0.000 000 000 000 000 000 008 533 6 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 000 000 008 533 6(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
0.000 000 000 000 000 000 008 533 6(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: 0.000 000 000 000 000 000 008 533 6.

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) 0.000 000 000 000 000 000 008 533 6 × 2 = 0 + 0.000 000 000 000 000 000 017 067 2;
  • 2) 0.000 000 000 000 000 000 017 067 2 × 2 = 0 + 0.000 000 000 000 000 000 034 134 4;
  • 3) 0.000 000 000 000 000 000 034 134 4 × 2 = 0 + 0.000 000 000 000 000 000 068 268 8;
  • 4) 0.000 000 000 000 000 000 068 268 8 × 2 = 0 + 0.000 000 000 000 000 000 136 537 6;
  • 5) 0.000 000 000 000 000 000 136 537 6 × 2 = 0 + 0.000 000 000 000 000 000 273 075 2;
  • 6) 0.000 000 000 000 000 000 273 075 2 × 2 = 0 + 0.000 000 000 000 000 000 546 150 4;
  • 7) 0.000 000 000 000 000 000 546 150 4 × 2 = 0 + 0.000 000 000 000 000 001 092 300 8;
  • 8) 0.000 000 000 000 000 001 092 300 8 × 2 = 0 + 0.000 000 000 000 000 002 184 601 6;
  • 9) 0.000 000 000 000 000 002 184 601 6 × 2 = 0 + 0.000 000 000 000 000 004 369 203 2;
  • 10) 0.000 000 000 000 000 004 369 203 2 × 2 = 0 + 0.000 000 000 000 000 008 738 406 4;
  • 11) 0.000 000 000 000 000 008 738 406 4 × 2 = 0 + 0.000 000 000 000 000 017 476 812 8;
  • 12) 0.000 000 000 000 000 017 476 812 8 × 2 = 0 + 0.000 000 000 000 000 034 953 625 6;
  • 13) 0.000 000 000 000 000 034 953 625 6 × 2 = 0 + 0.000 000 000 000 000 069 907 251 2;
  • 14) 0.000 000 000 000 000 069 907 251 2 × 2 = 0 + 0.000 000 000 000 000 139 814 502 4;
  • 15) 0.000 000 000 000 000 139 814 502 4 × 2 = 0 + 0.000 000 000 000 000 279 629 004 8;
  • 16) 0.000 000 000 000 000 279 629 004 8 × 2 = 0 + 0.000 000 000 000 000 559 258 009 6;
  • 17) 0.000 000 000 000 000 559 258 009 6 × 2 = 0 + 0.000 000 000 000 001 118 516 019 2;
  • 18) 0.000 000 000 000 001 118 516 019 2 × 2 = 0 + 0.000 000 000 000 002 237 032 038 4;
  • 19) 0.000 000 000 000 002 237 032 038 4 × 2 = 0 + 0.000 000 000 000 004 474 064 076 8;
  • 20) 0.000 000 000 000 004 474 064 076 8 × 2 = 0 + 0.000 000 000 000 008 948 128 153 6;
  • 21) 0.000 000 000 000 008 948 128 153 6 × 2 = 0 + 0.000 000 000 000 017 896 256 307 2;
  • 22) 0.000 000 000 000 017 896 256 307 2 × 2 = 0 + 0.000 000 000 000 035 792 512 614 4;
  • 23) 0.000 000 000 000 035 792 512 614 4 × 2 = 0 + 0.000 000 000 000 071 585 025 228 8;
  • 24) 0.000 000 000 000 071 585 025 228 8 × 2 = 0 + 0.000 000 000 000 143 170 050 457 6;
  • 25) 0.000 000 000 000 143 170 050 457 6 × 2 = 0 + 0.000 000 000 000 286 340 100 915 2;
  • 26) 0.000 000 000 000 286 340 100 915 2 × 2 = 0 + 0.000 000 000 000 572 680 201 830 4;
  • 27) 0.000 000 000 000 572 680 201 830 4 × 2 = 0 + 0.000 000 000 001 145 360 403 660 8;
  • 28) 0.000 000 000 001 145 360 403 660 8 × 2 = 0 + 0.000 000 000 002 290 720 807 321 6;
  • 29) 0.000 000 000 002 290 720 807 321 6 × 2 = 0 + 0.000 000 000 004 581 441 614 643 2;
  • 30) 0.000 000 000 004 581 441 614 643 2 × 2 = 0 + 0.000 000 000 009 162 883 229 286 4;
  • 31) 0.000 000 000 009 162 883 229 286 4 × 2 = 0 + 0.000 000 000 018 325 766 458 572 8;
  • 32) 0.000 000 000 018 325 766 458 572 8 × 2 = 0 + 0.000 000 000 036 651 532 917 145 6;
  • 33) 0.000 000 000 036 651 532 917 145 6 × 2 = 0 + 0.000 000 000 073 303 065 834 291 2;
  • 34) 0.000 000 000 073 303 065 834 291 2 × 2 = 0 + 0.000 000 000 146 606 131 668 582 4;
  • 35) 0.000 000 000 146 606 131 668 582 4 × 2 = 0 + 0.000 000 000 293 212 263 337 164 8;
  • 36) 0.000 000 000 293 212 263 337 164 8 × 2 = 0 + 0.000 000 000 586 424 526 674 329 6;
  • 37) 0.000 000 000 586 424 526 674 329 6 × 2 = 0 + 0.000 000 001 172 849 053 348 659 2;
  • 38) 0.000 000 001 172 849 053 348 659 2 × 2 = 0 + 0.000 000 002 345 698 106 697 318 4;
  • 39) 0.000 000 002 345 698 106 697 318 4 × 2 = 0 + 0.000 000 004 691 396 213 394 636 8;
  • 40) 0.000 000 004 691 396 213 394 636 8 × 2 = 0 + 0.000 000 009 382 792 426 789 273 6;
  • 41) 0.000 000 009 382 792 426 789 273 6 × 2 = 0 + 0.000 000 018 765 584 853 578 547 2;
  • 42) 0.000 000 018 765 584 853 578 547 2 × 2 = 0 + 0.000 000 037 531 169 707 157 094 4;
  • 43) 0.000 000 037 531 169 707 157 094 4 × 2 = 0 + 0.000 000 075 062 339 414 314 188 8;
  • 44) 0.000 000 075 062 339 414 314 188 8 × 2 = 0 + 0.000 000 150 124 678 828 628 377 6;
  • 45) 0.000 000 150 124 678 828 628 377 6 × 2 = 0 + 0.000 000 300 249 357 657 256 755 2;
  • 46) 0.000 000 300 249 357 657 256 755 2 × 2 = 0 + 0.000 000 600 498 715 314 513 510 4;
  • 47) 0.000 000 600 498 715 314 513 510 4 × 2 = 0 + 0.000 001 200 997 430 629 027 020 8;
  • 48) 0.000 001 200 997 430 629 027 020 8 × 2 = 0 + 0.000 002 401 994 861 258 054 041 6;
  • 49) 0.000 002 401 994 861 258 054 041 6 × 2 = 0 + 0.000 004 803 989 722 516 108 083 2;
  • 50) 0.000 004 803 989 722 516 108 083 2 × 2 = 0 + 0.000 009 607 979 445 032 216 166 4;
  • 51) 0.000 009 607 979 445 032 216 166 4 × 2 = 0 + 0.000 019 215 958 890 064 432 332 8;
  • 52) 0.000 019 215 958 890 064 432 332 8 × 2 = 0 + 0.000 038 431 917 780 128 864 665 6;
  • 53) 0.000 038 431 917 780 128 864 665 6 × 2 = 0 + 0.000 076 863 835 560 257 729 331 2;
  • 54) 0.000 076 863 835 560 257 729 331 2 × 2 = 0 + 0.000 153 727 671 120 515 458 662 4;
  • 55) 0.000 153 727 671 120 515 458 662 4 × 2 = 0 + 0.000 307 455 342 241 030 917 324 8;
  • 56) 0.000 307 455 342 241 030 917 324 8 × 2 = 0 + 0.000 614 910 684 482 061 834 649 6;
  • 57) 0.000 614 910 684 482 061 834 649 6 × 2 = 0 + 0.001 229 821 368 964 123 669 299 2;
  • 58) 0.001 229 821 368 964 123 669 299 2 × 2 = 0 + 0.002 459 642 737 928 247 338 598 4;
  • 59) 0.002 459 642 737 928 247 338 598 4 × 2 = 0 + 0.004 919 285 475 856 494 677 196 8;
  • 60) 0.004 919 285 475 856 494 677 196 8 × 2 = 0 + 0.009 838 570 951 712 989 354 393 6;
  • 61) 0.009 838 570 951 712 989 354 393 6 × 2 = 0 + 0.019 677 141 903 425 978 708 787 2;
  • 62) 0.019 677 141 903 425 978 708 787 2 × 2 = 0 + 0.039 354 283 806 851 957 417 574 4;
  • 63) 0.039 354 283 806 851 957 417 574 4 × 2 = 0 + 0.078 708 567 613 703 914 835 148 8;
  • 64) 0.078 708 567 613 703 914 835 148 8 × 2 = 0 + 0.157 417 135 227 407 829 670 297 6;
  • 65) 0.157 417 135 227 407 829 670 297 6 × 2 = 0 + 0.314 834 270 454 815 659 340 595 2;
  • 66) 0.314 834 270 454 815 659 340 595 2 × 2 = 0 + 0.629 668 540 909 631 318 681 190 4;
  • 67) 0.629 668 540 909 631 318 681 190 4 × 2 = 1 + 0.259 337 081 819 262 637 362 380 8;
  • 68) 0.259 337 081 819 262 637 362 380 8 × 2 = 0 + 0.518 674 163 638 525 274 724 761 6;
  • 69) 0.518 674 163 638 525 274 724 761 6 × 2 = 1 + 0.037 348 327 277 050 549 449 523 2;
  • 70) 0.037 348 327 277 050 549 449 523 2 × 2 = 0 + 0.074 696 654 554 101 098 899 046 4;
  • 71) 0.074 696 654 554 101 098 899 046 4 × 2 = 0 + 0.149 393 309 108 202 197 798 092 8;
  • 72) 0.149 393 309 108 202 197 798 092 8 × 2 = 0 + 0.298 786 618 216 404 395 596 185 6;
  • 73) 0.298 786 618 216 404 395 596 185 6 × 2 = 0 + 0.597 573 236 432 808 791 192 371 2;
  • 74) 0.597 573 236 432 808 791 192 371 2 × 2 = 1 + 0.195 146 472 865 617 582 384 742 4;
  • 75) 0.195 146 472 865 617 582 384 742 4 × 2 = 0 + 0.390 292 945 731 235 164 769 484 8;
  • 76) 0.390 292 945 731 235 164 769 484 8 × 2 = 0 + 0.780 585 891 462 470 329 538 969 6;
  • 77) 0.780 585 891 462 470 329 538 969 6 × 2 = 1 + 0.561 171 782 924 940 659 077 939 2;
  • 78) 0.561 171 782 924 940 659 077 939 2 × 2 = 1 + 0.122 343 565 849 881 318 155 878 4;
  • 79) 0.122 343 565 849 881 318 155 878 4 × 2 = 0 + 0.244 687 131 699 762 636 311 756 8;
  • 80) 0.244 687 131 699 762 636 311 756 8 × 2 = 0 + 0.489 374 263 399 525 272 623 513 6;
  • 81) 0.489 374 263 399 525 272 623 513 6 × 2 = 0 + 0.978 748 526 799 050 545 247 027 2;
  • 82) 0.978 748 526 799 050 545 247 027 2 × 2 = 1 + 0.957 497 053 598 101 090 494 054 4;
  • 83) 0.957 497 053 598 101 090 494 054 4 × 2 = 1 + 0.914 994 107 196 202 180 988 108 8;
  • 84) 0.914 994 107 196 202 180 988 108 8 × 2 = 1 + 0.829 988 214 392 404 361 976 217 6;
  • 85) 0.829 988 214 392 404 361 976 217 6 × 2 = 1 + 0.659 976 428 784 808 723 952 435 2;
  • 86) 0.659 976 428 784 808 723 952 435 2 × 2 = 1 + 0.319 952 857 569 617 447 904 870 4;
  • 87) 0.319 952 857 569 617 447 904 870 4 × 2 = 0 + 0.639 905 715 139 234 895 809 740 8;
  • 88) 0.639 905 715 139 234 895 809 740 8 × 2 = 1 + 0.279 811 430 278 469 791 619 481 6;
  • 89) 0.279 811 430 278 469 791 619 481 6 × 2 = 0 + 0.559 622 860 556 939 583 238 963 2;
  • 90) 0.559 622 860 556 939 583 238 963 2 × 2 = 1 + 0.119 245 721 113 879 166 477 926 4;
  • 91) 0.119 245 721 113 879 166 477 926 4 × 2 = 0 + 0.238 491 442 227 758 332 955 852 8;
  • 92) 0.238 491 442 227 758 332 955 852 8 × 2 = 0 + 0.476 982 884 455 516 665 911 705 6;
  • 93) 0.476 982 884 455 516 665 911 705 6 × 2 = 0 + 0.953 965 768 911 033 331 823 411 2;
  • 94) 0.953 965 768 911 033 331 823 411 2 × 2 = 1 + 0.907 931 537 822 066 663 646 822 4;
  • 95) 0.907 931 537 822 066 663 646 822 4 × 2 = 1 + 0.815 863 075 644 133 327 293 644 8;
  • 96) 0.815 863 075 644 133 327 293 644 8 × 2 = 1 + 0.631 726 151 288 266 654 587 289 6;
  • 97) 0.631 726 151 288 266 654 587 289 6 × 2 = 1 + 0.263 452 302 576 533 309 174 579 2;
  • 98) 0.263 452 302 576 533 309 174 579 2 × 2 = 0 + 0.526 904 605 153 066 618 349 158 4;
  • 99) 0.526 904 605 153 066 618 349 158 4 × 2 = 1 + 0.053 809 210 306 133 236 698 316 8;
  • 100) 0.053 809 210 306 133 236 698 316 8 × 2 = 0 + 0.107 618 420 612 266 473 396 633 6;
  • 101) 0.107 618 420 612 266 473 396 633 6 × 2 = 0 + 0.215 236 841 224 532 946 793 267 2;
  • 102) 0.215 236 841 224 532 946 793 267 2 × 2 = 0 + 0.430 473 682 449 065 893 586 534 4;
  • 103) 0.430 473 682 449 065 893 586 534 4 × 2 = 0 + 0.860 947 364 898 131 787 173 068 8;
  • 104) 0.860 947 364 898 131 787 173 068 8 × 2 = 1 + 0.721 894 729 796 263 574 346 137 6;
  • 105) 0.721 894 729 796 263 574 346 137 6 × 2 = 1 + 0.443 789 459 592 527 148 692 275 2;
  • 106) 0.443 789 459 592 527 148 692 275 2 × 2 = 0 + 0.887 578 919 185 054 297 384 550 4;
  • 107) 0.887 578 919 185 054 297 384 550 4 × 2 = 1 + 0.775 157 838 370 108 594 769 100 8;
  • 108) 0.775 157 838 370 108 594 769 100 8 × 2 = 1 + 0.550 315 676 740 217 189 538 201 6;
  • 109) 0.550 315 676 740 217 189 538 201 6 × 2 = 1 + 0.100 631 353 480 434 379 076 403 2;
  • 110) 0.100 631 353 480 434 379 076 403 2 × 2 = 0 + 0.201 262 706 960 868 758 152 806 4;
  • 111) 0.201 262 706 960 868 758 152 806 4 × 2 = 0 + 0.402 525 413 921 737 516 305 612 8;
  • 112) 0.402 525 413 921 737 516 305 612 8 × 2 = 0 + 0.805 050 827 843 475 032 611 225 6;
  • 113) 0.805 050 827 843 475 032 611 225 6 × 2 = 1 + 0.610 101 655 686 950 065 222 451 2;
  • 114) 0.610 101 655 686 950 065 222 451 2 × 2 = 1 + 0.220 203 311 373 900 130 444 902 4;
  • 115) 0.220 203 311 373 900 130 444 902 4 × 2 = 0 + 0.440 406 622 747 800 260 889 804 8;
  • 116) 0.440 406 622 747 800 260 889 804 8 × 2 = 0 + 0.880 813 245 495 600 521 779 609 6;
  • 117) 0.880 813 245 495 600 521 779 609 6 × 2 = 1 + 0.761 626 490 991 201 043 559 219 2;
  • 118) 0.761 626 490 991 201 043 559 219 2 × 2 = 1 + 0.523 252 981 982 402 087 118 438 4;
  • 119) 0.523 252 981 982 402 087 118 438 4 × 2 = 1 + 0.046 505 963 964 804 174 236 876 8;

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:


0.000 000 000 000 000 000 008 533 6(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1100 0111 1101 0100 0111 1010 0001 1011 1000 1100 111(2)

5. Positive number before normalization:

0.000 000 000 000 000 000 008 533 6(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1100 0111 1101 0100 0111 1010 0001 1011 1000 1100 111(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:


0.000 000 000 000 000 000 008 533 6(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1100 0111 1101 0100 0111 1010 0001 1011 1000 1100 111(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1100 0111 1101 0100 0111 1010 0001 1011 1000 1100 111(2) × 20 =

1.0100 0010 0110 0011 1110 1010 0011 1101 0000 1101 1100 0110 0111(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 0110 0011 1110 1010 0011 1101 0000 1101 1100 0110 0111


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 0110 0011 1110 1010 0011 1101 0000 1101 1100 0110 0111 =

0100 0010 0110 0011 1110 1010 0011 1101 0000 1101 1100 0110 0111


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 0110 0011 1110 1010 0011 1101 0000 1101 1100 0110 0111


Decimal number 0.000 000 000 000 000 000 008 533 6 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1011 1100 - 0100 0010 0110 0011 1110 1010 0011 1101 0000 1101 1100 0110 0111

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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