0.000 000 000 000 000 000 008 532 4 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 000 000 008 532 4(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
0.000 000 000 000 000 000 008 532 4(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: 0.000 000 000 000 000 000 008 532 4.

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) 0.000 000 000 000 000 000 008 532 4 × 2 = 0 + 0.000 000 000 000 000 000 017 064 8;
  • 2) 0.000 000 000 000 000 000 017 064 8 × 2 = 0 + 0.000 000 000 000 000 000 034 129 6;
  • 3) 0.000 000 000 000 000 000 034 129 6 × 2 = 0 + 0.000 000 000 000 000 000 068 259 2;
  • 4) 0.000 000 000 000 000 000 068 259 2 × 2 = 0 + 0.000 000 000 000 000 000 136 518 4;
  • 5) 0.000 000 000 000 000 000 136 518 4 × 2 = 0 + 0.000 000 000 000 000 000 273 036 8;
  • 6) 0.000 000 000 000 000 000 273 036 8 × 2 = 0 + 0.000 000 000 000 000 000 546 073 6;
  • 7) 0.000 000 000 000 000 000 546 073 6 × 2 = 0 + 0.000 000 000 000 000 001 092 147 2;
  • 8) 0.000 000 000 000 000 001 092 147 2 × 2 = 0 + 0.000 000 000 000 000 002 184 294 4;
  • 9) 0.000 000 000 000 000 002 184 294 4 × 2 = 0 + 0.000 000 000 000 000 004 368 588 8;
  • 10) 0.000 000 000 000 000 004 368 588 8 × 2 = 0 + 0.000 000 000 000 000 008 737 177 6;
  • 11) 0.000 000 000 000 000 008 737 177 6 × 2 = 0 + 0.000 000 000 000 000 017 474 355 2;
  • 12) 0.000 000 000 000 000 017 474 355 2 × 2 = 0 + 0.000 000 000 000 000 034 948 710 4;
  • 13) 0.000 000 000 000 000 034 948 710 4 × 2 = 0 + 0.000 000 000 000 000 069 897 420 8;
  • 14) 0.000 000 000 000 000 069 897 420 8 × 2 = 0 + 0.000 000 000 000 000 139 794 841 6;
  • 15) 0.000 000 000 000 000 139 794 841 6 × 2 = 0 + 0.000 000 000 000 000 279 589 683 2;
  • 16) 0.000 000 000 000 000 279 589 683 2 × 2 = 0 + 0.000 000 000 000 000 559 179 366 4;
  • 17) 0.000 000 000 000 000 559 179 366 4 × 2 = 0 + 0.000 000 000 000 001 118 358 732 8;
  • 18) 0.000 000 000 000 001 118 358 732 8 × 2 = 0 + 0.000 000 000 000 002 236 717 465 6;
  • 19) 0.000 000 000 000 002 236 717 465 6 × 2 = 0 + 0.000 000 000 000 004 473 434 931 2;
  • 20) 0.000 000 000 000 004 473 434 931 2 × 2 = 0 + 0.000 000 000 000 008 946 869 862 4;
  • 21) 0.000 000 000 000 008 946 869 862 4 × 2 = 0 + 0.000 000 000 000 017 893 739 724 8;
  • 22) 0.000 000 000 000 017 893 739 724 8 × 2 = 0 + 0.000 000 000 000 035 787 479 449 6;
  • 23) 0.000 000 000 000 035 787 479 449 6 × 2 = 0 + 0.000 000 000 000 071 574 958 899 2;
  • 24) 0.000 000 000 000 071 574 958 899 2 × 2 = 0 + 0.000 000 000 000 143 149 917 798 4;
  • 25) 0.000 000 000 000 143 149 917 798 4 × 2 = 0 + 0.000 000 000 000 286 299 835 596 8;
  • 26) 0.000 000 000 000 286 299 835 596 8 × 2 = 0 + 0.000 000 000 000 572 599 671 193 6;
  • 27) 0.000 000 000 000 572 599 671 193 6 × 2 = 0 + 0.000 000 000 001 145 199 342 387 2;
  • 28) 0.000 000 000 001 145 199 342 387 2 × 2 = 0 + 0.000 000 000 002 290 398 684 774 4;
  • 29) 0.000 000 000 002 290 398 684 774 4 × 2 = 0 + 0.000 000 000 004 580 797 369 548 8;
  • 30) 0.000 000 000 004 580 797 369 548 8 × 2 = 0 + 0.000 000 000 009 161 594 739 097 6;
  • 31) 0.000 000 000 009 161 594 739 097 6 × 2 = 0 + 0.000 000 000 018 323 189 478 195 2;
  • 32) 0.000 000 000 018 323 189 478 195 2 × 2 = 0 + 0.000 000 000 036 646 378 956 390 4;
  • 33) 0.000 000 000 036 646 378 956 390 4 × 2 = 0 + 0.000 000 000 073 292 757 912 780 8;
  • 34) 0.000 000 000 073 292 757 912 780 8 × 2 = 0 + 0.000 000 000 146 585 515 825 561 6;
  • 35) 0.000 000 000 146 585 515 825 561 6 × 2 = 0 + 0.000 000 000 293 171 031 651 123 2;
  • 36) 0.000 000 000 293 171 031 651 123 2 × 2 = 0 + 0.000 000 000 586 342 063 302 246 4;
  • 37) 0.000 000 000 586 342 063 302 246 4 × 2 = 0 + 0.000 000 001 172 684 126 604 492 8;
  • 38) 0.000 000 001 172 684 126 604 492 8 × 2 = 0 + 0.000 000 002 345 368 253 208 985 6;
  • 39) 0.000 000 002 345 368 253 208 985 6 × 2 = 0 + 0.000 000 004 690 736 506 417 971 2;
  • 40) 0.000 000 004 690 736 506 417 971 2 × 2 = 0 + 0.000 000 009 381 473 012 835 942 4;
  • 41) 0.000 000 009 381 473 012 835 942 4 × 2 = 0 + 0.000 000 018 762 946 025 671 884 8;
  • 42) 0.000 000 018 762 946 025 671 884 8 × 2 = 0 + 0.000 000 037 525 892 051 343 769 6;
  • 43) 0.000 000 037 525 892 051 343 769 6 × 2 = 0 + 0.000 000 075 051 784 102 687 539 2;
  • 44) 0.000 000 075 051 784 102 687 539 2 × 2 = 0 + 0.000 000 150 103 568 205 375 078 4;
  • 45) 0.000 000 150 103 568 205 375 078 4 × 2 = 0 + 0.000 000 300 207 136 410 750 156 8;
  • 46) 0.000 000 300 207 136 410 750 156 8 × 2 = 0 + 0.000 000 600 414 272 821 500 313 6;
  • 47) 0.000 000 600 414 272 821 500 313 6 × 2 = 0 + 0.000 001 200 828 545 643 000 627 2;
  • 48) 0.000 001 200 828 545 643 000 627 2 × 2 = 0 + 0.000 002 401 657 091 286 001 254 4;
  • 49) 0.000 002 401 657 091 286 001 254 4 × 2 = 0 + 0.000 004 803 314 182 572 002 508 8;
  • 50) 0.000 004 803 314 182 572 002 508 8 × 2 = 0 + 0.000 009 606 628 365 144 005 017 6;
  • 51) 0.000 009 606 628 365 144 005 017 6 × 2 = 0 + 0.000 019 213 256 730 288 010 035 2;
  • 52) 0.000 019 213 256 730 288 010 035 2 × 2 = 0 + 0.000 038 426 513 460 576 020 070 4;
  • 53) 0.000 038 426 513 460 576 020 070 4 × 2 = 0 + 0.000 076 853 026 921 152 040 140 8;
  • 54) 0.000 076 853 026 921 152 040 140 8 × 2 = 0 + 0.000 153 706 053 842 304 080 281 6;
  • 55) 0.000 153 706 053 842 304 080 281 6 × 2 = 0 + 0.000 307 412 107 684 608 160 563 2;
  • 56) 0.000 307 412 107 684 608 160 563 2 × 2 = 0 + 0.000 614 824 215 369 216 321 126 4;
  • 57) 0.000 614 824 215 369 216 321 126 4 × 2 = 0 + 0.001 229 648 430 738 432 642 252 8;
  • 58) 0.001 229 648 430 738 432 642 252 8 × 2 = 0 + 0.002 459 296 861 476 865 284 505 6;
  • 59) 0.002 459 296 861 476 865 284 505 6 × 2 = 0 + 0.004 918 593 722 953 730 569 011 2;
  • 60) 0.004 918 593 722 953 730 569 011 2 × 2 = 0 + 0.009 837 187 445 907 461 138 022 4;
  • 61) 0.009 837 187 445 907 461 138 022 4 × 2 = 0 + 0.019 674 374 891 814 922 276 044 8;
  • 62) 0.019 674 374 891 814 922 276 044 8 × 2 = 0 + 0.039 348 749 783 629 844 552 089 6;
  • 63) 0.039 348 749 783 629 844 552 089 6 × 2 = 0 + 0.078 697 499 567 259 689 104 179 2;
  • 64) 0.078 697 499 567 259 689 104 179 2 × 2 = 0 + 0.157 394 999 134 519 378 208 358 4;
  • 65) 0.157 394 999 134 519 378 208 358 4 × 2 = 0 + 0.314 789 998 269 038 756 416 716 8;
  • 66) 0.314 789 998 269 038 756 416 716 8 × 2 = 0 + 0.629 579 996 538 077 512 833 433 6;
  • 67) 0.629 579 996 538 077 512 833 433 6 × 2 = 1 + 0.259 159 993 076 155 025 666 867 2;
  • 68) 0.259 159 993 076 155 025 666 867 2 × 2 = 0 + 0.518 319 986 152 310 051 333 734 4;
  • 69) 0.518 319 986 152 310 051 333 734 4 × 2 = 1 + 0.036 639 972 304 620 102 667 468 8;
  • 70) 0.036 639 972 304 620 102 667 468 8 × 2 = 0 + 0.073 279 944 609 240 205 334 937 6;
  • 71) 0.073 279 944 609 240 205 334 937 6 × 2 = 0 + 0.146 559 889 218 480 410 669 875 2;
  • 72) 0.146 559 889 218 480 410 669 875 2 × 2 = 0 + 0.293 119 778 436 960 821 339 750 4;
  • 73) 0.293 119 778 436 960 821 339 750 4 × 2 = 0 + 0.586 239 556 873 921 642 679 500 8;
  • 74) 0.586 239 556 873 921 642 679 500 8 × 2 = 1 + 0.172 479 113 747 843 285 359 001 6;
  • 75) 0.172 479 113 747 843 285 359 001 6 × 2 = 0 + 0.344 958 227 495 686 570 718 003 2;
  • 76) 0.344 958 227 495 686 570 718 003 2 × 2 = 0 + 0.689 916 454 991 373 141 436 006 4;
  • 77) 0.689 916 454 991 373 141 436 006 4 × 2 = 1 + 0.379 832 909 982 746 282 872 012 8;
  • 78) 0.379 832 909 982 746 282 872 012 8 × 2 = 0 + 0.759 665 819 965 492 565 744 025 6;
  • 79) 0.759 665 819 965 492 565 744 025 6 × 2 = 1 + 0.519 331 639 930 985 131 488 051 2;
  • 80) 0.519 331 639 930 985 131 488 051 2 × 2 = 1 + 0.038 663 279 861 970 262 976 102 4;
  • 81) 0.038 663 279 861 970 262 976 102 4 × 2 = 0 + 0.077 326 559 723 940 525 952 204 8;
  • 82) 0.077 326 559 723 940 525 952 204 8 × 2 = 0 + 0.154 653 119 447 881 051 904 409 6;
  • 83) 0.154 653 119 447 881 051 904 409 6 × 2 = 0 + 0.309 306 238 895 762 103 808 819 2;
  • 84) 0.309 306 238 895 762 103 808 819 2 × 2 = 0 + 0.618 612 477 791 524 207 617 638 4;
  • 85) 0.618 612 477 791 524 207 617 638 4 × 2 = 1 + 0.237 224 955 583 048 415 235 276 8;
  • 86) 0.237 224 955 583 048 415 235 276 8 × 2 = 0 + 0.474 449 911 166 096 830 470 553 6;
  • 87) 0.474 449 911 166 096 830 470 553 6 × 2 = 0 + 0.948 899 822 332 193 660 941 107 2;
  • 88) 0.948 899 822 332 193 660 941 107 2 × 2 = 1 + 0.897 799 644 664 387 321 882 214 4;
  • 89) 0.897 799 644 664 387 321 882 214 4 × 2 = 1 + 0.795 599 289 328 774 643 764 428 8;
  • 90) 0.795 599 289 328 774 643 764 428 8 × 2 = 1 + 0.591 198 578 657 549 287 528 857 6;
  • 91) 0.591 198 578 657 549 287 528 857 6 × 2 = 1 + 0.182 397 157 315 098 575 057 715 2;
  • 92) 0.182 397 157 315 098 575 057 715 2 × 2 = 0 + 0.364 794 314 630 197 150 115 430 4;
  • 93) 0.364 794 314 630 197 150 115 430 4 × 2 = 0 + 0.729 588 629 260 394 300 230 860 8;
  • 94) 0.729 588 629 260 394 300 230 860 8 × 2 = 1 + 0.459 177 258 520 788 600 461 721 6;
  • 95) 0.459 177 258 520 788 600 461 721 6 × 2 = 0 + 0.918 354 517 041 577 200 923 443 2;
  • 96) 0.918 354 517 041 577 200 923 443 2 × 2 = 1 + 0.836 709 034 083 154 401 846 886 4;
  • 97) 0.836 709 034 083 154 401 846 886 4 × 2 = 1 + 0.673 418 068 166 308 803 693 772 8;
  • 98) 0.673 418 068 166 308 803 693 772 8 × 2 = 1 + 0.346 836 136 332 617 607 387 545 6;
  • 99) 0.346 836 136 332 617 607 387 545 6 × 2 = 0 + 0.693 672 272 665 235 214 775 091 2;
  • 100) 0.693 672 272 665 235 214 775 091 2 × 2 = 1 + 0.387 344 545 330 470 429 550 182 4;
  • 101) 0.387 344 545 330 470 429 550 182 4 × 2 = 0 + 0.774 689 090 660 940 859 100 364 8;
  • 102) 0.774 689 090 660 940 859 100 364 8 × 2 = 1 + 0.549 378 181 321 881 718 200 729 6;
  • 103) 0.549 378 181 321 881 718 200 729 6 × 2 = 1 + 0.098 756 362 643 763 436 401 459 2;
  • 104) 0.098 756 362 643 763 436 401 459 2 × 2 = 0 + 0.197 512 725 287 526 872 802 918 4;
  • 105) 0.197 512 725 287 526 872 802 918 4 × 2 = 0 + 0.395 025 450 575 053 745 605 836 8;
  • 106) 0.395 025 450 575 053 745 605 836 8 × 2 = 0 + 0.790 050 901 150 107 491 211 673 6;
  • 107) 0.790 050 901 150 107 491 211 673 6 × 2 = 1 + 0.580 101 802 300 214 982 423 347 2;
  • 108) 0.580 101 802 300 214 982 423 347 2 × 2 = 1 + 0.160 203 604 600 429 964 846 694 4;
  • 109) 0.160 203 604 600 429 964 846 694 4 × 2 = 0 + 0.320 407 209 200 859 929 693 388 8;
  • 110) 0.320 407 209 200 859 929 693 388 8 × 2 = 0 + 0.640 814 418 401 719 859 386 777 6;
  • 111) 0.640 814 418 401 719 859 386 777 6 × 2 = 1 + 0.281 628 836 803 439 718 773 555 2;
  • 112) 0.281 628 836 803 439 718 773 555 2 × 2 = 0 + 0.563 257 673 606 879 437 547 110 4;
  • 113) 0.563 257 673 606 879 437 547 110 4 × 2 = 1 + 0.126 515 347 213 758 875 094 220 8;
  • 114) 0.126 515 347 213 758 875 094 220 8 × 2 = 0 + 0.253 030 694 427 517 750 188 441 6;
  • 115) 0.253 030 694 427 517 750 188 441 6 × 2 = 0 + 0.506 061 388 855 035 500 376 883 2;
  • 116) 0.506 061 388 855 035 500 376 883 2 × 2 = 1 + 0.012 122 777 710 071 000 753 766 4;
  • 117) 0.012 122 777 710 071 000 753 766 4 × 2 = 0 + 0.024 245 555 420 142 001 507 532 8;
  • 118) 0.024 245 555 420 142 001 507 532 8 × 2 = 0 + 0.048 491 110 840 284 003 015 065 6;
  • 119) 0.048 491 110 840 284 003 015 065 6 × 2 = 0 + 0.096 982 221 680 568 006 030 131 2;

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:


0.000 000 000 000 000 000 008 532 4(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1011 0000 1001 1110 0101 1101 0110 0011 0010 1001 000(2)

5. Positive number before normalization:

0.000 000 000 000 000 000 008 532 4(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1011 0000 1001 1110 0101 1101 0110 0011 0010 1001 000(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:


0.000 000 000 000 000 000 008 532 4(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1011 0000 1001 1110 0101 1101 0110 0011 0010 1001 000(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0100 1011 0000 1001 1110 0101 1101 0110 0011 0010 1001 000(2) × 20 =

1.0100 0010 0101 1000 0100 1111 0010 1110 1011 0001 1001 0100 1000(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 0101 1000 0100 1111 0010 1110 1011 0001 1001 0100 1000


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 0101 1000 0100 1111 0010 1110 1011 0001 1001 0100 1000 =

0100 0010 0101 1000 0100 1111 0010 1110 1011 0001 1001 0100 1000


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 0101 1000 0100 1111 0010 1110 1011 0001 1001 0100 1000


Decimal number 0.000 000 000 000 000 000 008 532 4 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1011 1100 - 0100 0010 0101 1000 0100 1111 0010 1110 1011 0001 1001 0100 1000

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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