0.000 000 000 000 000 000 008 535 78 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 000 000 008 535 78(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
0.000 000 000 000 000 000 008 535 78(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: 0.000 000 000 000 000 000 008 535 78.

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) 0.000 000 000 000 000 000 008 535 78 × 2 = 0 + 0.000 000 000 000 000 000 017 071 56;
  • 2) 0.000 000 000 000 000 000 017 071 56 × 2 = 0 + 0.000 000 000 000 000 000 034 143 12;
  • 3) 0.000 000 000 000 000 000 034 143 12 × 2 = 0 + 0.000 000 000 000 000 000 068 286 24;
  • 4) 0.000 000 000 000 000 000 068 286 24 × 2 = 0 + 0.000 000 000 000 000 000 136 572 48;
  • 5) 0.000 000 000 000 000 000 136 572 48 × 2 = 0 + 0.000 000 000 000 000 000 273 144 96;
  • 6) 0.000 000 000 000 000 000 273 144 96 × 2 = 0 + 0.000 000 000 000 000 000 546 289 92;
  • 7) 0.000 000 000 000 000 000 546 289 92 × 2 = 0 + 0.000 000 000 000 000 001 092 579 84;
  • 8) 0.000 000 000 000 000 001 092 579 84 × 2 = 0 + 0.000 000 000 000 000 002 185 159 68;
  • 9) 0.000 000 000 000 000 002 185 159 68 × 2 = 0 + 0.000 000 000 000 000 004 370 319 36;
  • 10) 0.000 000 000 000 000 004 370 319 36 × 2 = 0 + 0.000 000 000 000 000 008 740 638 72;
  • 11) 0.000 000 000 000 000 008 740 638 72 × 2 = 0 + 0.000 000 000 000 000 017 481 277 44;
  • 12) 0.000 000 000 000 000 017 481 277 44 × 2 = 0 + 0.000 000 000 000 000 034 962 554 88;
  • 13) 0.000 000 000 000 000 034 962 554 88 × 2 = 0 + 0.000 000 000 000 000 069 925 109 76;
  • 14) 0.000 000 000 000 000 069 925 109 76 × 2 = 0 + 0.000 000 000 000 000 139 850 219 52;
  • 15) 0.000 000 000 000 000 139 850 219 52 × 2 = 0 + 0.000 000 000 000 000 279 700 439 04;
  • 16) 0.000 000 000 000 000 279 700 439 04 × 2 = 0 + 0.000 000 000 000 000 559 400 878 08;
  • 17) 0.000 000 000 000 000 559 400 878 08 × 2 = 0 + 0.000 000 000 000 001 118 801 756 16;
  • 18) 0.000 000 000 000 001 118 801 756 16 × 2 = 0 + 0.000 000 000 000 002 237 603 512 32;
  • 19) 0.000 000 000 000 002 237 603 512 32 × 2 = 0 + 0.000 000 000 000 004 475 207 024 64;
  • 20) 0.000 000 000 000 004 475 207 024 64 × 2 = 0 + 0.000 000 000 000 008 950 414 049 28;
  • 21) 0.000 000 000 000 008 950 414 049 28 × 2 = 0 + 0.000 000 000 000 017 900 828 098 56;
  • 22) 0.000 000 000 000 017 900 828 098 56 × 2 = 0 + 0.000 000 000 000 035 801 656 197 12;
  • 23) 0.000 000 000 000 035 801 656 197 12 × 2 = 0 + 0.000 000 000 000 071 603 312 394 24;
  • 24) 0.000 000 000 000 071 603 312 394 24 × 2 = 0 + 0.000 000 000 000 143 206 624 788 48;
  • 25) 0.000 000 000 000 143 206 624 788 48 × 2 = 0 + 0.000 000 000 000 286 413 249 576 96;
  • 26) 0.000 000 000 000 286 413 249 576 96 × 2 = 0 + 0.000 000 000 000 572 826 499 153 92;
  • 27) 0.000 000 000 000 572 826 499 153 92 × 2 = 0 + 0.000 000 000 001 145 652 998 307 84;
  • 28) 0.000 000 000 001 145 652 998 307 84 × 2 = 0 + 0.000 000 000 002 291 305 996 615 68;
  • 29) 0.000 000 000 002 291 305 996 615 68 × 2 = 0 + 0.000 000 000 004 582 611 993 231 36;
  • 30) 0.000 000 000 004 582 611 993 231 36 × 2 = 0 + 0.000 000 000 009 165 223 986 462 72;
  • 31) 0.000 000 000 009 165 223 986 462 72 × 2 = 0 + 0.000 000 000 018 330 447 972 925 44;
  • 32) 0.000 000 000 018 330 447 972 925 44 × 2 = 0 + 0.000 000 000 036 660 895 945 850 88;
  • 33) 0.000 000 000 036 660 895 945 850 88 × 2 = 0 + 0.000 000 000 073 321 791 891 701 76;
  • 34) 0.000 000 000 073 321 791 891 701 76 × 2 = 0 + 0.000 000 000 146 643 583 783 403 52;
  • 35) 0.000 000 000 146 643 583 783 403 52 × 2 = 0 + 0.000 000 000 293 287 167 566 807 04;
  • 36) 0.000 000 000 293 287 167 566 807 04 × 2 = 0 + 0.000 000 000 586 574 335 133 614 08;
  • 37) 0.000 000 000 586 574 335 133 614 08 × 2 = 0 + 0.000 000 001 173 148 670 267 228 16;
  • 38) 0.000 000 001 173 148 670 267 228 16 × 2 = 0 + 0.000 000 002 346 297 340 534 456 32;
  • 39) 0.000 000 002 346 297 340 534 456 32 × 2 = 0 + 0.000 000 004 692 594 681 068 912 64;
  • 40) 0.000 000 004 692 594 681 068 912 64 × 2 = 0 + 0.000 000 009 385 189 362 137 825 28;
  • 41) 0.000 000 009 385 189 362 137 825 28 × 2 = 0 + 0.000 000 018 770 378 724 275 650 56;
  • 42) 0.000 000 018 770 378 724 275 650 56 × 2 = 0 + 0.000 000 037 540 757 448 551 301 12;
  • 43) 0.000 000 037 540 757 448 551 301 12 × 2 = 0 + 0.000 000 075 081 514 897 102 602 24;
  • 44) 0.000 000 075 081 514 897 102 602 24 × 2 = 0 + 0.000 000 150 163 029 794 205 204 48;
  • 45) 0.000 000 150 163 029 794 205 204 48 × 2 = 0 + 0.000 000 300 326 059 588 410 408 96;
  • 46) 0.000 000 300 326 059 588 410 408 96 × 2 = 0 + 0.000 000 600 652 119 176 820 817 92;
  • 47) 0.000 000 600 652 119 176 820 817 92 × 2 = 0 + 0.000 001 201 304 238 353 641 635 84;
  • 48) 0.000 001 201 304 238 353 641 635 84 × 2 = 0 + 0.000 002 402 608 476 707 283 271 68;
  • 49) 0.000 002 402 608 476 707 283 271 68 × 2 = 0 + 0.000 004 805 216 953 414 566 543 36;
  • 50) 0.000 004 805 216 953 414 566 543 36 × 2 = 0 + 0.000 009 610 433 906 829 133 086 72;
  • 51) 0.000 009 610 433 906 829 133 086 72 × 2 = 0 + 0.000 019 220 867 813 658 266 173 44;
  • 52) 0.000 019 220 867 813 658 266 173 44 × 2 = 0 + 0.000 038 441 735 627 316 532 346 88;
  • 53) 0.000 038 441 735 627 316 532 346 88 × 2 = 0 + 0.000 076 883 471 254 633 064 693 76;
  • 54) 0.000 076 883 471 254 633 064 693 76 × 2 = 0 + 0.000 153 766 942 509 266 129 387 52;
  • 55) 0.000 153 766 942 509 266 129 387 52 × 2 = 0 + 0.000 307 533 885 018 532 258 775 04;
  • 56) 0.000 307 533 885 018 532 258 775 04 × 2 = 0 + 0.000 615 067 770 037 064 517 550 08;
  • 57) 0.000 615 067 770 037 064 517 550 08 × 2 = 0 + 0.001 230 135 540 074 129 035 100 16;
  • 58) 0.001 230 135 540 074 129 035 100 16 × 2 = 0 + 0.002 460 271 080 148 258 070 200 32;
  • 59) 0.002 460 271 080 148 258 070 200 32 × 2 = 0 + 0.004 920 542 160 296 516 140 400 64;
  • 60) 0.004 920 542 160 296 516 140 400 64 × 2 = 0 + 0.009 841 084 320 593 032 280 801 28;
  • 61) 0.009 841 084 320 593 032 280 801 28 × 2 = 0 + 0.019 682 168 641 186 064 561 602 56;
  • 62) 0.019 682 168 641 186 064 561 602 56 × 2 = 0 + 0.039 364 337 282 372 129 123 205 12;
  • 63) 0.039 364 337 282 372 129 123 205 12 × 2 = 0 + 0.078 728 674 564 744 258 246 410 24;
  • 64) 0.078 728 674 564 744 258 246 410 24 × 2 = 0 + 0.157 457 349 129 488 516 492 820 48;
  • 65) 0.157 457 349 129 488 516 492 820 48 × 2 = 0 + 0.314 914 698 258 977 032 985 640 96;
  • 66) 0.314 914 698 258 977 032 985 640 96 × 2 = 0 + 0.629 829 396 517 954 065 971 281 92;
  • 67) 0.629 829 396 517 954 065 971 281 92 × 2 = 1 + 0.259 658 793 035 908 131 942 563 84;
  • 68) 0.259 658 793 035 908 131 942 563 84 × 2 = 0 + 0.519 317 586 071 816 263 885 127 68;
  • 69) 0.519 317 586 071 816 263 885 127 68 × 2 = 1 + 0.038 635 172 143 632 527 770 255 36;
  • 70) 0.038 635 172 143 632 527 770 255 36 × 2 = 0 + 0.077 270 344 287 265 055 540 510 72;
  • 71) 0.077 270 344 287 265 055 540 510 72 × 2 = 0 + 0.154 540 688 574 530 111 081 021 44;
  • 72) 0.154 540 688 574 530 111 081 021 44 × 2 = 0 + 0.309 081 377 149 060 222 162 042 88;
  • 73) 0.309 081 377 149 060 222 162 042 88 × 2 = 0 + 0.618 162 754 298 120 444 324 085 76;
  • 74) 0.618 162 754 298 120 444 324 085 76 × 2 = 1 + 0.236 325 508 596 240 888 648 171 52;
  • 75) 0.236 325 508 596 240 888 648 171 52 × 2 = 0 + 0.472 651 017 192 481 777 296 343 04;
  • 76) 0.472 651 017 192 481 777 296 343 04 × 2 = 0 + 0.945 302 034 384 963 554 592 686 08;
  • 77) 0.945 302 034 384 963 554 592 686 08 × 2 = 1 + 0.890 604 068 769 927 109 185 372 16;
  • 78) 0.890 604 068 769 927 109 185 372 16 × 2 = 1 + 0.781 208 137 539 854 218 370 744 32;
  • 79) 0.781 208 137 539 854 218 370 744 32 × 2 = 1 + 0.562 416 275 079 708 436 741 488 64;
  • 80) 0.562 416 275 079 708 436 741 488 64 × 2 = 1 + 0.124 832 550 159 416 873 482 977 28;
  • 81) 0.124 832 550 159 416 873 482 977 28 × 2 = 0 + 0.249 665 100 318 833 746 965 954 56;
  • 82) 0.249 665 100 318 833 746 965 954 56 × 2 = 0 + 0.499 330 200 637 667 493 931 909 12;
  • 83) 0.499 330 200 637 667 493 931 909 12 × 2 = 0 + 0.998 660 401 275 334 987 863 818 24;
  • 84) 0.998 660 401 275 334 987 863 818 24 × 2 = 1 + 0.997 320 802 550 669 975 727 636 48;
  • 85) 0.997 320 802 550 669 975 727 636 48 × 2 = 1 + 0.994 641 605 101 339 951 455 272 96;
  • 86) 0.994 641 605 101 339 951 455 272 96 × 2 = 1 + 0.989 283 210 202 679 902 910 545 92;
  • 87) 0.989 283 210 202 679 902 910 545 92 × 2 = 1 + 0.978 566 420 405 359 805 821 091 84;
  • 88) 0.978 566 420 405 359 805 821 091 84 × 2 = 1 + 0.957 132 840 810 719 611 642 183 68;
  • 89) 0.957 132 840 810 719 611 642 183 68 × 2 = 1 + 0.914 265 681 621 439 223 284 367 36;
  • 90) 0.914 265 681 621 439 223 284 367 36 × 2 = 1 + 0.828 531 363 242 878 446 568 734 72;
  • 91) 0.828 531 363 242 878 446 568 734 72 × 2 = 1 + 0.657 062 726 485 756 893 137 469 44;
  • 92) 0.657 062 726 485 756 893 137 469 44 × 2 = 1 + 0.314 125 452 971 513 786 274 938 88;
  • 93) 0.314 125 452 971 513 786 274 938 88 × 2 = 0 + 0.628 250 905 943 027 572 549 877 76;
  • 94) 0.628 250 905 943 027 572 549 877 76 × 2 = 1 + 0.256 501 811 886 055 145 099 755 52;
  • 95) 0.256 501 811 886 055 145 099 755 52 × 2 = 0 + 0.513 003 623 772 110 290 199 511 04;
  • 96) 0.513 003 623 772 110 290 199 511 04 × 2 = 1 + 0.026 007 247 544 220 580 399 022 08;
  • 97) 0.026 007 247 544 220 580 399 022 08 × 2 = 0 + 0.052 014 495 088 441 160 798 044 16;
  • 98) 0.052 014 495 088 441 160 798 044 16 × 2 = 0 + 0.104 028 990 176 882 321 596 088 32;
  • 99) 0.104 028 990 176 882 321 596 088 32 × 2 = 0 + 0.208 057 980 353 764 643 192 176 64;
  • 100) 0.208 057 980 353 764 643 192 176 64 × 2 = 0 + 0.416 115 960 707 529 286 384 353 28;
  • 101) 0.416 115 960 707 529 286 384 353 28 × 2 = 0 + 0.832 231 921 415 058 572 768 706 56;
  • 102) 0.832 231 921 415 058 572 768 706 56 × 2 = 1 + 0.664 463 842 830 117 145 537 413 12;
  • 103) 0.664 463 842 830 117 145 537 413 12 × 2 = 1 + 0.328 927 685 660 234 291 074 826 24;
  • 104) 0.328 927 685 660 234 291 074 826 24 × 2 = 0 + 0.657 855 371 320 468 582 149 652 48;
  • 105) 0.657 855 371 320 468 582 149 652 48 × 2 = 1 + 0.315 710 742 640 937 164 299 304 96;
  • 106) 0.315 710 742 640 937 164 299 304 96 × 2 = 0 + 0.631 421 485 281 874 328 598 609 92;
  • 107) 0.631 421 485 281 874 328 598 609 92 × 2 = 1 + 0.262 842 970 563 748 657 197 219 84;
  • 108) 0.262 842 970 563 748 657 197 219 84 × 2 = 0 + 0.525 685 941 127 497 314 394 439 68;
  • 109) 0.525 685 941 127 497 314 394 439 68 × 2 = 1 + 0.051 371 882 254 994 628 788 879 36;
  • 110) 0.051 371 882 254 994 628 788 879 36 × 2 = 0 + 0.102 743 764 509 989 257 577 758 72;
  • 111) 0.102 743 764 509 989 257 577 758 72 × 2 = 0 + 0.205 487 529 019 978 515 155 517 44;
  • 112) 0.205 487 529 019 978 515 155 517 44 × 2 = 0 + 0.410 975 058 039 957 030 311 034 88;
  • 113) 0.410 975 058 039 957 030 311 034 88 × 2 = 0 + 0.821 950 116 079 914 060 622 069 76;
  • 114) 0.821 950 116 079 914 060 622 069 76 × 2 = 1 + 0.643 900 232 159 828 121 244 139 52;
  • 115) 0.643 900 232 159 828 121 244 139 52 × 2 = 1 + 0.287 800 464 319 656 242 488 279 04;
  • 116) 0.287 800 464 319 656 242 488 279 04 × 2 = 0 + 0.575 600 928 639 312 484 976 558 08;
  • 117) 0.575 600 928 639 312 484 976 558 08 × 2 = 1 + 0.151 201 857 278 624 969 953 116 16;
  • 118) 0.151 201 857 278 624 969 953 116 16 × 2 = 0 + 0.302 403 714 557 249 939 906 232 32;
  • 119) 0.302 403 714 557 249 939 906 232 32 × 2 = 0 + 0.604 807 429 114 499 879 812 464 64;

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:


0.000 000 000 000 000 000 008 535 78(10) =

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

5. Positive number before normalization:

0.000 000 000 000 000 000 008 535 78(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1111 0001 1111 1111 0101 0000 0110 1010 1000 0110 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:


0.000 000 000 000 000 000 008 535 78(10) =

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

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

1.0100 0010 0111 1000 1111 1111 1010 1000 0011 0101 0100 0011 0100(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 0111 1000 1111 1111 1010 1000 0011 0101 0100 0011 0100


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 0111 1000 1111 1111 1010 1000 0011 0101 0100 0011 0100 =

0100 0010 0111 1000 1111 1111 1010 1000 0011 0101 0100 0011 0100


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 0111 1000 1111 1111 1010 1000 0011 0101 0100 0011 0100


Decimal number 0.000 000 000 000 000 000 008 535 78 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1011 1100 - 0100 0010 0111 1000 1111 1111 1010 1000 0011 0101 0100 0011 0100

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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