0.000 000 000 000 000 000 008 535 5 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 000 000 008 535 5(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 5(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 5.

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 5 × 2 = 0 + 0.000 000 000 000 000 000 017 071;
  • 2) 0.000 000 000 000 000 000 017 071 × 2 = 0 + 0.000 000 000 000 000 000 034 142;
  • 3) 0.000 000 000 000 000 000 034 142 × 2 = 0 + 0.000 000 000 000 000 000 068 284;
  • 4) 0.000 000 000 000 000 000 068 284 × 2 = 0 + 0.000 000 000 000 000 000 136 568;
  • 5) 0.000 000 000 000 000 000 136 568 × 2 = 0 + 0.000 000 000 000 000 000 273 136;
  • 6) 0.000 000 000 000 000 000 273 136 × 2 = 0 + 0.000 000 000 000 000 000 546 272;
  • 7) 0.000 000 000 000 000 000 546 272 × 2 = 0 + 0.000 000 000 000 000 001 092 544;
  • 8) 0.000 000 000 000 000 001 092 544 × 2 = 0 + 0.000 000 000 000 000 002 185 088;
  • 9) 0.000 000 000 000 000 002 185 088 × 2 = 0 + 0.000 000 000 000 000 004 370 176;
  • 10) 0.000 000 000 000 000 004 370 176 × 2 = 0 + 0.000 000 000 000 000 008 740 352;
  • 11) 0.000 000 000 000 000 008 740 352 × 2 = 0 + 0.000 000 000 000 000 017 480 704;
  • 12) 0.000 000 000 000 000 017 480 704 × 2 = 0 + 0.000 000 000 000 000 034 961 408;
  • 13) 0.000 000 000 000 000 034 961 408 × 2 = 0 + 0.000 000 000 000 000 069 922 816;
  • 14) 0.000 000 000 000 000 069 922 816 × 2 = 0 + 0.000 000 000 000 000 139 845 632;
  • 15) 0.000 000 000 000 000 139 845 632 × 2 = 0 + 0.000 000 000 000 000 279 691 264;
  • 16) 0.000 000 000 000 000 279 691 264 × 2 = 0 + 0.000 000 000 000 000 559 382 528;
  • 17) 0.000 000 000 000 000 559 382 528 × 2 = 0 + 0.000 000 000 000 001 118 765 056;
  • 18) 0.000 000 000 000 001 118 765 056 × 2 = 0 + 0.000 000 000 000 002 237 530 112;
  • 19) 0.000 000 000 000 002 237 530 112 × 2 = 0 + 0.000 000 000 000 004 475 060 224;
  • 20) 0.000 000 000 000 004 475 060 224 × 2 = 0 + 0.000 000 000 000 008 950 120 448;
  • 21) 0.000 000 000 000 008 950 120 448 × 2 = 0 + 0.000 000 000 000 017 900 240 896;
  • 22) 0.000 000 000 000 017 900 240 896 × 2 = 0 + 0.000 000 000 000 035 800 481 792;
  • 23) 0.000 000 000 000 035 800 481 792 × 2 = 0 + 0.000 000 000 000 071 600 963 584;
  • 24) 0.000 000 000 000 071 600 963 584 × 2 = 0 + 0.000 000 000 000 143 201 927 168;
  • 25) 0.000 000 000 000 143 201 927 168 × 2 = 0 + 0.000 000 000 000 286 403 854 336;
  • 26) 0.000 000 000 000 286 403 854 336 × 2 = 0 + 0.000 000 000 000 572 807 708 672;
  • 27) 0.000 000 000 000 572 807 708 672 × 2 = 0 + 0.000 000 000 001 145 615 417 344;
  • 28) 0.000 000 000 001 145 615 417 344 × 2 = 0 + 0.000 000 000 002 291 230 834 688;
  • 29) 0.000 000 000 002 291 230 834 688 × 2 = 0 + 0.000 000 000 004 582 461 669 376;
  • 30) 0.000 000 000 004 582 461 669 376 × 2 = 0 + 0.000 000 000 009 164 923 338 752;
  • 31) 0.000 000 000 009 164 923 338 752 × 2 = 0 + 0.000 000 000 018 329 846 677 504;
  • 32) 0.000 000 000 018 329 846 677 504 × 2 = 0 + 0.000 000 000 036 659 693 355 008;
  • 33) 0.000 000 000 036 659 693 355 008 × 2 = 0 + 0.000 000 000 073 319 386 710 016;
  • 34) 0.000 000 000 073 319 386 710 016 × 2 = 0 + 0.000 000 000 146 638 773 420 032;
  • 35) 0.000 000 000 146 638 773 420 032 × 2 = 0 + 0.000 000 000 293 277 546 840 064;
  • 36) 0.000 000 000 293 277 546 840 064 × 2 = 0 + 0.000 000 000 586 555 093 680 128;
  • 37) 0.000 000 000 586 555 093 680 128 × 2 = 0 + 0.000 000 001 173 110 187 360 256;
  • 38) 0.000 000 001 173 110 187 360 256 × 2 = 0 + 0.000 000 002 346 220 374 720 512;
  • 39) 0.000 000 002 346 220 374 720 512 × 2 = 0 + 0.000 000 004 692 440 749 441 024;
  • 40) 0.000 000 004 692 440 749 441 024 × 2 = 0 + 0.000 000 009 384 881 498 882 048;
  • 41) 0.000 000 009 384 881 498 882 048 × 2 = 0 + 0.000 000 018 769 762 997 764 096;
  • 42) 0.000 000 018 769 762 997 764 096 × 2 = 0 + 0.000 000 037 539 525 995 528 192;
  • 43) 0.000 000 037 539 525 995 528 192 × 2 = 0 + 0.000 000 075 079 051 991 056 384;
  • 44) 0.000 000 075 079 051 991 056 384 × 2 = 0 + 0.000 000 150 158 103 982 112 768;
  • 45) 0.000 000 150 158 103 982 112 768 × 2 = 0 + 0.000 000 300 316 207 964 225 536;
  • 46) 0.000 000 300 316 207 964 225 536 × 2 = 0 + 0.000 000 600 632 415 928 451 072;
  • 47) 0.000 000 600 632 415 928 451 072 × 2 = 0 + 0.000 001 201 264 831 856 902 144;
  • 48) 0.000 001 201 264 831 856 902 144 × 2 = 0 + 0.000 002 402 529 663 713 804 288;
  • 49) 0.000 002 402 529 663 713 804 288 × 2 = 0 + 0.000 004 805 059 327 427 608 576;
  • 50) 0.000 004 805 059 327 427 608 576 × 2 = 0 + 0.000 009 610 118 654 855 217 152;
  • 51) 0.000 009 610 118 654 855 217 152 × 2 = 0 + 0.000 019 220 237 309 710 434 304;
  • 52) 0.000 019 220 237 309 710 434 304 × 2 = 0 + 0.000 038 440 474 619 420 868 608;
  • 53) 0.000 038 440 474 619 420 868 608 × 2 = 0 + 0.000 076 880 949 238 841 737 216;
  • 54) 0.000 076 880 949 238 841 737 216 × 2 = 0 + 0.000 153 761 898 477 683 474 432;
  • 55) 0.000 153 761 898 477 683 474 432 × 2 = 0 + 0.000 307 523 796 955 366 948 864;
  • 56) 0.000 307 523 796 955 366 948 864 × 2 = 0 + 0.000 615 047 593 910 733 897 728;
  • 57) 0.000 615 047 593 910 733 897 728 × 2 = 0 + 0.001 230 095 187 821 467 795 456;
  • 58) 0.001 230 095 187 821 467 795 456 × 2 = 0 + 0.002 460 190 375 642 935 590 912;
  • 59) 0.002 460 190 375 642 935 590 912 × 2 = 0 + 0.004 920 380 751 285 871 181 824;
  • 60) 0.004 920 380 751 285 871 181 824 × 2 = 0 + 0.009 840 761 502 571 742 363 648;
  • 61) 0.009 840 761 502 571 742 363 648 × 2 = 0 + 0.019 681 523 005 143 484 727 296;
  • 62) 0.019 681 523 005 143 484 727 296 × 2 = 0 + 0.039 363 046 010 286 969 454 592;
  • 63) 0.039 363 046 010 286 969 454 592 × 2 = 0 + 0.078 726 092 020 573 938 909 184;
  • 64) 0.078 726 092 020 573 938 909 184 × 2 = 0 + 0.157 452 184 041 147 877 818 368;
  • 65) 0.157 452 184 041 147 877 818 368 × 2 = 0 + 0.314 904 368 082 295 755 636 736;
  • 66) 0.314 904 368 082 295 755 636 736 × 2 = 0 + 0.629 808 736 164 591 511 273 472;
  • 67) 0.629 808 736 164 591 511 273 472 × 2 = 1 + 0.259 617 472 329 183 022 546 944;
  • 68) 0.259 617 472 329 183 022 546 944 × 2 = 0 + 0.519 234 944 658 366 045 093 888;
  • 69) 0.519 234 944 658 366 045 093 888 × 2 = 1 + 0.038 469 889 316 732 090 187 776;
  • 70) 0.038 469 889 316 732 090 187 776 × 2 = 0 + 0.076 939 778 633 464 180 375 552;
  • 71) 0.076 939 778 633 464 180 375 552 × 2 = 0 + 0.153 879 557 266 928 360 751 104;
  • 72) 0.153 879 557 266 928 360 751 104 × 2 = 0 + 0.307 759 114 533 856 721 502 208;
  • 73) 0.307 759 114 533 856 721 502 208 × 2 = 0 + 0.615 518 229 067 713 443 004 416;
  • 74) 0.615 518 229 067 713 443 004 416 × 2 = 1 + 0.231 036 458 135 426 886 008 832;
  • 75) 0.231 036 458 135 426 886 008 832 × 2 = 0 + 0.462 072 916 270 853 772 017 664;
  • 76) 0.462 072 916 270 853 772 017 664 × 2 = 0 + 0.924 145 832 541 707 544 035 328;
  • 77) 0.924 145 832 541 707 544 035 328 × 2 = 1 + 0.848 291 665 083 415 088 070 656;
  • 78) 0.848 291 665 083 415 088 070 656 × 2 = 1 + 0.696 583 330 166 830 176 141 312;
  • 79) 0.696 583 330 166 830 176 141 312 × 2 = 1 + 0.393 166 660 333 660 352 282 624;
  • 80) 0.393 166 660 333 660 352 282 624 × 2 = 0 + 0.786 333 320 667 320 704 565 248;
  • 81) 0.786 333 320 667 320 704 565 248 × 2 = 1 + 0.572 666 641 334 641 409 130 496;
  • 82) 0.572 666 641 334 641 409 130 496 × 2 = 1 + 0.145 333 282 669 282 818 260 992;
  • 83) 0.145 333 282 669 282 818 260 992 × 2 = 0 + 0.290 666 565 338 565 636 521 984;
  • 84) 0.290 666 565 338 565 636 521 984 × 2 = 0 + 0.581 333 130 677 131 273 043 968;
  • 85) 0.581 333 130 677 131 273 043 968 × 2 = 1 + 0.162 666 261 354 262 546 087 936;
  • 86) 0.162 666 261 354 262 546 087 936 × 2 = 0 + 0.325 332 522 708 525 092 175 872;
  • 87) 0.325 332 522 708 525 092 175 872 × 2 = 0 + 0.650 665 045 417 050 184 351 744;
  • 88) 0.650 665 045 417 050 184 351 744 × 2 = 1 + 0.301 330 090 834 100 368 703 488;
  • 89) 0.301 330 090 834 100 368 703 488 × 2 = 0 + 0.602 660 181 668 200 737 406 976;
  • 90) 0.602 660 181 668 200 737 406 976 × 2 = 1 + 0.205 320 363 336 401 474 813 952;
  • 91) 0.205 320 363 336 401 474 813 952 × 2 = 0 + 0.410 640 726 672 802 949 627 904;
  • 92) 0.410 640 726 672 802 949 627 904 × 2 = 0 + 0.821 281 453 345 605 899 255 808;
  • 93) 0.821 281 453 345 605 899 255 808 × 2 = 1 + 0.642 562 906 691 211 798 511 616;
  • 94) 0.642 562 906 691 211 798 511 616 × 2 = 1 + 0.285 125 813 382 423 597 023 232;
  • 95) 0.285 125 813 382 423 597 023 232 × 2 = 0 + 0.570 251 626 764 847 194 046 464;
  • 96) 0.570 251 626 764 847 194 046 464 × 2 = 1 + 0.140 503 253 529 694 388 092 928;
  • 97) 0.140 503 253 529 694 388 092 928 × 2 = 0 + 0.281 006 507 059 388 776 185 856;
  • 98) 0.281 006 507 059 388 776 185 856 × 2 = 0 + 0.562 013 014 118 777 552 371 712;
  • 99) 0.562 013 014 118 777 552 371 712 × 2 = 1 + 0.124 026 028 237 555 104 743 424;
  • 100) 0.124 026 028 237 555 104 743 424 × 2 = 0 + 0.248 052 056 475 110 209 486 848;
  • 101) 0.248 052 056 475 110 209 486 848 × 2 = 0 + 0.496 104 112 950 220 418 973 696;
  • 102) 0.496 104 112 950 220 418 973 696 × 2 = 0 + 0.992 208 225 900 440 837 947 392;
  • 103) 0.992 208 225 900 440 837 947 392 × 2 = 1 + 0.984 416 451 800 881 675 894 784;
  • 104) 0.984 416 451 800 881 675 894 784 × 2 = 1 + 0.968 832 903 601 763 351 789 568;
  • 105) 0.968 832 903 601 763 351 789 568 × 2 = 1 + 0.937 665 807 203 526 703 579 136;
  • 106) 0.937 665 807 203 526 703 579 136 × 2 = 1 + 0.875 331 614 407 053 407 158 272;
  • 107) 0.875 331 614 407 053 407 158 272 × 2 = 1 + 0.750 663 228 814 106 814 316 544;
  • 108) 0.750 663 228 814 106 814 316 544 × 2 = 1 + 0.501 326 457 628 213 628 633 088;
  • 109) 0.501 326 457 628 213 628 633 088 × 2 = 1 + 0.002 652 915 256 427 257 266 176;
  • 110) 0.002 652 915 256 427 257 266 176 × 2 = 0 + 0.005 305 830 512 854 514 532 352;
  • 111) 0.005 305 830 512 854 514 532 352 × 2 = 0 + 0.010 611 661 025 709 029 064 704;
  • 112) 0.010 611 661 025 709 029 064 704 × 2 = 0 + 0.021 223 322 051 418 058 129 408;
  • 113) 0.021 223 322 051 418 058 129 408 × 2 = 0 + 0.042 446 644 102 836 116 258 816;
  • 114) 0.042 446 644 102 836 116 258 816 × 2 = 0 + 0.084 893 288 205 672 232 517 632;
  • 115) 0.084 893 288 205 672 232 517 632 × 2 = 0 + 0.169 786 576 411 344 465 035 264;
  • 116) 0.169 786 576 411 344 465 035 264 × 2 = 0 + 0.339 573 152 822 688 930 070 528;
  • 117) 0.339 573 152 822 688 930 070 528 × 2 = 0 + 0.679 146 305 645 377 860 141 056;
  • 118) 0.679 146 305 645 377 860 141 056 × 2 = 1 + 0.358 292 611 290 755 720 282 112;
  • 119) 0.358 292 611 290 755 720 282 112 × 2 = 0 + 0.716 585 222 581 511 440 564 224;

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 5(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1110 1100 1001 0100 1101 0010 0011 1111 1000 0000 010(2)

5. Positive number before normalization:

0.000 000 000 000 000 000 008 535 5(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1110 1100 1001 0100 1101 0010 0011 1111 1000 0000 010(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 5(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1110 1100 1001 0100 1101 0010 0011 1111 1000 0000 010(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1110 1100 1001 0100 1101 0010 0011 1111 1000 0000 010(2) × 20 =

1.0100 0010 0111 0110 0100 1010 0110 1001 0001 1111 1100 0000 0010(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 0110 0100 1010 0110 1001 0001 1111 1100 0000 0010


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 0110 0100 1010 0110 1001 0001 1111 1100 0000 0010 =

0100 0010 0111 0110 0100 1010 0110 1001 0001 1111 1100 0000 0010


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 0110 0100 1010 0110 1001 0001 1111 1100 0000 0010


Decimal number 0.000 000 000 000 000 000 008 535 5 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1011 1100 - 0100 0010 0111 0110 0100 1010 0110 1001 0001 1111 1100 0000 0010

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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