0.000 000 000 000 000 000 008 534 8 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 000 000 008 534 8(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
0.000 000 000 000 000 000 008 534 8(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: 0.000 000 000 000 000 000 008 534 8.

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) 0.000 000 000 000 000 000 008 534 8 × 2 = 0 + 0.000 000 000 000 000 000 017 069 6;
  • 2) 0.000 000 000 000 000 000 017 069 6 × 2 = 0 + 0.000 000 000 000 000 000 034 139 2;
  • 3) 0.000 000 000 000 000 000 034 139 2 × 2 = 0 + 0.000 000 000 000 000 000 068 278 4;
  • 4) 0.000 000 000 000 000 000 068 278 4 × 2 = 0 + 0.000 000 000 000 000 000 136 556 8;
  • 5) 0.000 000 000 000 000 000 136 556 8 × 2 = 0 + 0.000 000 000 000 000 000 273 113 6;
  • 6) 0.000 000 000 000 000 000 273 113 6 × 2 = 0 + 0.000 000 000 000 000 000 546 227 2;
  • 7) 0.000 000 000 000 000 000 546 227 2 × 2 = 0 + 0.000 000 000 000 000 001 092 454 4;
  • 8) 0.000 000 000 000 000 001 092 454 4 × 2 = 0 + 0.000 000 000 000 000 002 184 908 8;
  • 9) 0.000 000 000 000 000 002 184 908 8 × 2 = 0 + 0.000 000 000 000 000 004 369 817 6;
  • 10) 0.000 000 000 000 000 004 369 817 6 × 2 = 0 + 0.000 000 000 000 000 008 739 635 2;
  • 11) 0.000 000 000 000 000 008 739 635 2 × 2 = 0 + 0.000 000 000 000 000 017 479 270 4;
  • 12) 0.000 000 000 000 000 017 479 270 4 × 2 = 0 + 0.000 000 000 000 000 034 958 540 8;
  • 13) 0.000 000 000 000 000 034 958 540 8 × 2 = 0 + 0.000 000 000 000 000 069 917 081 6;
  • 14) 0.000 000 000 000 000 069 917 081 6 × 2 = 0 + 0.000 000 000 000 000 139 834 163 2;
  • 15) 0.000 000 000 000 000 139 834 163 2 × 2 = 0 + 0.000 000 000 000 000 279 668 326 4;
  • 16) 0.000 000 000 000 000 279 668 326 4 × 2 = 0 + 0.000 000 000 000 000 559 336 652 8;
  • 17) 0.000 000 000 000 000 559 336 652 8 × 2 = 0 + 0.000 000 000 000 001 118 673 305 6;
  • 18) 0.000 000 000 000 001 118 673 305 6 × 2 = 0 + 0.000 000 000 000 002 237 346 611 2;
  • 19) 0.000 000 000 000 002 237 346 611 2 × 2 = 0 + 0.000 000 000 000 004 474 693 222 4;
  • 20) 0.000 000 000 000 004 474 693 222 4 × 2 = 0 + 0.000 000 000 000 008 949 386 444 8;
  • 21) 0.000 000 000 000 008 949 386 444 8 × 2 = 0 + 0.000 000 000 000 017 898 772 889 6;
  • 22) 0.000 000 000 000 017 898 772 889 6 × 2 = 0 + 0.000 000 000 000 035 797 545 779 2;
  • 23) 0.000 000 000 000 035 797 545 779 2 × 2 = 0 + 0.000 000 000 000 071 595 091 558 4;
  • 24) 0.000 000 000 000 071 595 091 558 4 × 2 = 0 + 0.000 000 000 000 143 190 183 116 8;
  • 25) 0.000 000 000 000 143 190 183 116 8 × 2 = 0 + 0.000 000 000 000 286 380 366 233 6;
  • 26) 0.000 000 000 000 286 380 366 233 6 × 2 = 0 + 0.000 000 000 000 572 760 732 467 2;
  • 27) 0.000 000 000 000 572 760 732 467 2 × 2 = 0 + 0.000 000 000 001 145 521 464 934 4;
  • 28) 0.000 000 000 001 145 521 464 934 4 × 2 = 0 + 0.000 000 000 002 291 042 929 868 8;
  • 29) 0.000 000 000 002 291 042 929 868 8 × 2 = 0 + 0.000 000 000 004 582 085 859 737 6;
  • 30) 0.000 000 000 004 582 085 859 737 6 × 2 = 0 + 0.000 000 000 009 164 171 719 475 2;
  • 31) 0.000 000 000 009 164 171 719 475 2 × 2 = 0 + 0.000 000 000 018 328 343 438 950 4;
  • 32) 0.000 000 000 018 328 343 438 950 4 × 2 = 0 + 0.000 000 000 036 656 686 877 900 8;
  • 33) 0.000 000 000 036 656 686 877 900 8 × 2 = 0 + 0.000 000 000 073 313 373 755 801 6;
  • 34) 0.000 000 000 073 313 373 755 801 6 × 2 = 0 + 0.000 000 000 146 626 747 511 603 2;
  • 35) 0.000 000 000 146 626 747 511 603 2 × 2 = 0 + 0.000 000 000 293 253 495 023 206 4;
  • 36) 0.000 000 000 293 253 495 023 206 4 × 2 = 0 + 0.000 000 000 586 506 990 046 412 8;
  • 37) 0.000 000 000 586 506 990 046 412 8 × 2 = 0 + 0.000 000 001 173 013 980 092 825 6;
  • 38) 0.000 000 001 173 013 980 092 825 6 × 2 = 0 + 0.000 000 002 346 027 960 185 651 2;
  • 39) 0.000 000 002 346 027 960 185 651 2 × 2 = 0 + 0.000 000 004 692 055 920 371 302 4;
  • 40) 0.000 000 004 692 055 920 371 302 4 × 2 = 0 + 0.000 000 009 384 111 840 742 604 8;
  • 41) 0.000 000 009 384 111 840 742 604 8 × 2 = 0 + 0.000 000 018 768 223 681 485 209 6;
  • 42) 0.000 000 018 768 223 681 485 209 6 × 2 = 0 + 0.000 000 037 536 447 362 970 419 2;
  • 43) 0.000 000 037 536 447 362 970 419 2 × 2 = 0 + 0.000 000 075 072 894 725 940 838 4;
  • 44) 0.000 000 075 072 894 725 940 838 4 × 2 = 0 + 0.000 000 150 145 789 451 881 676 8;
  • 45) 0.000 000 150 145 789 451 881 676 8 × 2 = 0 + 0.000 000 300 291 578 903 763 353 6;
  • 46) 0.000 000 300 291 578 903 763 353 6 × 2 = 0 + 0.000 000 600 583 157 807 526 707 2;
  • 47) 0.000 000 600 583 157 807 526 707 2 × 2 = 0 + 0.000 001 201 166 315 615 053 414 4;
  • 48) 0.000 001 201 166 315 615 053 414 4 × 2 = 0 + 0.000 002 402 332 631 230 106 828 8;
  • 49) 0.000 002 402 332 631 230 106 828 8 × 2 = 0 + 0.000 004 804 665 262 460 213 657 6;
  • 50) 0.000 004 804 665 262 460 213 657 6 × 2 = 0 + 0.000 009 609 330 524 920 427 315 2;
  • 51) 0.000 009 609 330 524 920 427 315 2 × 2 = 0 + 0.000 019 218 661 049 840 854 630 4;
  • 52) 0.000 019 218 661 049 840 854 630 4 × 2 = 0 + 0.000 038 437 322 099 681 709 260 8;
  • 53) 0.000 038 437 322 099 681 709 260 8 × 2 = 0 + 0.000 076 874 644 199 363 418 521 6;
  • 54) 0.000 076 874 644 199 363 418 521 6 × 2 = 0 + 0.000 153 749 288 398 726 837 043 2;
  • 55) 0.000 153 749 288 398 726 837 043 2 × 2 = 0 + 0.000 307 498 576 797 453 674 086 4;
  • 56) 0.000 307 498 576 797 453 674 086 4 × 2 = 0 + 0.000 614 997 153 594 907 348 172 8;
  • 57) 0.000 614 997 153 594 907 348 172 8 × 2 = 0 + 0.001 229 994 307 189 814 696 345 6;
  • 58) 0.001 229 994 307 189 814 696 345 6 × 2 = 0 + 0.002 459 988 614 379 629 392 691 2;
  • 59) 0.002 459 988 614 379 629 392 691 2 × 2 = 0 + 0.004 919 977 228 759 258 785 382 4;
  • 60) 0.004 919 977 228 759 258 785 382 4 × 2 = 0 + 0.009 839 954 457 518 517 570 764 8;
  • 61) 0.009 839 954 457 518 517 570 764 8 × 2 = 0 + 0.019 679 908 915 037 035 141 529 6;
  • 62) 0.019 679 908 915 037 035 141 529 6 × 2 = 0 + 0.039 359 817 830 074 070 283 059 2;
  • 63) 0.039 359 817 830 074 070 283 059 2 × 2 = 0 + 0.078 719 635 660 148 140 566 118 4;
  • 64) 0.078 719 635 660 148 140 566 118 4 × 2 = 0 + 0.157 439 271 320 296 281 132 236 8;
  • 65) 0.157 439 271 320 296 281 132 236 8 × 2 = 0 + 0.314 878 542 640 592 562 264 473 6;
  • 66) 0.314 878 542 640 592 562 264 473 6 × 2 = 0 + 0.629 757 085 281 185 124 528 947 2;
  • 67) 0.629 757 085 281 185 124 528 947 2 × 2 = 1 + 0.259 514 170 562 370 249 057 894 4;
  • 68) 0.259 514 170 562 370 249 057 894 4 × 2 = 0 + 0.519 028 341 124 740 498 115 788 8;
  • 69) 0.519 028 341 124 740 498 115 788 8 × 2 = 1 + 0.038 056 682 249 480 996 231 577 6;
  • 70) 0.038 056 682 249 480 996 231 577 6 × 2 = 0 + 0.076 113 364 498 961 992 463 155 2;
  • 71) 0.076 113 364 498 961 992 463 155 2 × 2 = 0 + 0.152 226 728 997 923 984 926 310 4;
  • 72) 0.152 226 728 997 923 984 926 310 4 × 2 = 0 + 0.304 453 457 995 847 969 852 620 8;
  • 73) 0.304 453 457 995 847 969 852 620 8 × 2 = 0 + 0.608 906 915 991 695 939 705 241 6;
  • 74) 0.608 906 915 991 695 939 705 241 6 × 2 = 1 + 0.217 813 831 983 391 879 410 483 2;
  • 75) 0.217 813 831 983 391 879 410 483 2 × 2 = 0 + 0.435 627 663 966 783 758 820 966 4;
  • 76) 0.435 627 663 966 783 758 820 966 4 × 2 = 0 + 0.871 255 327 933 567 517 641 932 8;
  • 77) 0.871 255 327 933 567 517 641 932 8 × 2 = 1 + 0.742 510 655 867 135 035 283 865 6;
  • 78) 0.742 510 655 867 135 035 283 865 6 × 2 = 1 + 0.485 021 311 734 270 070 567 731 2;
  • 79) 0.485 021 311 734 270 070 567 731 2 × 2 = 0 + 0.970 042 623 468 540 141 135 462 4;
  • 80) 0.970 042 623 468 540 141 135 462 4 × 2 = 1 + 0.940 085 246 937 080 282 270 924 8;
  • 81) 0.940 085 246 937 080 282 270 924 8 × 2 = 1 + 0.880 170 493 874 160 564 541 849 6;
  • 82) 0.880 170 493 874 160 564 541 849 6 × 2 = 1 + 0.760 340 987 748 321 129 083 699 2;
  • 83) 0.760 340 987 748 321 129 083 699 2 × 2 = 1 + 0.520 681 975 496 642 258 167 398 4;
  • 84) 0.520 681 975 496 642 258 167 398 4 × 2 = 1 + 0.041 363 950 993 284 516 334 796 8;
  • 85) 0.041 363 950 993 284 516 334 796 8 × 2 = 0 + 0.082 727 901 986 569 032 669 593 6;
  • 86) 0.082 727 901 986 569 032 669 593 6 × 2 = 0 + 0.165 455 803 973 138 065 339 187 2;
  • 87) 0.165 455 803 973 138 065 339 187 2 × 2 = 0 + 0.330 911 607 946 276 130 678 374 4;
  • 88) 0.330 911 607 946 276 130 678 374 4 × 2 = 0 + 0.661 823 215 892 552 261 356 748 8;
  • 89) 0.661 823 215 892 552 261 356 748 8 × 2 = 1 + 0.323 646 431 785 104 522 713 497 6;
  • 90) 0.323 646 431 785 104 522 713 497 6 × 2 = 0 + 0.647 292 863 570 209 045 426 995 2;
  • 91) 0.647 292 863 570 209 045 426 995 2 × 2 = 1 + 0.294 585 727 140 418 090 853 990 4;
  • 92) 0.294 585 727 140 418 090 853 990 4 × 2 = 0 + 0.589 171 454 280 836 181 707 980 8;
  • 93) 0.589 171 454 280 836 181 707 980 8 × 2 = 1 + 0.178 342 908 561 672 363 415 961 6;
  • 94) 0.178 342 908 561 672 363 415 961 6 × 2 = 0 + 0.356 685 817 123 344 726 831 923 2;
  • 95) 0.356 685 817 123 344 726 831 923 2 × 2 = 0 + 0.713 371 634 246 689 453 663 846 4;
  • 96) 0.713 371 634 246 689 453 663 846 4 × 2 = 1 + 0.426 743 268 493 378 907 327 692 8;
  • 97) 0.426 743 268 493 378 907 327 692 8 × 2 = 0 + 0.853 486 536 986 757 814 655 385 6;
  • 98) 0.853 486 536 986 757 814 655 385 6 × 2 = 1 + 0.706 973 073 973 515 629 310 771 2;
  • 99) 0.706 973 073 973 515 629 310 771 2 × 2 = 1 + 0.413 946 147 947 031 258 621 542 4;
  • 100) 0.413 946 147 947 031 258 621 542 4 × 2 = 0 + 0.827 892 295 894 062 517 243 084 8;
  • 101) 0.827 892 295 894 062 517 243 084 8 × 2 = 1 + 0.655 784 591 788 125 034 486 169 6;
  • 102) 0.655 784 591 788 125 034 486 169 6 × 2 = 1 + 0.311 569 183 576 250 068 972 339 2;
  • 103) 0.311 569 183 576 250 068 972 339 2 × 2 = 0 + 0.623 138 367 152 500 137 944 678 4;
  • 104) 0.623 138 367 152 500 137 944 678 4 × 2 = 1 + 0.246 276 734 305 000 275 889 356 8;
  • 105) 0.246 276 734 305 000 275 889 356 8 × 2 = 0 + 0.492 553 468 610 000 551 778 713 6;
  • 106) 0.492 553 468 610 000 551 778 713 6 × 2 = 0 + 0.985 106 937 220 001 103 557 427 2;
  • 107) 0.985 106 937 220 001 103 557 427 2 × 2 = 1 + 0.970 213 874 440 002 207 114 854 4;
  • 108) 0.970 213 874 440 002 207 114 854 4 × 2 = 1 + 0.940 427 748 880 004 414 229 708 8;
  • 109) 0.940 427 748 880 004 414 229 708 8 × 2 = 1 + 0.880 855 497 760 008 828 459 417 6;
  • 110) 0.880 855 497 760 008 828 459 417 6 × 2 = 1 + 0.761 710 995 520 017 656 918 835 2;
  • 111) 0.761 710 995 520 017 656 918 835 2 × 2 = 1 + 0.523 421 991 040 035 313 837 670 4;
  • 112) 0.523 421 991 040 035 313 837 670 4 × 2 = 1 + 0.046 843 982 080 070 627 675 340 8;
  • 113) 0.046 843 982 080 070 627 675 340 8 × 2 = 0 + 0.093 687 964 160 141 255 350 681 6;
  • 114) 0.093 687 964 160 141 255 350 681 6 × 2 = 0 + 0.187 375 928 320 282 510 701 363 2;
  • 115) 0.187 375 928 320 282 510 701 363 2 × 2 = 0 + 0.374 751 856 640 565 021 402 726 4;
  • 116) 0.374 751 856 640 565 021 402 726 4 × 2 = 0 + 0.749 503 713 281 130 042 805 452 8;
  • 117) 0.749 503 713 281 130 042 805 452 8 × 2 = 1 + 0.499 007 426 562 260 085 610 905 6;
  • 118) 0.499 007 426 562 260 085 610 905 6 × 2 = 0 + 0.998 014 853 124 520 171 221 811 2;
  • 119) 0.998 014 853 124 520 171 221 811 2 × 2 = 1 + 0.996 029 706 249 040 342 443 622 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:


0.000 000 000 000 000 000 008 534 8(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1101 1111 0000 1010 1001 0110 1101 0011 1111 0000 101(2)

5. Positive number before normalization:

0.000 000 000 000 000 000 008 534 8(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1101 1111 0000 1010 1001 0110 1101 0011 1111 0000 101(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:


0.000 000 000 000 000 000 008 534 8(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1101 1111 0000 1010 1001 0110 1101 0011 1111 0000 101(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1101 1111 0000 1010 1001 0110 1101 0011 1111 0000 101(2) × 20 =

1.0100 0010 0110 1111 1000 0101 0100 1011 0110 1001 1111 1000 0101(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 1111 1000 0101 0100 1011 0110 1001 1111 1000 0101


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 1111 1000 0101 0100 1011 0110 1001 1111 1000 0101 =

0100 0010 0110 1111 1000 0101 0100 1011 0110 1001 1111 1000 0101


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 1111 1000 0101 0100 1011 0110 1001 1111 1000 0101


Decimal number 0.000 000 000 000 000 000 008 534 8 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1011 1100 - 0100 0010 0110 1111 1000 0101 0100 1011 0110 1001 1111 1000 0101

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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