0.000 000 000 000 000 000 008 552 1 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 000 000 008 552 1(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
0.000 000 000 000 000 000 008 552 1(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: 0.000 000 000 000 000 000 008 552 1.

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) 0.000 000 000 000 000 000 008 552 1 × 2 = 0 + 0.000 000 000 000 000 000 017 104 2;
  • 2) 0.000 000 000 000 000 000 017 104 2 × 2 = 0 + 0.000 000 000 000 000 000 034 208 4;
  • 3) 0.000 000 000 000 000 000 034 208 4 × 2 = 0 + 0.000 000 000 000 000 000 068 416 8;
  • 4) 0.000 000 000 000 000 000 068 416 8 × 2 = 0 + 0.000 000 000 000 000 000 136 833 6;
  • 5) 0.000 000 000 000 000 000 136 833 6 × 2 = 0 + 0.000 000 000 000 000 000 273 667 2;
  • 6) 0.000 000 000 000 000 000 273 667 2 × 2 = 0 + 0.000 000 000 000 000 000 547 334 4;
  • 7) 0.000 000 000 000 000 000 547 334 4 × 2 = 0 + 0.000 000 000 000 000 001 094 668 8;
  • 8) 0.000 000 000 000 000 001 094 668 8 × 2 = 0 + 0.000 000 000 000 000 002 189 337 6;
  • 9) 0.000 000 000 000 000 002 189 337 6 × 2 = 0 + 0.000 000 000 000 000 004 378 675 2;
  • 10) 0.000 000 000 000 000 004 378 675 2 × 2 = 0 + 0.000 000 000 000 000 008 757 350 4;
  • 11) 0.000 000 000 000 000 008 757 350 4 × 2 = 0 + 0.000 000 000 000 000 017 514 700 8;
  • 12) 0.000 000 000 000 000 017 514 700 8 × 2 = 0 + 0.000 000 000 000 000 035 029 401 6;
  • 13) 0.000 000 000 000 000 035 029 401 6 × 2 = 0 + 0.000 000 000 000 000 070 058 803 2;
  • 14) 0.000 000 000 000 000 070 058 803 2 × 2 = 0 + 0.000 000 000 000 000 140 117 606 4;
  • 15) 0.000 000 000 000 000 140 117 606 4 × 2 = 0 + 0.000 000 000 000 000 280 235 212 8;
  • 16) 0.000 000 000 000 000 280 235 212 8 × 2 = 0 + 0.000 000 000 000 000 560 470 425 6;
  • 17) 0.000 000 000 000 000 560 470 425 6 × 2 = 0 + 0.000 000 000 000 001 120 940 851 2;
  • 18) 0.000 000 000 000 001 120 940 851 2 × 2 = 0 + 0.000 000 000 000 002 241 881 702 4;
  • 19) 0.000 000 000 000 002 241 881 702 4 × 2 = 0 + 0.000 000 000 000 004 483 763 404 8;
  • 20) 0.000 000 000 000 004 483 763 404 8 × 2 = 0 + 0.000 000 000 000 008 967 526 809 6;
  • 21) 0.000 000 000 000 008 967 526 809 6 × 2 = 0 + 0.000 000 000 000 017 935 053 619 2;
  • 22) 0.000 000 000 000 017 935 053 619 2 × 2 = 0 + 0.000 000 000 000 035 870 107 238 4;
  • 23) 0.000 000 000 000 035 870 107 238 4 × 2 = 0 + 0.000 000 000 000 071 740 214 476 8;
  • 24) 0.000 000 000 000 071 740 214 476 8 × 2 = 0 + 0.000 000 000 000 143 480 428 953 6;
  • 25) 0.000 000 000 000 143 480 428 953 6 × 2 = 0 + 0.000 000 000 000 286 960 857 907 2;
  • 26) 0.000 000 000 000 286 960 857 907 2 × 2 = 0 + 0.000 000 000 000 573 921 715 814 4;
  • 27) 0.000 000 000 000 573 921 715 814 4 × 2 = 0 + 0.000 000 000 001 147 843 431 628 8;
  • 28) 0.000 000 000 001 147 843 431 628 8 × 2 = 0 + 0.000 000 000 002 295 686 863 257 6;
  • 29) 0.000 000 000 002 295 686 863 257 6 × 2 = 0 + 0.000 000 000 004 591 373 726 515 2;
  • 30) 0.000 000 000 004 591 373 726 515 2 × 2 = 0 + 0.000 000 000 009 182 747 453 030 4;
  • 31) 0.000 000 000 009 182 747 453 030 4 × 2 = 0 + 0.000 000 000 018 365 494 906 060 8;
  • 32) 0.000 000 000 018 365 494 906 060 8 × 2 = 0 + 0.000 000 000 036 730 989 812 121 6;
  • 33) 0.000 000 000 036 730 989 812 121 6 × 2 = 0 + 0.000 000 000 073 461 979 624 243 2;
  • 34) 0.000 000 000 073 461 979 624 243 2 × 2 = 0 + 0.000 000 000 146 923 959 248 486 4;
  • 35) 0.000 000 000 146 923 959 248 486 4 × 2 = 0 + 0.000 000 000 293 847 918 496 972 8;
  • 36) 0.000 000 000 293 847 918 496 972 8 × 2 = 0 + 0.000 000 000 587 695 836 993 945 6;
  • 37) 0.000 000 000 587 695 836 993 945 6 × 2 = 0 + 0.000 000 001 175 391 673 987 891 2;
  • 38) 0.000 000 001 175 391 673 987 891 2 × 2 = 0 + 0.000 000 002 350 783 347 975 782 4;
  • 39) 0.000 000 002 350 783 347 975 782 4 × 2 = 0 + 0.000 000 004 701 566 695 951 564 8;
  • 40) 0.000 000 004 701 566 695 951 564 8 × 2 = 0 + 0.000 000 009 403 133 391 903 129 6;
  • 41) 0.000 000 009 403 133 391 903 129 6 × 2 = 0 + 0.000 000 018 806 266 783 806 259 2;
  • 42) 0.000 000 018 806 266 783 806 259 2 × 2 = 0 + 0.000 000 037 612 533 567 612 518 4;
  • 43) 0.000 000 037 612 533 567 612 518 4 × 2 = 0 + 0.000 000 075 225 067 135 225 036 8;
  • 44) 0.000 000 075 225 067 135 225 036 8 × 2 = 0 + 0.000 000 150 450 134 270 450 073 6;
  • 45) 0.000 000 150 450 134 270 450 073 6 × 2 = 0 + 0.000 000 300 900 268 540 900 147 2;
  • 46) 0.000 000 300 900 268 540 900 147 2 × 2 = 0 + 0.000 000 601 800 537 081 800 294 4;
  • 47) 0.000 000 601 800 537 081 800 294 4 × 2 = 0 + 0.000 001 203 601 074 163 600 588 8;
  • 48) 0.000 001 203 601 074 163 600 588 8 × 2 = 0 + 0.000 002 407 202 148 327 201 177 6;
  • 49) 0.000 002 407 202 148 327 201 177 6 × 2 = 0 + 0.000 004 814 404 296 654 402 355 2;
  • 50) 0.000 004 814 404 296 654 402 355 2 × 2 = 0 + 0.000 009 628 808 593 308 804 710 4;
  • 51) 0.000 009 628 808 593 308 804 710 4 × 2 = 0 + 0.000 019 257 617 186 617 609 420 8;
  • 52) 0.000 019 257 617 186 617 609 420 8 × 2 = 0 + 0.000 038 515 234 373 235 218 841 6;
  • 53) 0.000 038 515 234 373 235 218 841 6 × 2 = 0 + 0.000 077 030 468 746 470 437 683 2;
  • 54) 0.000 077 030 468 746 470 437 683 2 × 2 = 0 + 0.000 154 060 937 492 940 875 366 4;
  • 55) 0.000 154 060 937 492 940 875 366 4 × 2 = 0 + 0.000 308 121 874 985 881 750 732 8;
  • 56) 0.000 308 121 874 985 881 750 732 8 × 2 = 0 + 0.000 616 243 749 971 763 501 465 6;
  • 57) 0.000 616 243 749 971 763 501 465 6 × 2 = 0 + 0.001 232 487 499 943 527 002 931 2;
  • 58) 0.001 232 487 499 943 527 002 931 2 × 2 = 0 + 0.002 464 974 999 887 054 005 862 4;
  • 59) 0.002 464 974 999 887 054 005 862 4 × 2 = 0 + 0.004 929 949 999 774 108 011 724 8;
  • 60) 0.004 929 949 999 774 108 011 724 8 × 2 = 0 + 0.009 859 899 999 548 216 023 449 6;
  • 61) 0.009 859 899 999 548 216 023 449 6 × 2 = 0 + 0.019 719 799 999 096 432 046 899 2;
  • 62) 0.019 719 799 999 096 432 046 899 2 × 2 = 0 + 0.039 439 599 998 192 864 093 798 4;
  • 63) 0.039 439 599 998 192 864 093 798 4 × 2 = 0 + 0.078 879 199 996 385 728 187 596 8;
  • 64) 0.078 879 199 996 385 728 187 596 8 × 2 = 0 + 0.157 758 399 992 771 456 375 193 6;
  • 65) 0.157 758 399 992 771 456 375 193 6 × 2 = 0 + 0.315 516 799 985 542 912 750 387 2;
  • 66) 0.315 516 799 985 542 912 750 387 2 × 2 = 0 + 0.631 033 599 971 085 825 500 774 4;
  • 67) 0.631 033 599 971 085 825 500 774 4 × 2 = 1 + 0.262 067 199 942 171 651 001 548 8;
  • 68) 0.262 067 199 942 171 651 001 548 8 × 2 = 0 + 0.524 134 399 884 343 302 003 097 6;
  • 69) 0.524 134 399 884 343 302 003 097 6 × 2 = 1 + 0.048 268 799 768 686 604 006 195 2;
  • 70) 0.048 268 799 768 686 604 006 195 2 × 2 = 0 + 0.096 537 599 537 373 208 012 390 4;
  • 71) 0.096 537 599 537 373 208 012 390 4 × 2 = 0 + 0.193 075 199 074 746 416 024 780 8;
  • 72) 0.193 075 199 074 746 416 024 780 8 × 2 = 0 + 0.386 150 398 149 492 832 049 561 6;
  • 73) 0.386 150 398 149 492 832 049 561 6 × 2 = 0 + 0.772 300 796 298 985 664 099 123 2;
  • 74) 0.772 300 796 298 985 664 099 123 2 × 2 = 1 + 0.544 601 592 597 971 328 198 246 4;
  • 75) 0.544 601 592 597 971 328 198 246 4 × 2 = 1 + 0.089 203 185 195 942 656 396 492 8;
  • 76) 0.089 203 185 195 942 656 396 492 8 × 2 = 0 + 0.178 406 370 391 885 312 792 985 6;
  • 77) 0.178 406 370 391 885 312 792 985 6 × 2 = 0 + 0.356 812 740 783 770 625 585 971 2;
  • 78) 0.356 812 740 783 770 625 585 971 2 × 2 = 0 + 0.713 625 481 567 541 251 171 942 4;
  • 79) 0.713 625 481 567 541 251 171 942 4 × 2 = 1 + 0.427 250 963 135 082 502 343 884 8;
  • 80) 0.427 250 963 135 082 502 343 884 8 × 2 = 0 + 0.854 501 926 270 165 004 687 769 6;
  • 81) 0.854 501 926 270 165 004 687 769 6 × 2 = 1 + 0.709 003 852 540 330 009 375 539 2;
  • 82) 0.709 003 852 540 330 009 375 539 2 × 2 = 1 + 0.418 007 705 080 660 018 751 078 4;
  • 83) 0.418 007 705 080 660 018 751 078 4 × 2 = 0 + 0.836 015 410 161 320 037 502 156 8;
  • 84) 0.836 015 410 161 320 037 502 156 8 × 2 = 1 + 0.672 030 820 322 640 075 004 313 6;
  • 85) 0.672 030 820 322 640 075 004 313 6 × 2 = 1 + 0.344 061 640 645 280 150 008 627 2;
  • 86) 0.344 061 640 645 280 150 008 627 2 × 2 = 0 + 0.688 123 281 290 560 300 017 254 4;
  • 87) 0.688 123 281 290 560 300 017 254 4 × 2 = 1 + 0.376 246 562 581 120 600 034 508 8;
  • 88) 0.376 246 562 581 120 600 034 508 8 × 2 = 0 + 0.752 493 125 162 241 200 069 017 6;
  • 89) 0.752 493 125 162 241 200 069 017 6 × 2 = 1 + 0.504 986 250 324 482 400 138 035 2;
  • 90) 0.504 986 250 324 482 400 138 035 2 × 2 = 1 + 0.009 972 500 648 964 800 276 070 4;
  • 91) 0.009 972 500 648 964 800 276 070 4 × 2 = 0 + 0.019 945 001 297 929 600 552 140 8;
  • 92) 0.019 945 001 297 929 600 552 140 8 × 2 = 0 + 0.039 890 002 595 859 201 104 281 6;
  • 93) 0.039 890 002 595 859 201 104 281 6 × 2 = 0 + 0.079 780 005 191 718 402 208 563 2;
  • 94) 0.079 780 005 191 718 402 208 563 2 × 2 = 0 + 0.159 560 010 383 436 804 417 126 4;
  • 95) 0.159 560 010 383 436 804 417 126 4 × 2 = 0 + 0.319 120 020 766 873 608 834 252 8;
  • 96) 0.319 120 020 766 873 608 834 252 8 × 2 = 0 + 0.638 240 041 533 747 217 668 505 6;
  • 97) 0.638 240 041 533 747 217 668 505 6 × 2 = 1 + 0.276 480 083 067 494 435 337 011 2;
  • 98) 0.276 480 083 067 494 435 337 011 2 × 2 = 0 + 0.552 960 166 134 988 870 674 022 4;
  • 99) 0.552 960 166 134 988 870 674 022 4 × 2 = 1 + 0.105 920 332 269 977 741 348 044 8;
  • 100) 0.105 920 332 269 977 741 348 044 8 × 2 = 0 + 0.211 840 664 539 955 482 696 089 6;
  • 101) 0.211 840 664 539 955 482 696 089 6 × 2 = 0 + 0.423 681 329 079 910 965 392 179 2;
  • 102) 0.423 681 329 079 910 965 392 179 2 × 2 = 0 + 0.847 362 658 159 821 930 784 358 4;
  • 103) 0.847 362 658 159 821 930 784 358 4 × 2 = 1 + 0.694 725 316 319 643 861 568 716 8;
  • 104) 0.694 725 316 319 643 861 568 716 8 × 2 = 1 + 0.389 450 632 639 287 723 137 433 6;
  • 105) 0.389 450 632 639 287 723 137 433 6 × 2 = 0 + 0.778 901 265 278 575 446 274 867 2;
  • 106) 0.778 901 265 278 575 446 274 867 2 × 2 = 1 + 0.557 802 530 557 150 892 549 734 4;
  • 107) 0.557 802 530 557 150 892 549 734 4 × 2 = 1 + 0.115 605 061 114 301 785 099 468 8;
  • 108) 0.115 605 061 114 301 785 099 468 8 × 2 = 0 + 0.231 210 122 228 603 570 198 937 6;
  • 109) 0.231 210 122 228 603 570 198 937 6 × 2 = 0 + 0.462 420 244 457 207 140 397 875 2;
  • 110) 0.462 420 244 457 207 140 397 875 2 × 2 = 0 + 0.924 840 488 914 414 280 795 750 4;
  • 111) 0.924 840 488 914 414 280 795 750 4 × 2 = 1 + 0.849 680 977 828 828 561 591 500 8;
  • 112) 0.849 680 977 828 828 561 591 500 8 × 2 = 1 + 0.699 361 955 657 657 123 183 001 6;
  • 113) 0.699 361 955 657 657 123 183 001 6 × 2 = 1 + 0.398 723 911 315 314 246 366 003 2;
  • 114) 0.398 723 911 315 314 246 366 003 2 × 2 = 0 + 0.797 447 822 630 628 492 732 006 4;
  • 115) 0.797 447 822 630 628 492 732 006 4 × 2 = 1 + 0.594 895 645 261 256 985 464 012 8;
  • 116) 0.594 895 645 261 256 985 464 012 8 × 2 = 1 + 0.189 791 290 522 513 970 928 025 6;
  • 117) 0.189 791 290 522 513 970 928 025 6 × 2 = 0 + 0.379 582 581 045 027 941 856 051 2;
  • 118) 0.379 582 581 045 027 941 856 051 2 × 2 = 0 + 0.759 165 162 090 055 883 712 102 4;
  • 119) 0.759 165 162 090 055 883 712 102 4 × 2 = 1 + 0.518 330 324 180 111 767 424 204 8;

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:


0.000 000 000 000 000 000 008 552 1(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0110 0010 1101 1010 1100 0000 1010 0011 0110 0011 1011 001(2)

5. Positive number before normalization:

0.000 000 000 000 000 000 008 552 1(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0110 0010 1101 1010 1100 0000 1010 0011 0110 0011 1011 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:


0.000 000 000 000 000 000 008 552 1(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0110 0010 1101 1010 1100 0000 1010 0011 0110 0011 1011 001(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1000 0110 0010 1101 1010 1100 0000 1010 0011 0110 0011 1011 001(2) × 20 =

1.0100 0011 0001 0110 1101 0110 0000 0101 0001 1011 0001 1101 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 0011 0001 0110 1101 0110 0000 0101 0001 1011 0001 1101 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 0011 0001 0110 1101 0110 0000 0101 0001 1011 0001 1101 1001 =

0100 0011 0001 0110 1101 0110 0000 0101 0001 1011 0001 1101 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 0011 0001 0110 1101 0110 0000 0101 0001 1011 0001 1101 1001


Decimal number 0.000 000 000 000 000 000 008 552 1 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1011 1100 - 0100 0011 0001 0110 1101 0110 0000 0101 0001 1011 0001 1101 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