64bit IEEE 754: Decimal ↗ Double Precision Floating Point Binary: -0.008 788 423 612 713 799 903 950 473 435 543 244 704 604 148 864 746 093 1 Convert the Number to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard, From a Base Ten Decimal System Number

Number -0.008 788 423 612 713 799 903 950 473 435 543 244 704 604 148 864 746 093 1(10) converted and written in 64 bit double precision IEEE 754 binary floating point representation (1 bit for sign, 11 bits for exponent, 52 bits for mantissa)

1. Start with the positive version of the number:

|-0.008 788 423 612 713 799 903 950 473 435 543 244 704 604 148 864 746 093 1| = 0.008 788 423 612 713 799 903 950 473 435 543 244 704 604 148 864 746 093 1

2. 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;

3. 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)


4. Convert to binary (base 2) the fractional part: 0.008 788 423 612 713 799 903 950 473 435 543 244 704 604 148 864 746 093 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) 0.008 788 423 612 713 799 903 950 473 435 543 244 704 604 148 864 746 093 1 × 2 = 0 + 0.017 576 847 225 427 599 807 900 946 871 086 489 409 208 297 729 492 186 2;
  • 2) 0.017 576 847 225 427 599 807 900 946 871 086 489 409 208 297 729 492 186 2 × 2 = 0 + 0.035 153 694 450 855 199 615 801 893 742 172 978 818 416 595 458 984 372 4;
  • 3) 0.035 153 694 450 855 199 615 801 893 742 172 978 818 416 595 458 984 372 4 × 2 = 0 + 0.070 307 388 901 710 399 231 603 787 484 345 957 636 833 190 917 968 744 8;
  • 4) 0.070 307 388 901 710 399 231 603 787 484 345 957 636 833 190 917 968 744 8 × 2 = 0 + 0.140 614 777 803 420 798 463 207 574 968 691 915 273 666 381 835 937 489 6;
  • 5) 0.140 614 777 803 420 798 463 207 574 968 691 915 273 666 381 835 937 489 6 × 2 = 0 + 0.281 229 555 606 841 596 926 415 149 937 383 830 547 332 763 671 874 979 2;
  • 6) 0.281 229 555 606 841 596 926 415 149 937 383 830 547 332 763 671 874 979 2 × 2 = 0 + 0.562 459 111 213 683 193 852 830 299 874 767 661 094 665 527 343 749 958 4;
  • 7) 0.562 459 111 213 683 193 852 830 299 874 767 661 094 665 527 343 749 958 4 × 2 = 1 + 0.124 918 222 427 366 387 705 660 599 749 535 322 189 331 054 687 499 916 8;
  • 8) 0.124 918 222 427 366 387 705 660 599 749 535 322 189 331 054 687 499 916 8 × 2 = 0 + 0.249 836 444 854 732 775 411 321 199 499 070 644 378 662 109 374 999 833 6;
  • 9) 0.249 836 444 854 732 775 411 321 199 499 070 644 378 662 109 374 999 833 6 × 2 = 0 + 0.499 672 889 709 465 550 822 642 398 998 141 288 757 324 218 749 999 667 2;
  • 10) 0.499 672 889 709 465 550 822 642 398 998 141 288 757 324 218 749 999 667 2 × 2 = 0 + 0.999 345 779 418 931 101 645 284 797 996 282 577 514 648 437 499 999 334 4;
  • 11) 0.999 345 779 418 931 101 645 284 797 996 282 577 514 648 437 499 999 334 4 × 2 = 1 + 0.998 691 558 837 862 203 290 569 595 992 565 155 029 296 874 999 998 668 8;
  • 12) 0.998 691 558 837 862 203 290 569 595 992 565 155 029 296 874 999 998 668 8 × 2 = 1 + 0.997 383 117 675 724 406 581 139 191 985 130 310 058 593 749 999 997 337 6;
  • 13) 0.997 383 117 675 724 406 581 139 191 985 130 310 058 593 749 999 997 337 6 × 2 = 1 + 0.994 766 235 351 448 813 162 278 383 970 260 620 117 187 499 999 994 675 2;
  • 14) 0.994 766 235 351 448 813 162 278 383 970 260 620 117 187 499 999 994 675 2 × 2 = 1 + 0.989 532 470 702 897 626 324 556 767 940 521 240 234 374 999 999 989 350 4;
  • 15) 0.989 532 470 702 897 626 324 556 767 940 521 240 234 374 999 999 989 350 4 × 2 = 1 + 0.979 064 941 405 795 252 649 113 535 881 042 480 468 749 999 999 978 700 8;
  • 16) 0.979 064 941 405 795 252 649 113 535 881 042 480 468 749 999 999 978 700 8 × 2 = 1 + 0.958 129 882 811 590 505 298 227 071 762 084 960 937 499 999 999 957 401 6;
  • 17) 0.958 129 882 811 590 505 298 227 071 762 084 960 937 499 999 999 957 401 6 × 2 = 1 + 0.916 259 765 623 181 010 596 454 143 524 169 921 874 999 999 999 914 803 2;
  • 18) 0.916 259 765 623 181 010 596 454 143 524 169 921 874 999 999 999 914 803 2 × 2 = 1 + 0.832 519 531 246 362 021 192 908 287 048 339 843 749 999 999 999 829 606 4;
  • 19) 0.832 519 531 246 362 021 192 908 287 048 339 843 749 999 999 999 829 606 4 × 2 = 1 + 0.665 039 062 492 724 042 385 816 574 096 679 687 499 999 999 999 659 212 8;
  • 20) 0.665 039 062 492 724 042 385 816 574 096 679 687 499 999 999 999 659 212 8 × 2 = 1 + 0.330 078 124 985 448 084 771 633 148 193 359 374 999 999 999 999 318 425 6;
  • 21) 0.330 078 124 985 448 084 771 633 148 193 359 374 999 999 999 999 318 425 6 × 2 = 0 + 0.660 156 249 970 896 169 543 266 296 386 718 749 999 999 999 998 636 851 2;
  • 22) 0.660 156 249 970 896 169 543 266 296 386 718 749 999 999 999 998 636 851 2 × 2 = 1 + 0.320 312 499 941 792 339 086 532 592 773 437 499 999 999 999 997 273 702 4;
  • 23) 0.320 312 499 941 792 339 086 532 592 773 437 499 999 999 999 997 273 702 4 × 2 = 0 + 0.640 624 999 883 584 678 173 065 185 546 874 999 999 999 999 994 547 404 8;
  • 24) 0.640 624 999 883 584 678 173 065 185 546 874 999 999 999 999 994 547 404 8 × 2 = 1 + 0.281 249 999 767 169 356 346 130 371 093 749 999 999 999 999 989 094 809 6;
  • 25) 0.281 249 999 767 169 356 346 130 371 093 749 999 999 999 999 989 094 809 6 × 2 = 0 + 0.562 499 999 534 338 712 692 260 742 187 499 999 999 999 999 978 189 619 2;
  • 26) 0.562 499 999 534 338 712 692 260 742 187 499 999 999 999 999 978 189 619 2 × 2 = 1 + 0.124 999 999 068 677 425 384 521 484 374 999 999 999 999 999 956 379 238 4;
  • 27) 0.124 999 999 068 677 425 384 521 484 374 999 999 999 999 999 956 379 238 4 × 2 = 0 + 0.249 999 998 137 354 850 769 042 968 749 999 999 999 999 999 912 758 476 8;
  • 28) 0.249 999 998 137 354 850 769 042 968 749 999 999 999 999 999 912 758 476 8 × 2 = 0 + 0.499 999 996 274 709 701 538 085 937 499 999 999 999 999 999 825 516 953 6;
  • 29) 0.499 999 996 274 709 701 538 085 937 499 999 999 999 999 999 825 516 953 6 × 2 = 0 + 0.999 999 992 549 419 403 076 171 874 999 999 999 999 999 999 651 033 907 2;
  • 30) 0.999 999 992 549 419 403 076 171 874 999 999 999 999 999 999 651 033 907 2 × 2 = 1 + 0.999 999 985 098 838 806 152 343 749 999 999 999 999 999 999 302 067 814 4;
  • 31) 0.999 999 985 098 838 806 152 343 749 999 999 999 999 999 999 302 067 814 4 × 2 = 1 + 0.999 999 970 197 677 612 304 687 499 999 999 999 999 999 998 604 135 628 8;
  • 32) 0.999 999 970 197 677 612 304 687 499 999 999 999 999 999 998 604 135 628 8 × 2 = 1 + 0.999 999 940 395 355 224 609 374 999 999 999 999 999 999 997 208 271 257 6;
  • 33) 0.999 999 940 395 355 224 609 374 999 999 999 999 999 999 997 208 271 257 6 × 2 = 1 + 0.999 999 880 790 710 449 218 749 999 999 999 999 999 999 994 416 542 515 2;
  • 34) 0.999 999 880 790 710 449 218 749 999 999 999 999 999 999 994 416 542 515 2 × 2 = 1 + 0.999 999 761 581 420 898 437 499 999 999 999 999 999 999 988 833 085 030 4;
  • 35) 0.999 999 761 581 420 898 437 499 999 999 999 999 999 999 988 833 085 030 4 × 2 = 1 + 0.999 999 523 162 841 796 874 999 999 999 999 999 999 999 977 666 170 060 8;
  • 36) 0.999 999 523 162 841 796 874 999 999 999 999 999 999 999 977 666 170 060 8 × 2 = 1 + 0.999 999 046 325 683 593 749 999 999 999 999 999 999 999 955 332 340 121 6;
  • 37) 0.999 999 046 325 683 593 749 999 999 999 999 999 999 999 955 332 340 121 6 × 2 = 1 + 0.999 998 092 651 367 187 499 999 999 999 999 999 999 999 910 664 680 243 2;
  • 38) 0.999 998 092 651 367 187 499 999 999 999 999 999 999 999 910 664 680 243 2 × 2 = 1 + 0.999 996 185 302 734 374 999 999 999 999 999 999 999 999 821 329 360 486 4;
  • 39) 0.999 996 185 302 734 374 999 999 999 999 999 999 999 999 821 329 360 486 4 × 2 = 1 + 0.999 992 370 605 468 749 999 999 999 999 999 999 999 999 642 658 720 972 8;
  • 40) 0.999 992 370 605 468 749 999 999 999 999 999 999 999 999 642 658 720 972 8 × 2 = 1 + 0.999 984 741 210 937 499 999 999 999 999 999 999 999 999 285 317 441 945 6;
  • 41) 0.999 984 741 210 937 499 999 999 999 999 999 999 999 999 285 317 441 945 6 × 2 = 1 + 0.999 969 482 421 874 999 999 999 999 999 999 999 999 998 570 634 883 891 2;
  • 42) 0.999 969 482 421 874 999 999 999 999 999 999 999 999 998 570 634 883 891 2 × 2 = 1 + 0.999 938 964 843 749 999 999 999 999 999 999 999 999 997 141 269 767 782 4;
  • 43) 0.999 938 964 843 749 999 999 999 999 999 999 999 999 997 141 269 767 782 4 × 2 = 1 + 0.999 877 929 687 499 999 999 999 999 999 999 999 999 994 282 539 535 564 8;
  • 44) 0.999 877 929 687 499 999 999 999 999 999 999 999 999 994 282 539 535 564 8 × 2 = 1 + 0.999 755 859 374 999 999 999 999 999 999 999 999 999 988 565 079 071 129 6;
  • 45) 0.999 755 859 374 999 999 999 999 999 999 999 999 999 988 565 079 071 129 6 × 2 = 1 + 0.999 511 718 749 999 999 999 999 999 999 999 999 999 977 130 158 142 259 2;
  • 46) 0.999 511 718 749 999 999 999 999 999 999 999 999 999 977 130 158 142 259 2 × 2 = 1 + 0.999 023 437 499 999 999 999 999 999 999 999 999 999 954 260 316 284 518 4;
  • 47) 0.999 023 437 499 999 999 999 999 999 999 999 999 999 954 260 316 284 518 4 × 2 = 1 + 0.998 046 874 999 999 999 999 999 999 999 999 999 999 908 520 632 569 036 8;
  • 48) 0.998 046 874 999 999 999 999 999 999 999 999 999 999 908 520 632 569 036 8 × 2 = 1 + 0.996 093 749 999 999 999 999 999 999 999 999 999 999 817 041 265 138 073 6;
  • 49) 0.996 093 749 999 999 999 999 999 999 999 999 999 999 817 041 265 138 073 6 × 2 = 1 + 0.992 187 499 999 999 999 999 999 999 999 999 999 999 634 082 530 276 147 2;
  • 50) 0.992 187 499 999 999 999 999 999 999 999 999 999 999 634 082 530 276 147 2 × 2 = 1 + 0.984 374 999 999 999 999 999 999 999 999 999 999 999 268 165 060 552 294 4;
  • 51) 0.984 374 999 999 999 999 999 999 999 999 999 999 999 268 165 060 552 294 4 × 2 = 1 + 0.968 749 999 999 999 999 999 999 999 999 999 999 998 536 330 121 104 588 8;
  • 52) 0.968 749 999 999 999 999 999 999 999 999 999 999 998 536 330 121 104 588 8 × 2 = 1 + 0.937 499 999 999 999 999 999 999 999 999 999 999 997 072 660 242 209 177 6;
  • 53) 0.937 499 999 999 999 999 999 999 999 999 999 999 997 072 660 242 209 177 6 × 2 = 1 + 0.874 999 999 999 999 999 999 999 999 999 999 999 994 145 320 484 418 355 2;
  • 54) 0.874 999 999 999 999 999 999 999 999 999 999 999 994 145 320 484 418 355 2 × 2 = 1 + 0.749 999 999 999 999 999 999 999 999 999 999 999 988 290 640 968 836 710 4;
  • 55) 0.749 999 999 999 999 999 999 999 999 999 999 999 988 290 640 968 836 710 4 × 2 = 1 + 0.499 999 999 999 999 999 999 999 999 999 999 999 976 581 281 937 673 420 8;
  • 56) 0.499 999 999 999 999 999 999 999 999 999 999 999 976 581 281 937 673 420 8 × 2 = 0 + 0.999 999 999 999 999 999 999 999 999 999 999 999 953 162 563 875 346 841 6;
  • 57) 0.999 999 999 999 999 999 999 999 999 999 999 999 953 162 563 875 346 841 6 × 2 = 1 + 0.999 999 999 999 999 999 999 999 999 999 999 999 906 325 127 750 693 683 2;
  • 58) 0.999 999 999 999 999 999 999 999 999 999 999 999 906 325 127 750 693 683 2 × 2 = 1 + 0.999 999 999 999 999 999 999 999 999 999 999 999 812 650 255 501 387 366 4;
  • 59) 0.999 999 999 999 999 999 999 999 999 999 999 999 812 650 255 501 387 366 4 × 2 = 1 + 0.999 999 999 999 999 999 999 999 999 999 999 999 625 300 511 002 774 732 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...)


5. 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:


0.008 788 423 612 713 799 903 950 473 435 543 244 704 604 148 864 746 093 1(10) =


0.0000 0010 0011 1111 1111 0101 0100 0111 1111 1111 1111 1111 1111 1110 111(2)


6. Positive number before normalization:

0.008 788 423 612 713 799 903 950 473 435 543 244 704 604 148 864 746 093 1(10) =


0.0000 0010 0011 1111 1111 0101 0100 0111 1111 1111 1111 1111 1111 1110 111(2)

7. Normalize the binary representation of the number.

Shift the decimal mark 7 positions to the right, so that only one non zero digit remains to the left of it:


0.008 788 423 612 713 799 903 950 473 435 543 244 704 604 148 864 746 093 1(10) =


0.0000 0010 0011 1111 1111 0101 0100 0111 1111 1111 1111 1111 1111 1110 111(2) =


0.0000 0010 0011 1111 1111 0101 0100 0111 1111 1111 1111 1111 1111 1110 111(2) × 20 =


1.0001 1111 1111 1010 1010 0011 1111 1111 1111 1111 1111 1111 0111(2) × 2-7


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): -7


Mantissa (not normalized):
1.0001 1111 1111 1010 1010 0011 1111 1111 1111 1111 1111 1111 0111


9. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =


Exponent (unadjusted) + 2(11-1) - 1 =


-7 + 2(11-1) - 1 =


(-7 + 1 023)(10) =


1 016(10)


10. 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;
  • 1 016 ÷ 2 = 508 + 0;
  • 508 ÷ 2 = 254 + 0;
  • 254 ÷ 2 = 127 + 0;
  • 127 ÷ 2 = 63 + 1;
  • 63 ÷ 2 = 31 + 1;
  • 31 ÷ 2 = 15 + 1;
  • 15 ÷ 2 = 7 + 1;
  • 7 ÷ 2 = 3 + 1;
  • 3 ÷ 2 = 1 + 1;
  • 1 ÷ 2 = 0 + 1;

11. Construct the base 2 representation of the adjusted exponent.

Take all the remainders starting from the bottom of the list constructed above.


Exponent (adjusted) =


1016(10) =


011 1111 1000(2)


12. 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. 0001 1111 1111 1010 1010 0011 1111 1111 1111 1111 1111 1111 0111 =


0001 1111 1111 1010 1010 0011 1111 1111 1111 1111 1111 1111 0111


13. The three elements that make up the number's 64 bit double precision IEEE 754 binary floating point representation:

Sign (1 bit) =
1 (a negative number)


Exponent (11 bits) =
011 1111 1000


Mantissa (52 bits) =
0001 1111 1111 1010 1010 0011 1111 1111 1111 1111 1111 1111 0111


The base ten decimal number -0.008 788 423 612 713 799 903 950 473 435 543 244 704 604 148 864 746 093 1 converted and written in 64 bit double precision IEEE 754 binary floating point representation:
1 - 011 1111 1000 - 0001 1111 1111 1010 1010 0011 1111 1111 1111 1111 1111 1111 0111

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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