0.000 000 000 000 000 000 008 524 2 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 000 000 008 524 2(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
0.000 000 000 000 000 000 008 524 2(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: 0.000 000 000 000 000 000 008 524 2.

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) 0.000 000 000 000 000 000 008 524 2 × 2 = 0 + 0.000 000 000 000 000 000 017 048 4;
  • 2) 0.000 000 000 000 000 000 017 048 4 × 2 = 0 + 0.000 000 000 000 000 000 034 096 8;
  • 3) 0.000 000 000 000 000 000 034 096 8 × 2 = 0 + 0.000 000 000 000 000 000 068 193 6;
  • 4) 0.000 000 000 000 000 000 068 193 6 × 2 = 0 + 0.000 000 000 000 000 000 136 387 2;
  • 5) 0.000 000 000 000 000 000 136 387 2 × 2 = 0 + 0.000 000 000 000 000 000 272 774 4;
  • 6) 0.000 000 000 000 000 000 272 774 4 × 2 = 0 + 0.000 000 000 000 000 000 545 548 8;
  • 7) 0.000 000 000 000 000 000 545 548 8 × 2 = 0 + 0.000 000 000 000 000 001 091 097 6;
  • 8) 0.000 000 000 000 000 001 091 097 6 × 2 = 0 + 0.000 000 000 000 000 002 182 195 2;
  • 9) 0.000 000 000 000 000 002 182 195 2 × 2 = 0 + 0.000 000 000 000 000 004 364 390 4;
  • 10) 0.000 000 000 000 000 004 364 390 4 × 2 = 0 + 0.000 000 000 000 000 008 728 780 8;
  • 11) 0.000 000 000 000 000 008 728 780 8 × 2 = 0 + 0.000 000 000 000 000 017 457 561 6;
  • 12) 0.000 000 000 000 000 017 457 561 6 × 2 = 0 + 0.000 000 000 000 000 034 915 123 2;
  • 13) 0.000 000 000 000 000 034 915 123 2 × 2 = 0 + 0.000 000 000 000 000 069 830 246 4;
  • 14) 0.000 000 000 000 000 069 830 246 4 × 2 = 0 + 0.000 000 000 000 000 139 660 492 8;
  • 15) 0.000 000 000 000 000 139 660 492 8 × 2 = 0 + 0.000 000 000 000 000 279 320 985 6;
  • 16) 0.000 000 000 000 000 279 320 985 6 × 2 = 0 + 0.000 000 000 000 000 558 641 971 2;
  • 17) 0.000 000 000 000 000 558 641 971 2 × 2 = 0 + 0.000 000 000 000 001 117 283 942 4;
  • 18) 0.000 000 000 000 001 117 283 942 4 × 2 = 0 + 0.000 000 000 000 002 234 567 884 8;
  • 19) 0.000 000 000 000 002 234 567 884 8 × 2 = 0 + 0.000 000 000 000 004 469 135 769 6;
  • 20) 0.000 000 000 000 004 469 135 769 6 × 2 = 0 + 0.000 000 000 000 008 938 271 539 2;
  • 21) 0.000 000 000 000 008 938 271 539 2 × 2 = 0 + 0.000 000 000 000 017 876 543 078 4;
  • 22) 0.000 000 000 000 017 876 543 078 4 × 2 = 0 + 0.000 000 000 000 035 753 086 156 8;
  • 23) 0.000 000 000 000 035 753 086 156 8 × 2 = 0 + 0.000 000 000 000 071 506 172 313 6;
  • 24) 0.000 000 000 000 071 506 172 313 6 × 2 = 0 + 0.000 000 000 000 143 012 344 627 2;
  • 25) 0.000 000 000 000 143 012 344 627 2 × 2 = 0 + 0.000 000 000 000 286 024 689 254 4;
  • 26) 0.000 000 000 000 286 024 689 254 4 × 2 = 0 + 0.000 000 000 000 572 049 378 508 8;
  • 27) 0.000 000 000 000 572 049 378 508 8 × 2 = 0 + 0.000 000 000 001 144 098 757 017 6;
  • 28) 0.000 000 000 001 144 098 757 017 6 × 2 = 0 + 0.000 000 000 002 288 197 514 035 2;
  • 29) 0.000 000 000 002 288 197 514 035 2 × 2 = 0 + 0.000 000 000 004 576 395 028 070 4;
  • 30) 0.000 000 000 004 576 395 028 070 4 × 2 = 0 + 0.000 000 000 009 152 790 056 140 8;
  • 31) 0.000 000 000 009 152 790 056 140 8 × 2 = 0 + 0.000 000 000 018 305 580 112 281 6;
  • 32) 0.000 000 000 018 305 580 112 281 6 × 2 = 0 + 0.000 000 000 036 611 160 224 563 2;
  • 33) 0.000 000 000 036 611 160 224 563 2 × 2 = 0 + 0.000 000 000 073 222 320 449 126 4;
  • 34) 0.000 000 000 073 222 320 449 126 4 × 2 = 0 + 0.000 000 000 146 444 640 898 252 8;
  • 35) 0.000 000 000 146 444 640 898 252 8 × 2 = 0 + 0.000 000 000 292 889 281 796 505 6;
  • 36) 0.000 000 000 292 889 281 796 505 6 × 2 = 0 + 0.000 000 000 585 778 563 593 011 2;
  • 37) 0.000 000 000 585 778 563 593 011 2 × 2 = 0 + 0.000 000 001 171 557 127 186 022 4;
  • 38) 0.000 000 001 171 557 127 186 022 4 × 2 = 0 + 0.000 000 002 343 114 254 372 044 8;
  • 39) 0.000 000 002 343 114 254 372 044 8 × 2 = 0 + 0.000 000 004 686 228 508 744 089 6;
  • 40) 0.000 000 004 686 228 508 744 089 6 × 2 = 0 + 0.000 000 009 372 457 017 488 179 2;
  • 41) 0.000 000 009 372 457 017 488 179 2 × 2 = 0 + 0.000 000 018 744 914 034 976 358 4;
  • 42) 0.000 000 018 744 914 034 976 358 4 × 2 = 0 + 0.000 000 037 489 828 069 952 716 8;
  • 43) 0.000 000 037 489 828 069 952 716 8 × 2 = 0 + 0.000 000 074 979 656 139 905 433 6;
  • 44) 0.000 000 074 979 656 139 905 433 6 × 2 = 0 + 0.000 000 149 959 312 279 810 867 2;
  • 45) 0.000 000 149 959 312 279 810 867 2 × 2 = 0 + 0.000 000 299 918 624 559 621 734 4;
  • 46) 0.000 000 299 918 624 559 621 734 4 × 2 = 0 + 0.000 000 599 837 249 119 243 468 8;
  • 47) 0.000 000 599 837 249 119 243 468 8 × 2 = 0 + 0.000 001 199 674 498 238 486 937 6;
  • 48) 0.000 001 199 674 498 238 486 937 6 × 2 = 0 + 0.000 002 399 348 996 476 973 875 2;
  • 49) 0.000 002 399 348 996 476 973 875 2 × 2 = 0 + 0.000 004 798 697 992 953 947 750 4;
  • 50) 0.000 004 798 697 992 953 947 750 4 × 2 = 0 + 0.000 009 597 395 985 907 895 500 8;
  • 51) 0.000 009 597 395 985 907 895 500 8 × 2 = 0 + 0.000 019 194 791 971 815 791 001 6;
  • 52) 0.000 019 194 791 971 815 791 001 6 × 2 = 0 + 0.000 038 389 583 943 631 582 003 2;
  • 53) 0.000 038 389 583 943 631 582 003 2 × 2 = 0 + 0.000 076 779 167 887 263 164 006 4;
  • 54) 0.000 076 779 167 887 263 164 006 4 × 2 = 0 + 0.000 153 558 335 774 526 328 012 8;
  • 55) 0.000 153 558 335 774 526 328 012 8 × 2 = 0 + 0.000 307 116 671 549 052 656 025 6;
  • 56) 0.000 307 116 671 549 052 656 025 6 × 2 = 0 + 0.000 614 233 343 098 105 312 051 2;
  • 57) 0.000 614 233 343 098 105 312 051 2 × 2 = 0 + 0.001 228 466 686 196 210 624 102 4;
  • 58) 0.001 228 466 686 196 210 624 102 4 × 2 = 0 + 0.002 456 933 372 392 421 248 204 8;
  • 59) 0.002 456 933 372 392 421 248 204 8 × 2 = 0 + 0.004 913 866 744 784 842 496 409 6;
  • 60) 0.004 913 866 744 784 842 496 409 6 × 2 = 0 + 0.009 827 733 489 569 684 992 819 2;
  • 61) 0.009 827 733 489 569 684 992 819 2 × 2 = 0 + 0.019 655 466 979 139 369 985 638 4;
  • 62) 0.019 655 466 979 139 369 985 638 4 × 2 = 0 + 0.039 310 933 958 278 739 971 276 8;
  • 63) 0.039 310 933 958 278 739 971 276 8 × 2 = 0 + 0.078 621 867 916 557 479 942 553 6;
  • 64) 0.078 621 867 916 557 479 942 553 6 × 2 = 0 + 0.157 243 735 833 114 959 885 107 2;
  • 65) 0.157 243 735 833 114 959 885 107 2 × 2 = 0 + 0.314 487 471 666 229 919 770 214 4;
  • 66) 0.314 487 471 666 229 919 770 214 4 × 2 = 0 + 0.628 974 943 332 459 839 540 428 8;
  • 67) 0.628 974 943 332 459 839 540 428 8 × 2 = 1 + 0.257 949 886 664 919 679 080 857 6;
  • 68) 0.257 949 886 664 919 679 080 857 6 × 2 = 0 + 0.515 899 773 329 839 358 161 715 2;
  • 69) 0.515 899 773 329 839 358 161 715 2 × 2 = 1 + 0.031 799 546 659 678 716 323 430 4;
  • 70) 0.031 799 546 659 678 716 323 430 4 × 2 = 0 + 0.063 599 093 319 357 432 646 860 8;
  • 71) 0.063 599 093 319 357 432 646 860 8 × 2 = 0 + 0.127 198 186 638 714 865 293 721 6;
  • 72) 0.127 198 186 638 714 865 293 721 6 × 2 = 0 + 0.254 396 373 277 429 730 587 443 2;
  • 73) 0.254 396 373 277 429 730 587 443 2 × 2 = 0 + 0.508 792 746 554 859 461 174 886 4;
  • 74) 0.508 792 746 554 859 461 174 886 4 × 2 = 1 + 0.017 585 493 109 718 922 349 772 8;
  • 75) 0.017 585 493 109 718 922 349 772 8 × 2 = 0 + 0.035 170 986 219 437 844 699 545 6;
  • 76) 0.035 170 986 219 437 844 699 545 6 × 2 = 0 + 0.070 341 972 438 875 689 399 091 2;
  • 77) 0.070 341 972 438 875 689 399 091 2 × 2 = 0 + 0.140 683 944 877 751 378 798 182 4;
  • 78) 0.140 683 944 877 751 378 798 182 4 × 2 = 0 + 0.281 367 889 755 502 757 596 364 8;
  • 79) 0.281 367 889 755 502 757 596 364 8 × 2 = 0 + 0.562 735 779 511 005 515 192 729 6;
  • 80) 0.562 735 779 511 005 515 192 729 6 × 2 = 1 + 0.125 471 559 022 011 030 385 459 2;
  • 81) 0.125 471 559 022 011 030 385 459 2 × 2 = 0 + 0.250 943 118 044 022 060 770 918 4;
  • 82) 0.250 943 118 044 022 060 770 918 4 × 2 = 0 + 0.501 886 236 088 044 121 541 836 8;
  • 83) 0.501 886 236 088 044 121 541 836 8 × 2 = 1 + 0.003 772 472 176 088 243 083 673 6;
  • 84) 0.003 772 472 176 088 243 083 673 6 × 2 = 0 + 0.007 544 944 352 176 486 167 347 2;
  • 85) 0.007 544 944 352 176 486 167 347 2 × 2 = 0 + 0.015 089 888 704 352 972 334 694 4;
  • 86) 0.015 089 888 704 352 972 334 694 4 × 2 = 0 + 0.030 179 777 408 705 944 669 388 8;
  • 87) 0.030 179 777 408 705 944 669 388 8 × 2 = 0 + 0.060 359 554 817 411 889 338 777 6;
  • 88) 0.060 359 554 817 411 889 338 777 6 × 2 = 0 + 0.120 719 109 634 823 778 677 555 2;
  • 89) 0.120 719 109 634 823 778 677 555 2 × 2 = 0 + 0.241 438 219 269 647 557 355 110 4;
  • 90) 0.241 438 219 269 647 557 355 110 4 × 2 = 0 + 0.482 876 438 539 295 114 710 220 8;
  • 91) 0.482 876 438 539 295 114 710 220 8 × 2 = 0 + 0.965 752 877 078 590 229 420 441 6;
  • 92) 0.965 752 877 078 590 229 420 441 6 × 2 = 1 + 0.931 505 754 157 180 458 840 883 2;
  • 93) 0.931 505 754 157 180 458 840 883 2 × 2 = 1 + 0.863 011 508 314 360 917 681 766 4;
  • 94) 0.863 011 508 314 360 917 681 766 4 × 2 = 1 + 0.726 023 016 628 721 835 363 532 8;
  • 95) 0.726 023 016 628 721 835 363 532 8 × 2 = 1 + 0.452 046 033 257 443 670 727 065 6;
  • 96) 0.452 046 033 257 443 670 727 065 6 × 2 = 0 + 0.904 092 066 514 887 341 454 131 2;
  • 97) 0.904 092 066 514 887 341 454 131 2 × 2 = 1 + 0.808 184 133 029 774 682 908 262 4;
  • 98) 0.808 184 133 029 774 682 908 262 4 × 2 = 1 + 0.616 368 266 059 549 365 816 524 8;
  • 99) 0.616 368 266 059 549 365 816 524 8 × 2 = 1 + 0.232 736 532 119 098 731 633 049 6;
  • 100) 0.232 736 532 119 098 731 633 049 6 × 2 = 0 + 0.465 473 064 238 197 463 266 099 2;
  • 101) 0.465 473 064 238 197 463 266 099 2 × 2 = 0 + 0.930 946 128 476 394 926 532 198 4;
  • 102) 0.930 946 128 476 394 926 532 198 4 × 2 = 1 + 0.861 892 256 952 789 853 064 396 8;
  • 103) 0.861 892 256 952 789 853 064 396 8 × 2 = 1 + 0.723 784 513 905 579 706 128 793 6;
  • 104) 0.723 784 513 905 579 706 128 793 6 × 2 = 1 + 0.447 569 027 811 159 412 257 587 2;
  • 105) 0.447 569 027 811 159 412 257 587 2 × 2 = 0 + 0.895 138 055 622 318 824 515 174 4;
  • 106) 0.895 138 055 622 318 824 515 174 4 × 2 = 1 + 0.790 276 111 244 637 649 030 348 8;
  • 107) 0.790 276 111 244 637 649 030 348 8 × 2 = 1 + 0.580 552 222 489 275 298 060 697 6;
  • 108) 0.580 552 222 489 275 298 060 697 6 × 2 = 1 + 0.161 104 444 978 550 596 121 395 2;
  • 109) 0.161 104 444 978 550 596 121 395 2 × 2 = 0 + 0.322 208 889 957 101 192 242 790 4;
  • 110) 0.322 208 889 957 101 192 242 790 4 × 2 = 0 + 0.644 417 779 914 202 384 485 580 8;
  • 111) 0.644 417 779 914 202 384 485 580 8 × 2 = 1 + 0.288 835 559 828 404 768 971 161 6;
  • 112) 0.288 835 559 828 404 768 971 161 6 × 2 = 0 + 0.577 671 119 656 809 537 942 323 2;
  • 113) 0.577 671 119 656 809 537 942 323 2 × 2 = 1 + 0.155 342 239 313 619 075 884 646 4;
  • 114) 0.155 342 239 313 619 075 884 646 4 × 2 = 0 + 0.310 684 478 627 238 151 769 292 8;
  • 115) 0.310 684 478 627 238 151 769 292 8 × 2 = 0 + 0.621 368 957 254 476 303 538 585 6;
  • 116) 0.621 368 957 254 476 303 538 585 6 × 2 = 1 + 0.242 737 914 508 952 607 077 171 2;
  • 117) 0.242 737 914 508 952 607 077 171 2 × 2 = 0 + 0.485 475 829 017 905 214 154 342 4;
  • 118) 0.485 475 829 017 905 214 154 342 4 × 2 = 0 + 0.970 951 658 035 810 428 308 684 8;
  • 119) 0.970 951 658 035 810 428 308 684 8 × 2 = 1 + 0.941 903 316 071 620 856 617 369 6;

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:


0.000 000 000 000 000 000 008 524 2(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 0001 0010 0000 0001 1110 1110 0111 0111 0010 1001 001(2)

5. Positive number before normalization:

0.000 000 000 000 000 000 008 524 2(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 0001 0010 0000 0001 1110 1110 0111 0111 0010 1001 001(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:


0.000 000 000 000 000 000 008 524 2(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 0001 0010 0000 0001 1110 1110 0111 0111 0010 1001 001(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 0001 0010 0000 0001 1110 1110 0111 0111 0010 1001 001(2) × 20 =

1.0100 0010 0000 1001 0000 0000 1111 0111 0011 1011 1001 0100 1001(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 0000 1001 0000 0000 1111 0111 0011 1011 1001 0100 1001


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 0000 1001 0000 0000 1111 0111 0011 1011 1001 0100 1001 =

0100 0010 0000 1001 0000 0000 1111 0111 0011 1011 1001 0100 1001


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 0000 1001 0000 0000 1111 0111 0011 1011 1001 0100 1001


Decimal number 0.000 000 000 000 000 000 008 524 2 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1011 1100 - 0100 0010 0000 1001 0000 0000 1111 0111 0011 1011 1001 0100 1001

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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