 2 = 1 + 0.208 785 614 080 848 029 338 828 8;
  • 86) 0.208 785 614 080 848 029 338 828 8 × 2 = 0 + 0.417 571 228 161 696 058 677 657 6;
  • 87) 0.417 571 228 161 696 058 677 657 6 × 2 = 0 + 0.835 142 456 323 392 117 355 315 2;
  • 88) 0.835 142 456 323 392 117 355 315 2 × 2 = 1 + 0.670 284 912 646 784 234 710 630 4;
  • 89) 0.670 284 912 646 784 234 710 630 4 × 2 = 1 + 0.340 569 825 293 568 469 421 260 8;
  • 90) 0.340 569 825 293 568 469 421 260 8 × 2 = 0 + 0.681 139 650 587 136 938 842 521 6;
  • 91) 0.681 139 650 587 136 938 842 521 6 × 2 = 1 + 0.362 279 301 174 273 877 685 043 2;
  • 92) 0.362 279 301 174 273 877 685 043 2 × 2 = 0 + 0.724 558 602 348 547 755 370 086 4;
  • 93) 0.724 558 602 348 547 755 370 086 4 × 2 = 1 + 0.449 117 204 697 095 510 740 172 8;
  • 94) 0.449 117 204 697 095 510 740 172 8 × 2 = 0 + 0.898 234 409 394 191 021 480 345 6;
  • 95) 0.898 234 409 394 191 021 480 345 6 × 2 = 1 + 0.796 468 818 788 382 042 960 691 2;
  • 96) 0.796 468 818 788 382 042 960 691 2 × 2 = 1 + 0.592 937 637 576 764 085 921 382 4;
  • 97) 0.592 937 637 576 764 085 921 382 4 × 2 = 1 + 0.185 875 275 153 528 171 842 764 8;
  • 98) 0.185 875 275 153 528 171 842 764 8 × 2 = 0 + 0.371 750 550 307 056 343 685 529 6;
  • 99) 0.371 750 550 307 056 343 685 529 6 × 2 = 0 + 0.743 501 100 614 112 687 371 059 2;
  • 100) 0.743 501 100 614 112 687 371 059 2 × 2 = 1 + 0.487 002 201 228 225 374 742 118 4;
  • 101) 0.487 002 201 228 225 374 742 118 4 × 2 = 0 + 0.974 004 402 456 450 749 484 236 8;
  • 102) 0.974 004 402 456 450 749 484 236 8 × 2 = 1 + 0.948 008 804 912 901 498 968 473 6;
  • 103) 0.948 008 804 912 901 498 968 473 6 × 2 = 1 + 0.896 017 609 825 802 997 936 947 2;
  • 104) 0.896 017 609 825 802 997 936 947 2 × 2 = 1 + 0.792 035 219 651 605 995 873 894 4;
  • 105) 0.792 035 219 651 605 995 873 894 4 × 2 = 1 + 0.584 070 439 303 211 991 747 788 8;
  • 106) 0.584 070 439 303 211 991 747 788 8 × 2 = 1 + 0.168 140 878 606 423 983 495 577 6;
  • 107) 0.168 140 878 606 423 983 495 577 6 × 2 = 0 + 0.336 281 757 212 847 966 991 155 2;
  • 108) 0.336 281 757 212 847 966 991 155 2 × 2 = 0 + 0.672 563 514 425 695 933 982 310 4;
  • 109) 0.672 563 514 425 695 933 982 310 4 × 2 = 1 + 0.345 127 028 851 391 867 964 620 8;
  • 110) 0.345 127 028 851 391 867 964 620 8 × 2 = 0 + 0.690 254 057 702 783 735 929 241 6;
  • 111) 0.690 254 057 702 783 735 929 241 6 × 2 = 1 + 0.380 508 115 405 567 471 858 483 2;
  • 112) 0.380 508 115 405 567 471 858 483 2 × 2 = 0 + 0.761 016 230 811 134 943 716 966 4;
  • 113) 0.761 016 230 811 134 943 716 966 4 × 2 = 1 + 0.522 032 461 622 269 887 433 932 8;
  • 114) 0.522 032 461 622 269 887 433 932 8 × 2 = 1 + 0.044 064 923 244 539 774 867 865 6;
  • 115) 0.044 064 923 244 539 774 867 865 6 × 2 = 0 + 0.088 129 846 489 079 549 735 731 2;
  • 116) 0.088 129 846 489 079 549 735 731 2 × 2 = 0 + 0.176 259 692 978 159 099 471 462 4;
  • 117) 0.176 259 692 978 159 099 471 462 4 × 2 = 0 + 0.352 519 385 956 318 198 942 924 8;
  • 118) 0.352 519 385 956 318 198 942 924 8 × 2 = 0 + 0.705 038 771 912 636 397 885 849 6;
  • 119) 0.705 038 771 912 636 397 885 849 6 × 2 = 1 + 0.410 077 543 825 272 795 771 699 2;

We didn't get any fractional part that was equal to zero. But we had enough iterations (over Mantissa limit) and at least one integer that was different from zero => FULL STOP (Losing precision - the converted number we get in the end will be just a very good approximation of the initial one).


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:


0.000 000 000 000 000 000 008 530 9(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1001 0011 1001 1010 1011 1001 0111 1100 1010 1100 001(2)

5. Positive number before normalization:

0.000 000 000 000 000 000 008 530 9(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1001 0011 1001 1010 1011 1001 0111 1100 1010 1100 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:


0.000 000 000 000 000 000 008 530 9(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1001 0011 1001 1010 1011 1001 0111 1100 1010 1100 001(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1001 0011 1001 1010 1011 1001 0111 1100 1010 1100 001(2) × 20 =

1.0100 0010 0100 1001 1100 1101 0101 1100 1011 1110 0101 0110 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 0010 0100 1001 1100 1101 0101 1100 1011 1110 0101 0110 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 0010 0100 1001 1100 1101 0101 1100 1011 1110 0101 0110 0001 =

0100 0010 0100 1001 1100 1101 0101 1100 1011 1110 0101 0110 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 0010 0100 1001 1100 1101 0101 1100 1011 1110 0101 0110 0001


Decimal number 0.000 000 000 000 000 000 008 530 9 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1011 1100 - 0100 0010 0100 1001 1100 1101 0101 1100 1011 1110 0101 0110 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