-2.211 829 052 383 358 300 119 548 661 699 653 280 265 920 329 838 991 165 158 6 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -2.211 829 052 383 358 300 119 548 661 699 653 280 265 920 329 838 991 165 158 6(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
-2.211 829 052 383 358 300 119 548 661 699 653 280 265 920 329 838 991 165 158 6(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:

|-2.211 829 052 383 358 300 119 548 661 699 653 280 265 920 329 838 991 165 158 6| = 2.211 829 052 383 358 300 119 548 661 699 653 280 265 920 329 838 991 165 158 6


2. First, convert to binary (in base 2) the integer part: 2.
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;
  • 2 ÷ 2 = 1 + 0;
  • 1 ÷ 2 = 0 + 1;

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.

2(10) =

10(2)


4. Convert to binary (base 2) the fractional part: 0.211 829 052 383 358 300 119 548 661 699 653 280 265 920 329 838 991 165 158 6.

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) 0.211 829 052 383 358 300 119 548 661 699 653 280 265 920 329 838 991 165 158 6 × 2 = 0 + 0.423 658 104 766 716 600 239 097 323 399 306 560 531 840 659 677 982 330 317 2;
  • 2) 0.423 658 104 766 716 600 239 097 323 399 306 560 531 840 659 677 982 330 317 2 × 2 = 0 + 0.847 316 209 533 433 200 478 194 646 798 613 121 063 681 319 355 964 660 634 4;
  • 3) 0.847 316 209 533 433 200 478 194 646 798 613 121 063 681 319 355 964 660 634 4 × 2 = 1 + 0.694 632 419 066 866 400 956 389 293 597 226 242 127 362 638 711 929 321 268 8;
  • 4) 0.694 632 419 066 866 400 956 389 293 597 226 242 127 362 638 711 929 321 268 8 × 2 = 1 + 0.389 264 838 133 732 801 912 778 587 194 452 484 254 725 277 423 858 642 537 6;
  • 5) 0.389 264 838 133 732 801 912 778 587 194 452 484 254 725 277 423 858 642 537 6 × 2 = 0 + 0.778 529 676 267 465 603 825 557 174 388 904 968 509 450 554 847 717 285 075 2;
  • 6) 0.778 529 676 267 465 603 825 557 174 388 904 968 509 450 554 847 717 285 075 2 × 2 = 1 + 0.557 059 352 534 931 207 651 114 348 777 809 937 018 901 109 695 434 570 150 4;
  • 7) 0.557 059 352 534 931 207 651 114 348 777 809 937 018 901 109 695 434 570 150 4 × 2 = 1 + 0.114 118 705 069 862 415 302 228 697 555 619 874 037 802 219 390 869 140 300 8;
  • 8) 0.114 118 705 069 862 415 302 228 697 555 619 874 037 802 219 390 869 140 300 8 × 2 = 0 + 0.228 237 410 139 724 830 604 457 395 111 239 748 075 604 438 781 738 280 601 6;
  • 9) 0.228 237 410 139 724 830 604 457 395 111 239 748 075 604 438 781 738 280 601 6 × 2 = 0 + 0.456 474 820 279 449 661 208 914 790 222 479 496 151 208 877 563 476 561 203 2;
  • 10) 0.456 474 820 279 449 661 208 914 790 222 479 496 151 208 877 563 476 561 203 2 × 2 = 0 + 0.912 949 640 558 899 322 417 829 580 444 958 992 302 417 755 126 953 122 406 4;
  • 11) 0.912 949 640 558 899 322 417 829 580 444 958 992 302 417 755 126 953 122 406 4 × 2 = 1 + 0.825 899 281 117 798 644 835 659 160 889 917 984 604 835 510 253 906 244 812 8;
  • 12) 0.825 899 281 117 798 644 835 659 160 889 917 984 604 835 510 253 906 244 812 8 × 2 = 1 + 0.651 798 562 235 597 289 671 318 321 779 835 969 209 671 020 507 812 489 625 6;
  • 13) 0.651 798 562 235 597 289 671 318 321 779 835 969 209 671 020 507 812 489 625 6 × 2 = 1 + 0.303 597 124 471 194 579 342 636 643 559 671 938 419 342 041 015 624 979 251 2;
  • 14) 0.303 597 124 471 194 579 342 636 643 559 671 938 419 342 041 015 624 979 251 2 × 2 = 0 + 0.607 194 248 942 389 158 685 273 287 119 343 876 838 684 082 031 249 958 502 4;
  • 15) 0.607 194 248 942 389 158 685 273 287 119 343 876 838 684 082 031 249 958 502 4 × 2 = 1 + 0.214 388 497 884 778 317 370 546 574 238 687 753 677 368 164 062 499 917 004 8;
  • 16) 0.214 388 497 884 778 317 370 546 574 238 687 753 677 368 164 062 499 917 004 8 × 2 = 0 + 0.428 776 995 769 556 634 741 093 148 477 375 507 354 736 328 124 999 834 009 6;
  • 17) 0.428 776 995 769 556 634 741 093 148 477 375 507 354 736 328 124 999 834 009 6 × 2 = 0 + 0.857 553 991 539 113 269 482 186 296 954 751 014 709 472 656 249 999 668 019 2;
  • 18) 0.857 553 991 539 113 269 482 186 296 954 751 014 709 472 656 249 999 668 019 2 × 2 = 1 + 0.715 107 983 078 226 538 964 372 593 909 502 029 418 945 312 499 999 336 038 4;
  • 19) 0.715 107 983 078 226 538 964 372 593 909 502 029 418 945 312 499 999 336 038 4 × 2 = 1 + 0.430 215 966 156 453 077 928 745 187 819 004 058 837 890 624 999 998 672 076 8;
  • 20) 0.430 215 966 156 453 077 928 745 187 819 004 058 837 890 624 999 998 672 076 8 × 2 = 0 + 0.860 431 932 312 906 155 857 490 375 638 008 117 675 781 249 999 997 344 153 6;
  • 21) 0.860 431 932 312 906 155 857 490 375 638 008 117 675 781 249 999 997 344 153 6 × 2 = 1 + 0.720 863 864 625 812 311 714 980 751 276 016 235 351 562 499 999 994 688 307 2;
  • 22) 0.720 863 864 625 812 311 714 980 751 276 016 235 351 562 499 999 994 688 307 2 × 2 = 1 + 0.441 727 729 251 624 623 429 961 502 552 032 470 703 124 999 999 989 376 614 4;
  • 23) 0.441 727 729 251 624 623 429 961 502 552 032 470 703 124 999 999 989 376 614 4 × 2 = 0 + 0.883 455 458 503 249 246 859 923 005 104 064 941 406 249 999 999 978 753 228 8;
  • 24) 0.883 455 458 503 249 246 859 923 005 104 064 941 406 249 999 999 978 753 228 8 × 2 = 1 + 0.766 910 917 006 498 493 719 846 010 208 129 882 812 499 999 999 957 506 457 6;
  • 25) 0.766 910 917 006 498 493 719 846 010 208 129 882 812 499 999 999 957 506 457 6 × 2 = 1 + 0.533 821 834 012 996 987 439 692 020 416 259 765 624 999 999 999 915 012 915 2;
  • 26) 0.533 821 834 012 996 987 439 692 020 416 259 765 624 999 999 999 915 012 915 2 × 2 = 1 + 0.067 643 668 025 993 974 879 384 040 832 519 531 249 999 999 999 830 025 830 4;
  • 27) 0.067 643 668 025 993 974 879 384 040 832 519 531 249 999 999 999 830 025 830 4 × 2 = 0 + 0.135 287 336 051 987 949 758 768 081 665 039 062 499 999 999 999 660 051 660 8;
  • 28) 0.135 287 336 051 987 949 758 768 081 665 039 062 499 999 999 999 660 051 660 8 × 2 = 0 + 0.270 574 672 103 975 899 517 536 163 330 078 124 999 999 999 999 320 103 321 6;
  • 29) 0.270 574 672 103 975 899 517 536 163 330 078 124 999 999 999 999 320 103 321 6 × 2 = 0 + 0.541 149 344 207 951 799 035 072 326 660 156 249 999 999 999 998 640 206 643 2;
  • 30) 0.541 149 344 207 951 799 035 072 326 660 156 249 999 999 999 998 640 206 643 2 × 2 = 1 + 0.082 298 688 415 903 598 070 144 653 320 312 499 999 999 999 997 280 413 286 4;
  • 31) 0.082 298 688 415 903 598 070 144 653 320 312 499 999 999 999 997 280 413 286 4 × 2 = 0 + 0.164 597 376 831 807 196 140 289 306 640 624 999 999 999 999 994 560 826 572 8;
  • 32) 0.164 597 376 831 807 196 140 289 306 640 624 999 999 999 999 994 560 826 572 8 × 2 = 0 + 0.329 194 753 663 614 392 280 578 613 281 249 999 999 999 999 989 121 653 145 6;
  • 33) 0.329 194 753 663 614 392 280 578 613 281 249 999 999 999 999 989 121 653 145 6 × 2 = 0 + 0.658 389 507 327 228 784 561 157 226 562 499 999 999 999 999 978 243 306 291 2;
  • 34) 0.658 389 507 327 228 784 561 157 226 562 499 999 999 999 999 978 243 306 291 2 × 2 = 1 + 0.316 779 014 654 457 569 122 314 453 124 999 999 999 999 999 956 486 612 582 4;
  • 35) 0.316 779 014 654 457 569 122 314 453 124 999 999 999 999 999 956 486 612 582 4 × 2 = 0 + 0.633 558 029 308 915 138 244 628 906 249 999 999 999 999 999 912 973 225 164 8;
  • 36) 0.633 558 029 308 915 138 244 628 906 249 999 999 999 999 999 912 973 225 164 8 × 2 = 1 + 0.267 116 058 617 830 276 489 257 812 499 999 999 999 999 999 825 946 450 329 6;
  • 37) 0.267 116 058 617 830 276 489 257 812 499 999 999 999 999 999 825 946 450 329 6 × 2 = 0 + 0.534 232 117 235 660 552 978 515 624 999 999 999 999 999 999 651 892 900 659 2;
  • 38) 0.534 232 117 235 660 552 978 515 624 999 999 999 999 999 999 651 892 900 659 2 × 2 = 1 + 0.068 464 234 471 321 105 957 031 249 999 999 999 999 999 999 303 785 801 318 4;
  • 39) 0.068 464 234 471 321 105 957 031 249 999 999 999 999 999 999 303 785 801 318 4 × 2 = 0 + 0.136 928 468 942 642 211 914 062 499 999 999 999 999 999 998 607 571 602 636 8;
  • 40) 0.136 928 468 942 642 211 914 062 499 999 999 999 999 999 998 607 571 602 636 8 × 2 = 0 + 0.273 856 937 885 284 423 828 124 999 999 999 999 999 999 997 215 143 205 273 6;
  • 41) 0.273 856 937 885 284 423 828 124 999 999 999 999 999 999 997 215 143 205 273 6 × 2 = 0 + 0.547 713 875 770 568 847 656 249 999 999 999 999 999 999 994 430 286 410 547 2;
  • 42) 0.547 713 875 770 568 847 656 249 999 999 999 999 999 999 994 430 286 410 547 2 × 2 = 1 + 0.095 427 751 541 137 695 312 499 999 999 999 999 999 999 988 860 572 821 094 4;
  • 43) 0.095 427 751 541 137 695 312 499 999 999 999 999 999 999 988 860 572 821 094 4 × 2 = 0 + 0.190 855 503 082 275 390 624 999 999 999 999 999 999 999 977 721 145 642 188 8;
  • 44) 0.190 855 503 082 275 390 624 999 999 999 999 999 999 999 977 721 145 642 188 8 × 2 = 0 + 0.381 711 006 164 550 781 249 999 999 999 999 999 999 999 955 442 291 284 377 6;
  • 45) 0.381 711 006 164 550 781 249 999 999 999 999 999 999 999 955 442 291 284 377 6 × 2 = 0 + 0.763 422 012 329 101 562 499 999 999 999 999 999 999 999 910 884 582 568 755 2;
  • 46) 0.763 422 012 329 101 562 499 999 999 999 999 999 999 999 910 884 582 568 755 2 × 2 = 1 + 0.526 844 024 658 203 124 999 999 999 999 999 999 999 999 821 769 165 137 510 4;
  • 47) 0.526 844 024 658 203 124 999 999 999 999 999 999 999 999 821 769 165 137 510 4 × 2 = 1 + 0.053 688 049 316 406 249 999 999 999 999 999 999 999 999 643 538 330 275 020 8;
  • 48) 0.053 688 049 316 406 249 999 999 999 999 999 999 999 999 643 538 330 275 020 8 × 2 = 0 + 0.107 376 098 632 812 499 999 999 999 999 999 999 999 999 287 076 660 550 041 6;
  • 49) 0.107 376 098 632 812 499 999 999 999 999 999 999 999 999 287 076 660 550 041 6 × 2 = 0 + 0.214 752 197 265 624 999 999 999 999 999 999 999 999 998 574 153 321 100 083 2;
  • 50) 0.214 752 197 265 624 999 999 999 999 999 999 999 999 998 574 153 321 100 083 2 × 2 = 0 + 0.429 504 394 531 249 999 999 999 999 999 999 999 999 997 148 306 642 200 166 4;
  • 51) 0.429 504 394 531 249 999 999 999 999 999 999 999 999 997 148 306 642 200 166 4 × 2 = 0 + 0.859 008 789 062 499 999 999 999 999 999 999 999 999 994 296 613 284 400 332 8;
  • 52) 0.859 008 789 062 499 999 999 999 999 999 999 999 999 994 296 613 284 400 332 8 × 2 = 1 + 0.718 017 578 124 999 999 999 999 999 999 999 999 999 988 593 226 568 800 665 6;
  • 53) 0.718 017 578 124 999 999 999 999 999 999 999 999 999 988 593 226 568 800 665 6 × 2 = 1 + 0.436 035 156 249 999 999 999 999 999 999 999 999 999 977 186 453 137 601 331 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).


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:


0.211 829 052 383 358 300 119 548 661 699 653 280 265 920 329 838 991 165 158 6(10) =

0.0011 0110 0011 1010 0110 1101 1100 0100 0101 0100 0100 0110 0001 1(2)

6. Positive number before normalization:

2.211 829 052 383 358 300 119 548 661 699 653 280 265 920 329 838 991 165 158 6(10) =

10.0011 0110 0011 1010 0110 1101 1100 0100 0101 0100 0100 0110 0001 1(2)

7. Normalize the binary representation of the number.

Shift the decimal mark 1 positions to the left, so that only one non zero digit remains to the left of it:


2.211 829 052 383 358 300 119 548 661 699 653 280 265 920 329 838 991 165 158 6(10) =

10.0011 0110 0011 1010 0110 1101 1100 0100 0101 0100 0100 0110 0001 1(2) =

10.0011 0110 0011 1010 0110 1101 1100 0100 0101 0100 0100 0110 0001 1(2) × 20 =

1.0001 1011 0001 1101 0011 0110 1110 0010 0010 1010 0010 0011 0000 11(2) × 21


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): 1

Mantissa (not normalized):
1.0001 1011 0001 1101 0011 0110 1110 0010 0010 1010 0010 0011 0000 11


9. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

1 + 2(11-1) - 1 =

(1 + 1 023)(10) =

1 024(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;
  • 1 024 ÷ 2 = 512 + 0;
  • 512 ÷ 2 = 256 + 0;
  • 256 ÷ 2 = 128 + 0;
  • 128 ÷ 2 = 64 + 0;
  • 64 ÷ 2 = 32 + 0;
  • 32 ÷ 2 = 16 + 0;
  • 16 ÷ 2 = 8 + 0;
  • 8 ÷ 2 = 4 + 0;
  • 4 ÷ 2 = 2 + 0;
  • 2 ÷ 2 = 1 + 0;
  • 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) =

1024(10) =

100 0000 0000(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, by removing the excess bits, from the right (if any of the excess bits is set on 1, we are losing precision...).


Mantissa (normalized) =

1. 0001 1011 0001 1101 0011 0110 1110 0010 0010 1010 0010 0011 0000 11 =

0001 1011 0001 1101 0011 0110 1110 0010 0010 1010 0010 0011 0000


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) =
100 0000 0000

Mantissa (52 bits) =
0001 1011 0001 1101 0011 0110 1110 0010 0010 1010 0010 0011 0000


Decimal number -2.211 829 052 383 358 300 119 548 661 699 653 280 265 920 329 838 991 165 158 6 converted to 64 bit double precision IEEE 754 binary floating point representation:

1 - 100 0000 0000 - 0001 1011 0001 1101 0011 0110 1110 0010 0010 1010 0010 0011 0000

Share

» Convert decimal -2.211 829 052 383 358 300 119 548 661 699 653 280 265 920 329 838 991 165 167 1 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 06, 2026 [June]: Decimal Numbers Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard. Calculations performed during the month of: June - by our visitors

» Improve your social skills: Reputation: Bridge Between Company's Actions and Market Value. NotebookLM YouTube Video of 6.55 minutes

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