0.000 000 000 000 000 000 008 542 3 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 000 000 008 542 3(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
0.000 000 000 000 000 000 008 542 3(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: 0.000 000 000 000 000 000 008 542 3.

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) 0.000 000 000 000 000 000 008 542 3 × 2 = 0 + 0.000 000 000 000 000 000 017 084 6;
  • 2) 0.000 000 000 000 000 000 017 084 6 × 2 = 0 + 0.000 000 000 000 000 000 034 169 2;
  • 3) 0.000 000 000 000 000 000 034 169 2 × 2 = 0 + 0.000 000 000 000 000 000 068 338 4;
  • 4) 0.000 000 000 000 000 000 068 338 4 × 2 = 0 + 0.000 000 000 000 000 000 136 676 8;
  • 5) 0.000 000 000 000 000 000 136 676 8 × 2 = 0 + 0.000 000 000 000 000 000 273 353 6;
  • 6) 0.000 000 000 000 000 000 273 353 6 × 2 = 0 + 0.000 000 000 000 000 000 546 707 2;
  • 7) 0.000 000 000 000 000 000 546 707 2 × 2 = 0 + 0.000 000 000 000 000 001 093 414 4;
  • 8) 0.000 000 000 000 000 001 093 414 4 × 2 = 0 + 0.000 000 000 000 000 002 186 828 8;
  • 9) 0.000 000 000 000 000 002 186 828 8 × 2 = 0 + 0.000 000 000 000 000 004 373 657 6;
  • 10) 0.000 000 000 000 000 004 373 657 6 × 2 = 0 + 0.000 000 000 000 000 008 747 315 2;
  • 11) 0.000 000 000 000 000 008 747 315 2 × 2 = 0 + 0.000 000 000 000 000 017 494 630 4;
  • 12) 0.000 000 000 000 000 017 494 630 4 × 2 = 0 + 0.000 000 000 000 000 034 989 260 8;
  • 13) 0.000 000 000 000 000 034 989 260 8 × 2 = 0 + 0.000 000 000 000 000 069 978 521 6;
  • 14) 0.000 000 000 000 000 069 978 521 6 × 2 = 0 + 0.000 000 000 000 000 139 957 043 2;
  • 15) 0.000 000 000 000 000 139 957 043 2 × 2 = 0 + 0.000 000 000 000 000 279 914 086 4;
  • 16) 0.000 000 000 000 000 279 914 086 4 × 2 = 0 + 0.000 000 000 000 000 559 828 172 8;
  • 17) 0.000 000 000 000 000 559 828 172 8 × 2 = 0 + 0.000 000 000 000 001 119 656 345 6;
  • 18) 0.000 000 000 000 001 119 656 345 6 × 2 = 0 + 0.000 000 000 000 002 239 312 691 2;
  • 19) 0.000 000 000 000 002 239 312 691 2 × 2 = 0 + 0.000 000 000 000 004 478 625 382 4;
  • 20) 0.000 000 000 000 004 478 625 382 4 × 2 = 0 + 0.000 000 000 000 008 957 250 764 8;
  • 21) 0.000 000 000 000 008 957 250 764 8 × 2 = 0 + 0.000 000 000 000 017 914 501 529 6;
  • 22) 0.000 000 000 000 017 914 501 529 6 × 2 = 0 + 0.000 000 000 000 035 829 003 059 2;
  • 23) 0.000 000 000 000 035 829 003 059 2 × 2 = 0 + 0.000 000 000 000 071 658 006 118 4;
  • 24) 0.000 000 000 000 071 658 006 118 4 × 2 = 0 + 0.000 000 000 000 143 316 012 236 8;
  • 25) 0.000 000 000 000 143 316 012 236 8 × 2 = 0 + 0.000 000 000 000 286 632 024 473 6;
  • 26) 0.000 000 000 000 286 632 024 473 6 × 2 = 0 + 0.000 000 000 000 573 264 048 947 2;
  • 27) 0.000 000 000 000 573 264 048 947 2 × 2 = 0 + 0.000 000 000 001 146 528 097 894 4;
  • 28) 0.000 000 000 001 146 528 097 894 4 × 2 = 0 + 0.000 000 000 002 293 056 195 788 8;
  • 29) 0.000 000 000 002 293 056 195 788 8 × 2 = 0 + 0.000 000 000 004 586 112 391 577 6;
  • 30) 0.000 000 000 004 586 112 391 577 6 × 2 = 0 + 0.000 000 000 009 172 224 783 155 2;
  • 31) 0.000 000 000 009 172 224 783 155 2 × 2 = 0 + 0.000 000 000 018 344 449 566 310 4;
  • 32) 0.000 000 000 018 344 449 566 310 4 × 2 = 0 + 0.000 000 000 036 688 899 132 620 8;
  • 33) 0.000 000 000 036 688 899 132 620 8 × 2 = 0 + 0.000 000 000 073 377 798 265 241 6;
  • 34) 0.000 000 000 073 377 798 265 241 6 × 2 = 0 + 0.000 000 000 146 755 596 530 483 2;
  • 35) 0.000 000 000 146 755 596 530 483 2 × 2 = 0 + 0.000 000 000 293 511 193 060 966 4;
  • 36) 0.000 000 000 293 511 193 060 966 4 × 2 = 0 + 0.000 000 000 587 022 386 121 932 8;
  • 37) 0.000 000 000 587 022 386 121 932 8 × 2 = 0 + 0.000 000 001 174 044 772 243 865 6;
  • 38) 0.000 000 001 174 044 772 243 865 6 × 2 = 0 + 0.000 000 002 348 089 544 487 731 2;
  • 39) 0.000 000 002 348 089 544 487 731 2 × 2 = 0 + 0.000 000 004 696 179 088 975 462 4;
  • 40) 0.000 000 004 696 179 088 975 462 4 × 2 = 0 + 0.000 000 009 392 358 177 950 924 8;
  • 41) 0.000 000 009 392 358 177 950 924 8 × 2 = 0 + 0.000 000 018 784 716 355 901 849 6;
  • 42) 0.000 000 018 784 716 355 901 849 6 × 2 = 0 + 0.000 000 037 569 432 711 803 699 2;
  • 43) 0.000 000 037 569 432 711 803 699 2 × 2 = 0 + 0.000 000 075 138 865 423 607 398 4;
  • 44) 0.000 000 075 138 865 423 607 398 4 × 2 = 0 + 0.000 000 150 277 730 847 214 796 8;
  • 45) 0.000 000 150 277 730 847 214 796 8 × 2 = 0 + 0.000 000 300 555 461 694 429 593 6;
  • 46) 0.000 000 300 555 461 694 429 593 6 × 2 = 0 + 0.000 000 601 110 923 388 859 187 2;
  • 47) 0.000 000 601 110 923 388 859 187 2 × 2 = 0 + 0.000 001 202 221 846 777 718 374 4;
  • 48) 0.000 001 202 221 846 777 718 374 4 × 2 = 0 + 0.000 002 404 443 693 555 436 748 8;
  • 49) 0.000 002 404 443 693 555 436 748 8 × 2 = 0 + 0.000 004 808 887 387 110 873 497 6;
  • 50) 0.000 004 808 887 387 110 873 497 6 × 2 = 0 + 0.000 009 617 774 774 221 746 995 2;
  • 51) 0.000 009 617 774 774 221 746 995 2 × 2 = 0 + 0.000 019 235 549 548 443 493 990 4;
  • 52) 0.000 019 235 549 548 443 493 990 4 × 2 = 0 + 0.000 038 471 099 096 886 987 980 8;
  • 53) 0.000 038 471 099 096 886 987 980 8 × 2 = 0 + 0.000 076 942 198 193 773 975 961 6;
  • 54) 0.000 076 942 198 193 773 975 961 6 × 2 = 0 + 0.000 153 884 396 387 547 951 923 2;
  • 55) 0.000 153 884 396 387 547 951 923 2 × 2 = 0 + 0.000 307 768 792 775 095 903 846 4;
  • 56) 0.000 307 768 792 775 095 903 846 4 × 2 = 0 + 0.000 615 537 585 550 191 807 692 8;
  • 57) 0.000 615 537 585 550 191 807 692 8 × 2 = 0 + 0.001 231 075 171 100 383 615 385 6;
  • 58) 0.001 231 075 171 100 383 615 385 6 × 2 = 0 + 0.002 462 150 342 200 767 230 771 2;
  • 59) 0.002 462 150 342 200 767 230 771 2 × 2 = 0 + 0.004 924 300 684 401 534 461 542 4;
  • 60) 0.004 924 300 684 401 534 461 542 4 × 2 = 0 + 0.009 848 601 368 803 068 923 084 8;
  • 61) 0.009 848 601 368 803 068 923 084 8 × 2 = 0 + 0.019 697 202 737 606 137 846 169 6;
  • 62) 0.019 697 202 737 606 137 846 169 6 × 2 = 0 + 0.039 394 405 475 212 275 692 339 2;
  • 63) 0.039 394 405 475 212 275 692 339 2 × 2 = 0 + 0.078 788 810 950 424 551 384 678 4;
  • 64) 0.078 788 810 950 424 551 384 678 4 × 2 = 0 + 0.157 577 621 900 849 102 769 356 8;
  • 65) 0.157 577 621 900 849 102 769 356 8 × 2 = 0 + 0.315 155 243 801 698 205 538 713 6;
  • 66) 0.315 155 243 801 698 205 538 713 6 × 2 = 0 + 0.630 310 487 603 396 411 077 427 2;
  • 67) 0.630 310 487 603 396 411 077 427 2 × 2 = 1 + 0.260 620 975 206 792 822 154 854 4;
  • 68) 0.260 620 975 206 792 822 154 854 4 × 2 = 0 + 0.521 241 950 413 585 644 309 708 8;
  • 69) 0.521 241 950 413 585 644 309 708 8 × 2 = 1 + 0.042 483 900 827 171 288 619 417 6;
  • 70) 0.042 483 900 827 171 288 619 417 6 × 2 = 0 + 0.084 967 801 654 342 577 238 835 2;
  • 71) 0.084 967 801 654 342 577 238 835 2 × 2 = 0 + 0.169 935 603 308 685 154 477 670 4;
  • 72) 0.169 935 603 308 685 154 477 670 4 × 2 = 0 + 0.339 871 206 617 370 308 955 340 8;
  • 73) 0.339 871 206 617 370 308 955 340 8 × 2 = 0 + 0.679 742 413 234 740 617 910 681 6;
  • 74) 0.679 742 413 234 740 617 910 681 6 × 2 = 1 + 0.359 484 826 469 481 235 821 363 2;
  • 75) 0.359 484 826 469 481 235 821 363 2 × 2 = 0 + 0.718 969 652 938 962 471 642 726 4;
  • 76) 0.718 969 652 938 962 471 642 726 4 × 2 = 1 + 0.437 939 305 877 924 943 285 452 8;
  • 77) 0.437 939 305 877 924 943 285 452 8 × 2 = 0 + 0.875 878 611 755 849 886 570 905 6;
  • 78) 0.875 878 611 755 849 886 570 905 6 × 2 = 1 + 0.751 757 223 511 699 773 141 811 2;
  • 79) 0.751 757 223 511 699 773 141 811 2 × 2 = 1 + 0.503 514 447 023 399 546 283 622 4;
  • 80) 0.503 514 447 023 399 546 283 622 4 × 2 = 1 + 0.007 028 894 046 799 092 567 244 8;
  • 81) 0.007 028 894 046 799 092 567 244 8 × 2 = 0 + 0.014 057 788 093 598 185 134 489 6;
  • 82) 0.014 057 788 093 598 185 134 489 6 × 2 = 0 + 0.028 115 576 187 196 370 268 979 2;
  • 83) 0.028 115 576 187 196 370 268 979 2 × 2 = 0 + 0.056 231 152 374 392 740 537 958 4;
  • 84) 0.056 231 152 374 392 740 537 958 4 × 2 = 0 + 0.112 462 304 748 785 481 075 916 8;
  • 85) 0.112 462 304 748 785 481 075 916 8 × 2 = 0 + 0.224 924 609 497 570 962 151 833 6;
  • 86) 0.224 924 609 497 570 962 151 833 6 × 2 = 0 + 0.449 849 218 995 141 924 303 667 2;
  • 87) 0.449 849 218 995 141 924 303 667 2 × 2 = 0 + 0.899 698 437 990 283 848 607 334 4;
  • 88) 0.899 698 437 990 283 848 607 334 4 × 2 = 1 + 0.799 396 875 980 567 697 214 668 8;
  • 89) 0.799 396 875 980 567 697 214 668 8 × 2 = 1 + 0.598 793 751 961 135 394 429 337 6;
  • 90) 0.598 793 751 961 135 394 429 337 6 × 2 = 1 + 0.197 587 503 922 270 788 858 675 2;
  • 91) 0.197 587 503 922 270 788 858 675 2 × 2 = 0 + 0.395 175 007 844 541 577 717 350 4;
  • 92) 0.395 175 007 844 541 577 717 350 4 × 2 = 0 + 0.790 350 015 689 083 155 434 700 8;
  • 93) 0.790 350 015 689 083 155 434 700 8 × 2 = 1 + 0.580 700 031 378 166 310 869 401 6;
  • 94) 0.580 700 031 378 166 310 869 401 6 × 2 = 1 + 0.161 400 062 756 332 621 738 803 2;
  • 95) 0.161 400 062 756 332 621 738 803 2 × 2 = 0 + 0.322 800 125 512 665 243 477 606 4;
  • 96) 0.322 800 125 512 665 243 477 606 4 × 2 = 0 + 0.645 600 251 025 330 486 955 212 8;
  • 97) 0.645 600 251 025 330 486 955 212 8 × 2 = 1 + 0.291 200 502 050 660 973 910 425 6;
  • 98) 0.291 200 502 050 660 973 910 425 6 × 2 = 0 + 0.582 401 004 101 321 947 820 851 2;
  • 99) 0.582 401 004 101 321 947 820 851 2 × 2 = 1 + 0.164 802 008 202 643 895 641 702 4;
  • 100) 0.164 802 008 202 643 895 641 702 4 × 2 = 0 + 0.329 604 016 405 287 791 283 404 8;
  • 101) 0.329 604 016 405 287 791 283 404 8 × 2 = 0 + 0.659 208 032 810 575 582 566 809 6;
  • 102) 0.659 208 032 810 575 582 566 809 6 × 2 = 1 + 0.318 416 065 621 151 165 133 619 2;
  • 103) 0.318 416 065 621 151 165 133 619 2 × 2 = 0 + 0.636 832 131 242 302 330 267 238 4;
  • 104) 0.636 832 131 242 302 330 267 238 4 × 2 = 1 + 0.273 664 262 484 604 660 534 476 8;
  • 105) 0.273 664 262 484 604 660 534 476 8 × 2 = 0 + 0.547 328 524 969 209 321 068 953 6;
  • 106) 0.547 328 524 969 209 321 068 953 6 × 2 = 1 + 0.094 657 049 938 418 642 137 907 2;
  • 107) 0.094 657 049 938 418 642 137 907 2 × 2 = 0 + 0.189 314 099 876 837 284 275 814 4;
  • 108) 0.189 314 099 876 837 284 275 814 4 × 2 = 0 + 0.378 628 199 753 674 568 551 628 8;
  • 109) 0.378 628 199 753 674 568 551 628 8 × 2 = 0 + 0.757 256 399 507 349 137 103 257 6;
  • 110) 0.757 256 399 507 349 137 103 257 6 × 2 = 1 + 0.514 512 799 014 698 274 206 515 2;
  • 111) 0.514 512 799 014 698 274 206 515 2 × 2 = 1 + 0.029 025 598 029 396 548 413 030 4;
  • 112) 0.029 025 598 029 396 548 413 030 4 × 2 = 0 + 0.058 051 196 058 793 096 826 060 8;
  • 113) 0.058 051 196 058 793 096 826 060 8 × 2 = 0 + 0.116 102 392 117 586 193 652 121 6;
  • 114) 0.116 102 392 117 586 193 652 121 6 × 2 = 0 + 0.232 204 784 235 172 387 304 243 2;
  • 115) 0.232 204 784 235 172 387 304 243 2 × 2 = 0 + 0.464 409 568 470 344 774 608 486 4;
  • 116) 0.464 409 568 470 344 774 608 486 4 × 2 = 0 + 0.928 819 136 940 689 549 216 972 8;
  • 117) 0.928 819 136 940 689 549 216 972 8 × 2 = 1 + 0.857 638 273 881 379 098 433 945 6;
  • 118) 0.857 638 273 881 379 098 433 945 6 × 2 = 1 + 0.715 276 547 762 758 196 867 891 2;
  • 119) 0.715 276 547 762 758 196 867 891 2 × 2 = 1 + 0.430 553 095 525 516 393 735 782 4;

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:


0.000 000 000 000 000 000 008 542 3(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0101 0111 0000 0001 1100 1100 1010 0101 0100 0110 0000 111(2)

5. Positive number before normalization:

0.000 000 000 000 000 000 008 542 3(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0101 0111 0000 0001 1100 1100 1010 0101 0100 0110 0000 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:


0.000 000 000 000 000 000 008 542 3(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0101 0111 0000 0001 1100 1100 1010 0101 0100 0110 0000 111(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0101 0111 0000 0001 1100 1100 1010 0101 0100 0110 0000 111(2) × 20 =

1.0100 0010 1011 1000 0000 1110 0110 0101 0010 1010 0011 0000 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 1011 1000 0000 1110 0110 0101 0010 1010 0011 0000 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 1011 1000 0000 1110 0110 0101 0010 1010 0011 0000 0111 =

0100 0010 1011 1000 0000 1110 0110 0101 0010 1010 0011 0000 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 1011 1000 0000 1110 0110 0101 0010 1010 0011 0000 0111


Decimal number 0.000 000 000 000 000 000 008 542 3 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1011 1100 - 0100 0010 1011 1000 0000 1110 0110 0101 0010 1010 0011 0000 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