0.000 000 000 000 000 000 008 536 9 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 000 000 008 536 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
0.000 000 000 000 000 000 008 536 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: 0.000 000 000 000 000 000 008 536 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) 0.000 000 000 000 000 000 008 536 9 × 2 = 0 + 0.000 000 000 000 000 000 017 073 8;
  • 2) 0.000 000 000 000 000 000 017 073 8 × 2 = 0 + 0.000 000 000 000 000 000 034 147 6;
  • 3) 0.000 000 000 000 000 000 034 147 6 × 2 = 0 + 0.000 000 000 000 000 000 068 295 2;
  • 4) 0.000 000 000 000 000 000 068 295 2 × 2 = 0 + 0.000 000 000 000 000 000 136 590 4;
  • 5) 0.000 000 000 000 000 000 136 590 4 × 2 = 0 + 0.000 000 000 000 000 000 273 180 8;
  • 6) 0.000 000 000 000 000 000 273 180 8 × 2 = 0 + 0.000 000 000 000 000 000 546 361 6;
  • 7) 0.000 000 000 000 000 000 546 361 6 × 2 = 0 + 0.000 000 000 000 000 001 092 723 2;
  • 8) 0.000 000 000 000 000 001 092 723 2 × 2 = 0 + 0.000 000 000 000 000 002 185 446 4;
  • 9) 0.000 000 000 000 000 002 185 446 4 × 2 = 0 + 0.000 000 000 000 000 004 370 892 8;
  • 10) 0.000 000 000 000 000 004 370 892 8 × 2 = 0 + 0.000 000 000 000 000 008 741 785 6;
  • 11) 0.000 000 000 000 000 008 741 785 6 × 2 = 0 + 0.000 000 000 000 000 017 483 571 2;
  • 12) 0.000 000 000 000 000 017 483 571 2 × 2 = 0 + 0.000 000 000 000 000 034 967 142 4;
  • 13) 0.000 000 000 000 000 034 967 142 4 × 2 = 0 + 0.000 000 000 000 000 069 934 284 8;
  • 14) 0.000 000 000 000 000 069 934 284 8 × 2 = 0 + 0.000 000 000 000 000 139 868 569 6;
  • 15) 0.000 000 000 000 000 139 868 569 6 × 2 = 0 + 0.000 000 000 000 000 279 737 139 2;
  • 16) 0.000 000 000 000 000 279 737 139 2 × 2 = 0 + 0.000 000 000 000 000 559 474 278 4;
  • 17) 0.000 000 000 000 000 559 474 278 4 × 2 = 0 + 0.000 000 000 000 001 118 948 556 8;
  • 18) 0.000 000 000 000 001 118 948 556 8 × 2 = 0 + 0.000 000 000 000 002 237 897 113 6;
  • 19) 0.000 000 000 000 002 237 897 113 6 × 2 = 0 + 0.000 000 000 000 004 475 794 227 2;
  • 20) 0.000 000 000 000 004 475 794 227 2 × 2 = 0 + 0.000 000 000 000 008 951 588 454 4;
  • 21) 0.000 000 000 000 008 951 588 454 4 × 2 = 0 + 0.000 000 000 000 017 903 176 908 8;
  • 22) 0.000 000 000 000 017 903 176 908 8 × 2 = 0 + 0.000 000 000 000 035 806 353 817 6;
  • 23) 0.000 000 000 000 035 806 353 817 6 × 2 = 0 + 0.000 000 000 000 071 612 707 635 2;
  • 24) 0.000 000 000 000 071 612 707 635 2 × 2 = 0 + 0.000 000 000 000 143 225 415 270 4;
  • 25) 0.000 000 000 000 143 225 415 270 4 × 2 = 0 + 0.000 000 000 000 286 450 830 540 8;
  • 26) 0.000 000 000 000 286 450 830 540 8 × 2 = 0 + 0.000 000 000 000 572 901 661 081 6;
  • 27) 0.000 000 000 000 572 901 661 081 6 × 2 = 0 + 0.000 000 000 001 145 803 322 163 2;
  • 28) 0.000 000 000 001 145 803 322 163 2 × 2 = 0 + 0.000 000 000 002 291 606 644 326 4;
  • 29) 0.000 000 000 002 291 606 644 326 4 × 2 = 0 + 0.000 000 000 004 583 213 288 652 8;
  • 30) 0.000 000 000 004 583 213 288 652 8 × 2 = 0 + 0.000 000 000 009 166 426 577 305 6;
  • 31) 0.000 000 000 009 166 426 577 305 6 × 2 = 0 + 0.000 000 000 018 332 853 154 611 2;
  • 32) 0.000 000 000 018 332 853 154 611 2 × 2 = 0 + 0.000 000 000 036 665 706 309 222 4;
  • 33) 0.000 000 000 036 665 706 309 222 4 × 2 = 0 + 0.000 000 000 073 331 412 618 444 8;
  • 34) 0.000 000 000 073 331 412 618 444 8 × 2 = 0 + 0.000 000 000 146 662 825 236 889 6;
  • 35) 0.000 000 000 146 662 825 236 889 6 × 2 = 0 + 0.000 000 000 293 325 650 473 779 2;
  • 36) 0.000 000 000 293 325 650 473 779 2 × 2 = 0 + 0.000 000 000 586 651 300 947 558 4;
  • 37) 0.000 000 000 586 651 300 947 558 4 × 2 = 0 + 0.000 000 001 173 302 601 895 116 8;
  • 38) 0.000 000 001 173 302 601 895 116 8 × 2 = 0 + 0.000 000 002 346 605 203 790 233 6;
  • 39) 0.000 000 002 346 605 203 790 233 6 × 2 = 0 + 0.000 000 004 693 210 407 580 467 2;
  • 40) 0.000 000 004 693 210 407 580 467 2 × 2 = 0 + 0.000 000 009 386 420 815 160 934 4;
  • 41) 0.000 000 009 386 420 815 160 934 4 × 2 = 0 + 0.000 000 018 772 841 630 321 868 8;
  • 42) 0.000 000 018 772 841 630 321 868 8 × 2 = 0 + 0.000 000 037 545 683 260 643 737 6;
  • 43) 0.000 000 037 545 683 260 643 737 6 × 2 = 0 + 0.000 000 075 091 366 521 287 475 2;
  • 44) 0.000 000 075 091 366 521 287 475 2 × 2 = 0 + 0.000 000 150 182 733 042 574 950 4;
  • 45) 0.000 000 150 182 733 042 574 950 4 × 2 = 0 + 0.000 000 300 365 466 085 149 900 8;
  • 46) 0.000 000 300 365 466 085 149 900 8 × 2 = 0 + 0.000 000 600 730 932 170 299 801 6;
  • 47) 0.000 000 600 730 932 170 299 801 6 × 2 = 0 + 0.000 001 201 461 864 340 599 603 2;
  • 48) 0.000 001 201 461 864 340 599 603 2 × 2 = 0 + 0.000 002 402 923 728 681 199 206 4;
  • 49) 0.000 002 402 923 728 681 199 206 4 × 2 = 0 + 0.000 004 805 847 457 362 398 412 8;
  • 50) 0.000 004 805 847 457 362 398 412 8 × 2 = 0 + 0.000 009 611 694 914 724 796 825 6;
  • 51) 0.000 009 611 694 914 724 796 825 6 × 2 = 0 + 0.000 019 223 389 829 449 593 651 2;
  • 52) 0.000 019 223 389 829 449 593 651 2 × 2 = 0 + 0.000 038 446 779 658 899 187 302 4;
  • 53) 0.000 038 446 779 658 899 187 302 4 × 2 = 0 + 0.000 076 893 559 317 798 374 604 8;
  • 54) 0.000 076 893 559 317 798 374 604 8 × 2 = 0 + 0.000 153 787 118 635 596 749 209 6;
  • 55) 0.000 153 787 118 635 596 749 209 6 × 2 = 0 + 0.000 307 574 237 271 193 498 419 2;
  • 56) 0.000 307 574 237 271 193 498 419 2 × 2 = 0 + 0.000 615 148 474 542 386 996 838 4;
  • 57) 0.000 615 148 474 542 386 996 838 4 × 2 = 0 + 0.001 230 296 949 084 773 993 676 8;
  • 58) 0.001 230 296 949 084 773 993 676 8 × 2 = 0 + 0.002 460 593 898 169 547 987 353 6;
  • 59) 0.002 460 593 898 169 547 987 353 6 × 2 = 0 + 0.004 921 187 796 339 095 974 707 2;
  • 60) 0.004 921 187 796 339 095 974 707 2 × 2 = 0 + 0.009 842 375 592 678 191 949 414 4;
  • 61) 0.009 842 375 592 678 191 949 414 4 × 2 = 0 + 0.019 684 751 185 356 383 898 828 8;
  • 62) 0.019 684 751 185 356 383 898 828 8 × 2 = 0 + 0.039 369 502 370 712 767 797 657 6;
  • 63) 0.039 369 502 370 712 767 797 657 6 × 2 = 0 + 0.078 739 004 741 425 535 595 315 2;
  • 64) 0.078 739 004 741 425 535 595 315 2 × 2 = 0 + 0.157 478 009 482 851 071 190 630 4;
  • 65) 0.157 478 009 482 851 071 190 630 4 × 2 = 0 + 0.314 956 018 965 702 142 381 260 8;
  • 66) 0.314 956 018 965 702 142 381 260 8 × 2 = 0 + 0.629 912 037 931 404 284 762 521 6;
  • 67) 0.629 912 037 931 404 284 762 521 6 × 2 = 1 + 0.259 824 075 862 808 569 525 043 2;
  • 68) 0.259 824 075 862 808 569 525 043 2 × 2 = 0 + 0.519 648 151 725 617 139 050 086 4;
  • 69) 0.519 648 151 725 617 139 050 086 4 × 2 = 1 + 0.039 296 303 451 234 278 100 172 8;
  • 70) 0.039 296 303 451 234 278 100 172 8 × 2 = 0 + 0.078 592 606 902 468 556 200 345 6;
  • 71) 0.078 592 606 902 468 556 200 345 6 × 2 = 0 + 0.157 185 213 804 937 112 400 691 2;
  • 72) 0.157 185 213 804 937 112 400 691 2 × 2 = 0 + 0.314 370 427 609 874 224 801 382 4;
  • 73) 0.314 370 427 609 874 224 801 382 4 × 2 = 0 + 0.628 740 855 219 748 449 602 764 8;
  • 74) 0.628 740 855 219 748 449 602 764 8 × 2 = 1 + 0.257 481 710 439 496 899 205 529 6;
  • 75) 0.257 481 710 439 496 899 205 529 6 × 2 = 0 + 0.514 963 420 878 993 798 411 059 2;
  • 76) 0.514 963 420 878 993 798 411 059 2 × 2 = 1 + 0.029 926 841 757 987 596 822 118 4;
  • 77) 0.029 926 841 757 987 596 822 118 4 × 2 = 0 + 0.059 853 683 515 975 193 644 236 8;
  • 78) 0.059 853 683 515 975 193 644 236 8 × 2 = 0 + 0.119 707 367 031 950 387 288 473 6;
  • 79) 0.119 707 367 031 950 387 288 473 6 × 2 = 0 + 0.239 414 734 063 900 774 576 947 2;
  • 80) 0.239 414 734 063 900 774 576 947 2 × 2 = 0 + 0.478 829 468 127 801 549 153 894 4;
  • 81) 0.478 829 468 127 801 549 153 894 4 × 2 = 0 + 0.957 658 936 255 603 098 307 788 8;
  • 82) 0.957 658 936 255 603 098 307 788 8 × 2 = 1 + 0.915 317 872 511 206 196 615 577 6;
  • 83) 0.915 317 872 511 206 196 615 577 6 × 2 = 1 + 0.830 635 745 022 412 393 231 155 2;
  • 84) 0.830 635 745 022 412 393 231 155 2 × 2 = 1 + 0.661 271 490 044 824 786 462 310 4;
  • 85) 0.661 271 490 044 824 786 462 310 4 × 2 = 1 + 0.322 542 980 089 649 572 924 620 8;
  • 86) 0.322 542 980 089 649 572 924 620 8 × 2 = 0 + 0.645 085 960 179 299 145 849 241 6;
  • 87) 0.645 085 960 179 299 145 849 241 6 × 2 = 1 + 0.290 171 920 358 598 291 698 483 2;
  • 88) 0.290 171 920 358 598 291 698 483 2 × 2 = 0 + 0.580 343 840 717 196 583 396 966 4;
  • 89) 0.580 343 840 717 196 583 396 966 4 × 2 = 1 + 0.160 687 681 434 393 166 793 932 8;
  • 90) 0.160 687 681 434 393 166 793 932 8 × 2 = 0 + 0.321 375 362 868 786 333 587 865 6;
  • 91) 0.321 375 362 868 786 333 587 865 6 × 2 = 0 + 0.642 750 725 737 572 667 175 731 2;
  • 92) 0.642 750 725 737 572 667 175 731 2 × 2 = 1 + 0.285 501 451 475 145 334 351 462 4;
  • 93) 0.285 501 451 475 145 334 351 462 4 × 2 = 0 + 0.571 002 902 950 290 668 702 924 8;
  • 94) 0.571 002 902 950 290 668 702 924 8 × 2 = 1 + 0.142 005 805 900 581 337 405 849 6;
  • 95) 0.142 005 805 900 581 337 405 849 6 × 2 = 0 + 0.284 011 611 801 162 674 811 699 2;
  • 96) 0.284 011 611 801 162 674 811 699 2 × 2 = 0 + 0.568 023 223 602 325 349 623 398 4;
  • 97) 0.568 023 223 602 325 349 623 398 4 × 2 = 1 + 0.136 046 447 204 650 699 246 796 8;
  • 98) 0.136 046 447 204 650 699 246 796 8 × 2 = 0 + 0.272 092 894 409 301 398 493 593 6;
  • 99) 0.272 092 894 409 301 398 493 593 6 × 2 = 0 + 0.544 185 788 818 602 796 987 187 2;
  • 100) 0.544 185 788 818 602 796 987 187 2 × 2 = 1 + 0.088 371 577 637 205 593 974 374 4;
  • 101) 0.088 371 577 637 205 593 974 374 4 × 2 = 0 + 0.176 743 155 274 411 187 948 748 8;
  • 102) 0.176 743 155 274 411 187 948 748 8 × 2 = 0 + 0.353 486 310 548 822 375 897 497 6;
  • 103) 0.353 486 310 548 822 375 897 497 6 × 2 = 0 + 0.706 972 621 097 644 751 794 995 2;
  • 104) 0.706 972 621 097 644 751 794 995 2 × 2 = 1 + 0.413 945 242 195 289 503 589 990 4;
  • 105) 0.413 945 242 195 289 503 589 990 4 × 2 = 0 + 0.827 890 484 390 579 007 179 980 8;
  • 106) 0.827 890 484 390 579 007 179 980 8 × 2 = 1 + 0.655 780 968 781 158 014 359 961 6;
  • 107) 0.655 780 968 781 158 014 359 961 6 × 2 = 1 + 0.311 561 937 562 316 028 719 923 2;
  • 108) 0.311 561 937 562 316 028 719 923 2 × 2 = 0 + 0.623 123 875 124 632 057 439 846 4;
  • 109) 0.623 123 875 124 632 057 439 846 4 × 2 = 1 + 0.246 247 750 249 264 114 879 692 8;
  • 110) 0.246 247 750 249 264 114 879 692 8 × 2 = 0 + 0.492 495 500 498 528 229 759 385 6;
  • 111) 0.492 495 500 498 528 229 759 385 6 × 2 = 0 + 0.984 991 000 997 056 459 518 771 2;
  • 112) 0.984 991 000 997 056 459 518 771 2 × 2 = 1 + 0.969 982 001 994 112 919 037 542 4;
  • 113) 0.969 982 001 994 112 919 037 542 4 × 2 = 1 + 0.939 964 003 988 225 838 075 084 8;
  • 114) 0.939 964 003 988 225 838 075 084 8 × 2 = 1 + 0.879 928 007 976 451 676 150 169 6;
  • 115) 0.879 928 007 976 451 676 150 169 6 × 2 = 1 + 0.759 856 015 952 903 352 300 339 2;
  • 116) 0.759 856 015 952 903 352 300 339 2 × 2 = 1 + 0.519 712 031 905 806 704 600 678 4;
  • 117) 0.519 712 031 905 806 704 600 678 4 × 2 = 1 + 0.039 424 063 811 613 409 201 356 8;
  • 118) 0.039 424 063 811 613 409 201 356 8 × 2 = 0 + 0.078 848 127 623 226 818 402 713 6;
  • 119) 0.078 848 127 623 226 818 402 713 6 × 2 = 0 + 0.157 696 255 246 453 636 805 427 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:


0.000 000 000 000 000 000 008 536 9(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0101 0000 0111 1010 1001 0100 1001 0001 0110 1001 1111 100(2)

5. Positive number before normalization:

0.000 000 000 000 000 000 008 536 9(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0101 0000 0111 1010 1001 0100 1001 0001 0110 1001 1111 100(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:


0.000 000 000 000 000 000 008 536 9(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0101 0000 0111 1010 1001 0100 1001 0001 0110 1001 1111 100(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0101 0000 0111 1010 1001 0100 1001 0001 0110 1001 1111 100(2) × 20 =

1.0100 0010 1000 0011 1101 0100 1010 0100 1000 1011 0100 1111 1100(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 1000 0011 1101 0100 1010 0100 1000 1011 0100 1111 1100


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 1000 0011 1101 0100 1010 0100 1000 1011 0100 1111 1100 =

0100 0010 1000 0011 1101 0100 1010 0100 1000 1011 0100 1111 1100


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 1000 0011 1101 0100 1010 0100 1000 1011 0100 1111 1100


Decimal number 0.000 000 000 000 000 000 008 536 9 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1011 1100 - 0100 0010 1000 0011 1101 0100 1010 0100 1000 1011 0100 1111 1100

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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