-0.016 738 891 601 562 531 225 022 567 582 527 699 414 640 665 054 320 59 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.016 738 891 601 562 531 225 022 567 582 527 699 414 640 665 054 320 59(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
-0.016 738 891 601 562 531 225 022 567 582 527 699 414 640 665 054 320 59(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. Start with the positive version of the number:

|-0.016 738 891 601 562 531 225 022 567 582 527 699 414 640 665 054 320 59| = 0.016 738 891 601 562 531 225 022 567 582 527 699 414 640 665 054 320 59


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: 0.016 738 891 601 562 531 225 022 567 582 527 699 414 640 665 054 320 59.

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) 0.016 738 891 601 562 531 225 022 567 582 527 699 414 640 665 054 320 59 × 2 = 0 + 0.033 477 783 203 125 062 450 045 135 165 055 398 829 281 330 108 641 18;
  • 2) 0.033 477 783 203 125 062 450 045 135 165 055 398 829 281 330 108 641 18 × 2 = 0 + 0.066 955 566 406 250 124 900 090 270 330 110 797 658 562 660 217 282 36;
  • 3) 0.066 955 566 406 250 124 900 090 270 330 110 797 658 562 660 217 282 36 × 2 = 0 + 0.133 911 132 812 500 249 800 180 540 660 221 595 317 125 320 434 564 72;
  • 4) 0.133 911 132 812 500 249 800 180 540 660 221 595 317 125 320 434 564 72 × 2 = 0 + 0.267 822 265 625 000 499 600 361 081 320 443 190 634 250 640 869 129 44;
  • 5) 0.267 822 265 625 000 499 600 361 081 320 443 190 634 250 640 869 129 44 × 2 = 0 + 0.535 644 531 250 000 999 200 722 162 640 886 381 268 501 281 738 258 88;
  • 6) 0.535 644 531 250 000 999 200 722 162 640 886 381 268 501 281 738 258 88 × 2 = 1 + 0.071 289 062 500 001 998 401 444 325 281 772 762 537 002 563 476 517 76;
  • 7) 0.071 289 062 500 001 998 401 444 325 281 772 762 537 002 563 476 517 76 × 2 = 0 + 0.142 578 125 000 003 996 802 888 650 563 545 525 074 005 126 953 035 52;
  • 8) 0.142 578 125 000 003 996 802 888 650 563 545 525 074 005 126 953 035 52 × 2 = 0 + 0.285 156 250 000 007 993 605 777 301 127 091 050 148 010 253 906 071 04;
  • 9) 0.285 156 250 000 007 993 605 777 301 127 091 050 148 010 253 906 071 04 × 2 = 0 + 0.570 312 500 000 015 987 211 554 602 254 182 100 296 020 507 812 142 08;
  • 10) 0.570 312 500 000 015 987 211 554 602 254 182 100 296 020 507 812 142 08 × 2 = 1 + 0.140 625 000 000 031 974 423 109 204 508 364 200 592 041 015 624 284 16;
  • 11) 0.140 625 000 000 031 974 423 109 204 508 364 200 592 041 015 624 284 16 × 2 = 0 + 0.281 250 000 000 063 948 846 218 409 016 728 401 184 082 031 248 568 32;
  • 12) 0.281 250 000 000 063 948 846 218 409 016 728 401 184 082 031 248 568 32 × 2 = 0 + 0.562 500 000 000 127 897 692 436 818 033 456 802 368 164 062 497 136 64;
  • 13) 0.562 500 000 000 127 897 692 436 818 033 456 802 368 164 062 497 136 64 × 2 = 1 + 0.125 000 000 000 255 795 384 873 636 066 913 604 736 328 124 994 273 28;
  • 14) 0.125 000 000 000 255 795 384 873 636 066 913 604 736 328 124 994 273 28 × 2 = 0 + 0.250 000 000 000 511 590 769 747 272 133 827 209 472 656 249 988 546 56;
  • 15) 0.250 000 000 000 511 590 769 747 272 133 827 209 472 656 249 988 546 56 × 2 = 0 + 0.500 000 000 001 023 181 539 494 544 267 654 418 945 312 499 977 093 12;
  • 16) 0.500 000 000 001 023 181 539 494 544 267 654 418 945 312 499 977 093 12 × 2 = 1 + 0.000 000 000 002 046 363 078 989 088 535 308 837 890 624 999 954 186 24;
  • 17) 0.000 000 000 002 046 363 078 989 088 535 308 837 890 624 999 954 186 24 × 2 = 0 + 0.000 000 000 004 092 726 157 978 177 070 617 675 781 249 999 908 372 48;
  • 18) 0.000 000 000 004 092 726 157 978 177 070 617 675 781 249 999 908 372 48 × 2 = 0 + 0.000 000 000 008 185 452 315 956 354 141 235 351 562 499 999 816 744 96;
  • 19) 0.000 000 000 008 185 452 315 956 354 141 235 351 562 499 999 816 744 96 × 2 = 0 + 0.000 000 000 016 370 904 631 912 708 282 470 703 124 999 999 633 489 92;
  • 20) 0.000 000 000 016 370 904 631 912 708 282 470 703 124 999 999 633 489 92 × 2 = 0 + 0.000 000 000 032 741 809 263 825 416 564 941 406 249 999 999 266 979 84;
  • 21) 0.000 000 000 032 741 809 263 825 416 564 941 406 249 999 999 266 979 84 × 2 = 0 + 0.000 000 000 065 483 618 527 650 833 129 882 812 499 999 998 533 959 68;
  • 22) 0.000 000 000 065 483 618 527 650 833 129 882 812 499 999 998 533 959 68 × 2 = 0 + 0.000 000 000 130 967 237 055 301 666 259 765 624 999 999 997 067 919 36;
  • 23) 0.000 000 000 130 967 237 055 301 666 259 765 624 999 999 997 067 919 36 × 2 = 0 + 0.000 000 000 261 934 474 110 603 332 519 531 249 999 999 994 135 838 72;
  • 24) 0.000 000 000 261 934 474 110 603 332 519 531 249 999 999 994 135 838 72 × 2 = 0 + 0.000 000 000 523 868 948 221 206 665 039 062 499 999 999 988 271 677 44;
  • 25) 0.000 000 000 523 868 948 221 206 665 039 062 499 999 999 988 271 677 44 × 2 = 0 + 0.000 000 001 047 737 896 442 413 330 078 124 999 999 999 976 543 354 88;
  • 26) 0.000 000 001 047 737 896 442 413 330 078 124 999 999 999 976 543 354 88 × 2 = 0 + 0.000 000 002 095 475 792 884 826 660 156 249 999 999 999 953 086 709 76;
  • 27) 0.000 000 002 095 475 792 884 826 660 156 249 999 999 999 953 086 709 76 × 2 = 0 + 0.000 000 004 190 951 585 769 653 320 312 499 999 999 999 906 173 419 52;
  • 28) 0.000 000 004 190 951 585 769 653 320 312 499 999 999 999 906 173 419 52 × 2 = 0 + 0.000 000 008 381 903 171 539 306 640 624 999 999 999 999 812 346 839 04;
  • 29) 0.000 000 008 381 903 171 539 306 640 624 999 999 999 999 812 346 839 04 × 2 = 0 + 0.000 000 016 763 806 343 078 613 281 249 999 999 999 999 624 693 678 08;
  • 30) 0.000 000 016 763 806 343 078 613 281 249 999 999 999 999 624 693 678 08 × 2 = 0 + 0.000 000 033 527 612 686 157 226 562 499 999 999 999 999 249 387 356 16;
  • 31) 0.000 000 033 527 612 686 157 226 562 499 999 999 999 999 249 387 356 16 × 2 = 0 + 0.000 000 067 055 225 372 314 453 124 999 999 999 999 998 498 774 712 32;
  • 32) 0.000 000 067 055 225 372 314 453 124 999 999 999 999 998 498 774 712 32 × 2 = 0 + 0.000 000 134 110 450 744 628 906 249 999 999 999 999 996 997 549 424 64;
  • 33) 0.000 000 134 110 450 744 628 906 249 999 999 999 999 996 997 549 424 64 × 2 = 0 + 0.000 000 268 220 901 489 257 812 499 999 999 999 999 993 995 098 849 28;
  • 34) 0.000 000 268 220 901 489 257 812 499 999 999 999 999 993 995 098 849 28 × 2 = 0 + 0.000 000 536 441 802 978 515 624 999 999 999 999 999 987 990 197 698 56;
  • 35) 0.000 000 536 441 802 978 515 624 999 999 999 999 999 987 990 197 698 56 × 2 = 0 + 0.000 001 072 883 605 957 031 249 999 999 999 999 999 975 980 395 397 12;
  • 36) 0.000 001 072 883 605 957 031 249 999 999 999 999 999 975 980 395 397 12 × 2 = 0 + 0.000 002 145 767 211 914 062 499 999 999 999 999 999 951 960 790 794 24;
  • 37) 0.000 002 145 767 211 914 062 499 999 999 999 999 999 951 960 790 794 24 × 2 = 0 + 0.000 004 291 534 423 828 124 999 999 999 999 999 999 903 921 581 588 48;
  • 38) 0.000 004 291 534 423 828 124 999 999 999 999 999 999 903 921 581 588 48 × 2 = 0 + 0.000 008 583 068 847 656 249 999 999 999 999 999 999 807 843 163 176 96;
  • 39) 0.000 008 583 068 847 656 249 999 999 999 999 999 999 807 843 163 176 96 × 2 = 0 + 0.000 017 166 137 695 312 499 999 999 999 999 999 999 615 686 326 353 92;
  • 40) 0.000 017 166 137 695 312 499 999 999 999 999 999 999 615 686 326 353 92 × 2 = 0 + 0.000 034 332 275 390 624 999 999 999 999 999 999 999 231 372 652 707 84;
  • 41) 0.000 034 332 275 390 624 999 999 999 999 999 999 999 231 372 652 707 84 × 2 = 0 + 0.000 068 664 550 781 249 999 999 999 999 999 999 998 462 745 305 415 68;
  • 42) 0.000 068 664 550 781 249 999 999 999 999 999 999 998 462 745 305 415 68 × 2 = 0 + 0.000 137 329 101 562 499 999 999 999 999 999 999 996 925 490 610 831 36;
  • 43) 0.000 137 329 101 562 499 999 999 999 999 999 999 996 925 490 610 831 36 × 2 = 0 + 0.000 274 658 203 124 999 999 999 999 999 999 999 993 850 981 221 662 72;
  • 44) 0.000 274 658 203 124 999 999 999 999 999 999 999 993 850 981 221 662 72 × 2 = 0 + 0.000 549 316 406 249 999 999 999 999 999 999 999 987 701 962 443 325 44;
  • 45) 0.000 549 316 406 249 999 999 999 999 999 999 999 987 701 962 443 325 44 × 2 = 0 + 0.001 098 632 812 499 999 999 999 999 999 999 999 975 403 924 886 650 88;
  • 46) 0.001 098 632 812 499 999 999 999 999 999 999 999 975 403 924 886 650 88 × 2 = 0 + 0.002 197 265 624 999 999 999 999 999 999 999 999 950 807 849 773 301 76;
  • 47) 0.002 197 265 624 999 999 999 999 999 999 999 999 950 807 849 773 301 76 × 2 = 0 + 0.004 394 531 249 999 999 999 999 999 999 999 999 901 615 699 546 603 52;
  • 48) 0.004 394 531 249 999 999 999 999 999 999 999 999 901 615 699 546 603 52 × 2 = 0 + 0.008 789 062 499 999 999 999 999 999 999 999 999 803 231 399 093 207 04;
  • 49) 0.008 789 062 499 999 999 999 999 999 999 999 999 803 231 399 093 207 04 × 2 = 0 + 0.017 578 124 999 999 999 999 999 999 999 999 999 606 462 798 186 414 08;
  • 50) 0.017 578 124 999 999 999 999 999 999 999 999 999 606 462 798 186 414 08 × 2 = 0 + 0.035 156 249 999 999 999 999 999 999 999 999 999 212 925 596 372 828 16;
  • 51) 0.035 156 249 999 999 999 999 999 999 999 999 999 212 925 596 372 828 16 × 2 = 0 + 0.070 312 499 999 999 999 999 999 999 999 999 998 425 851 192 745 656 32;
  • 52) 0.070 312 499 999 999 999 999 999 999 999 999 998 425 851 192 745 656 32 × 2 = 0 + 0.140 624 999 999 999 999 999 999 999 999 999 996 851 702 385 491 312 64;
  • 53) 0.140 624 999 999 999 999 999 999 999 999 999 996 851 702 385 491 312 64 × 2 = 0 + 0.281 249 999 999 999 999 999 999 999 999 999 993 703 404 770 982 625 28;
  • 54) 0.281 249 999 999 999 999 999 999 999 999 999 993 703 404 770 982 625 28 × 2 = 0 + 0.562 499 999 999 999 999 999 999 999 999 999 987 406 809 541 965 250 56;
  • 55) 0.562 499 999 999 999 999 999 999 999 999 999 987 406 809 541 965 250 56 × 2 = 1 + 0.124 999 999 999 999 999 999 999 999 999 999 974 813 619 083 930 501 12;
  • 56) 0.124 999 999 999 999 999 999 999 999 999 999 974 813 619 083 930 501 12 × 2 = 0 + 0.249 999 999 999 999 999 999 999 999 999 999 949 627 238 167 861 002 24;
  • 57) 0.249 999 999 999 999 999 999 999 999 999 999 949 627 238 167 861 002 24 × 2 = 0 + 0.499 999 999 999 999 999 999 999 999 999 999 899 254 476 335 722 004 48;
  • 58) 0.499 999 999 999 999 999 999 999 999 999 999 899 254 476 335 722 004 48 × 2 = 0 + 0.999 999 999 999 999 999 999 999 999 999 999 798 508 952 671 444 008 96;

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).


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:


0.016 738 891 601 562 531 225 022 567 582 527 699 414 640 665 054 320 59(10) =

0.0000 0100 0100 1001 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 00(2)

6. Positive number before normalization:

0.016 738 891 601 562 531 225 022 567 582 527 699 414 640 665 054 320 59(10) =

0.0000 0100 0100 1001 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 00(2)

7. Normalize the binary representation of the number.

Shift the decimal mark 6 positions to the right, so that only one non zero digit remains to the left of it:


0.016 738 891 601 562 531 225 022 567 582 527 699 414 640 665 054 320 59(10) =

0.0000 0100 0100 1001 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 00(2) =

0.0000 0100 0100 1001 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 00(2) × 20 =

1.0001 0010 0100 0000 0000 0000 0000 0000 0000 0000 0000 0000 1000(2) × 2-6


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): -6

Mantissa (not normalized):
1.0001 0010 0100 0000 0000 0000 0000 0000 0000 0000 0000 0000 1000


9. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

-6 + 2(11-1) - 1 =

(-6 + 1 023)(10) =

1 017(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;
  • 1 017 ÷ 2 = 508 + 1;
  • 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) =

1017(10) =

011 1111 1001(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 0010 0100 0000 0000 0000 0000 0000 0000 0000 0000 0000 1000 =

0001 0010 0100 0000 0000 0000 0000 0000 0000 0000 0000 0000 1000


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 1001

Mantissa (52 bits) =
0001 0010 0100 0000 0000 0000 0000 0000 0000 0000 0000 0000 1000


Decimal number -0.016 738 891 601 562 531 225 022 567 582 527 699 414 640 665 054 320 59 converted to 64 bit double precision IEEE 754 binary floating point representation:

1 - 011 1111 1001 - 0001 0010 0100 0000 0000 0000 0000 0000 0000 0000 0000 0000 1000

» Convert decimal -0.016 738 891 601 562 531 225 022 567 582 527 699 414 640 665 054 321 27 to 64 bit double precision IEEE 754 binary floating point representation standard

» Calculations Performed by Our Visitors: Decimal Numbers Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard. Data organized on a Monthly Basis

» Month 10, 2026 [October]: Decimal Numbers Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard. Calculations performed during the month of: October - by our visitors

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