64bit IEEE 754: Decimal ↗ Double Precision Floating Point Binary: 43 627 849 036 223 746 239 757 362 364 923 236 237 Convert the Number to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard, From a Base Ten Decimal System Number

Number 43 627 849 036 223 746 239 757 362 364 923 236 237(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. 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;
  • 43 627 849 036 223 746 239 757 362 364 923 236 237 ÷ 2 = 21 813 924 518 111 873 119 878 681 182 461 618 118 + 1;
  • 21 813 924 518 111 873 119 878 681 182 461 618 118 ÷ 2 = 10 906 962 259 055 936 559 939 340 591 230 809 059 + 0;
  • 10 906 962 259 055 936 559 939 340 591 230 809 059 ÷ 2 = 5 453 481 129 527 968 279 969 670 295 615 404 529 + 1;
  • 5 453 481 129 527 968 279 969 670 295 615 404 529 ÷ 2 = 2 726 740 564 763 984 139 984 835 147 807 702 264 + 1;
  • 2 726 740 564 763 984 139 984 835 147 807 702 264 ÷ 2 = 1 363 370 282 381 992 069 992 417 573 903 851 132 + 0;
  • 1 363 370 282 381 992 069 992 417 573 903 851 132 ÷ 2 = 681 685 141 190 996 034 996 208 786 951 925 566 + 0;
  • 681 685 141 190 996 034 996 208 786 951 925 566 ÷ 2 = 340 842 570 595 498 017 498 104 393 475 962 783 + 0;
  • 340 842 570 595 498 017 498 104 393 475 962 783 ÷ 2 = 170 421 285 297 749 008 749 052 196 737 981 391 + 1;
  • 170 421 285 297 749 008 749 052 196 737 981 391 ÷ 2 = 85 210 642 648 874 504 374 526 098 368 990 695 + 1;
  • 85 210 642 648 874 504 374 526 098 368 990 695 ÷ 2 = 42 605 321 324 437 252 187 263 049 184 495 347 + 1;
  • 42 605 321 324 437 252 187 263 049 184 495 347 ÷ 2 = 21 302 660 662 218 626 093 631 524 592 247 673 + 1;
  • 21 302 660 662 218 626 093 631 524 592 247 673 ÷ 2 = 10 651 330 331 109 313 046 815 762 296 123 836 + 1;
  • 10 651 330 331 109 313 046 815 762 296 123 836 ÷ 2 = 5 325 665 165 554 656 523 407 881 148 061 918 + 0;
  • 5 325 665 165 554 656 523 407 881 148 061 918 ÷ 2 = 2 662 832 582 777 328 261 703 940 574 030 959 + 0;
  • 2 662 832 582 777 328 261 703 940 574 030 959 ÷ 2 = 1 331 416 291 388 664 130 851 970 287 015 479 + 1;
  • 1 331 416 291 388 664 130 851 970 287 015 479 ÷ 2 = 665 708 145 694 332 065 425 985 143 507 739 + 1;
  • 665 708 145 694 332 065 425 985 143 507 739 ÷ 2 = 332 854 072 847 166 032 712 992 571 753 869 + 1;
  • 332 854 072 847 166 032 712 992 571 753 869 ÷ 2 = 166 427 036 423 583 016 356 496 285 876 934 + 1;
  • 166 427 036 423 583 016 356 496 285 876 934 ÷ 2 = 83 213 518 211 791 508 178 248 142 938 467 + 0;
  • 83 213 518 211 791 508 178 248 142 938 467 ÷ 2 = 41 606 759 105 895 754 089 124 071 469 233 + 1;
  • 41 606 759 105 895 754 089 124 071 469 233 ÷ 2 = 20 803 379 552 947 877 044 562 035 734 616 + 1;
  • 20 803 379 552 947 877 044 562 035 734 616 ÷ 2 = 10 401 689 776 473 938 522 281 017 867 308 + 0;
  • 10 401 689 776 473 938 522 281 017 867 308 ÷ 2 = 5 200 844 888 236 969 261 140 508 933 654 + 0;
  • 5 200 844 888 236 969 261 140 508 933 654 ÷ 2 = 2 600 422 444 118 484 630 570 254 466 827 + 0;
  • 2 600 422 444 118 484 630 570 254 466 827 ÷ 2 = 1 300 211 222 059 242 315 285 127 233 413 + 1;
  • 1 300 211 222 059 242 315 285 127 233 413 ÷ 2 = 650 105 611 029 621 157 642 563 616 706 + 1;
  • 650 105 611 029 621 157 642 563 616 706 ÷ 2 = 325 052 805 514 810 578 821 281 808 353 + 0;
  • 325 052 805 514 810 578 821 281 808 353 ÷ 2 = 162 526 402 757 405 289 410 640 904 176 + 1;
  • 162 526 402 757 405 289 410 640 904 176 ÷ 2 = 81 263 201 378 702 644 705 320 452 088 + 0;
  • 81 263 201 378 702 644 705 320 452 088 ÷ 2 = 40 631 600 689 351 322 352 660 226 044 + 0;
  • 40 631 600 689 351 322 352 660 226 044 ÷ 2 = 20 315 800 344 675 661 176 330 113 022 + 0;
  • 20 315 800 344 675 661 176 330 113 022 ÷ 2 = 10 157 900 172 337 830 588 165 056 511 + 0;
  • 10 157 900 172 337 830 588 165 056 511 ÷ 2 = 5 078 950 086 168 915 294 082 528 255 + 1;
  • 5 078 950 086 168 915 294 082 528 255 ÷ 2 = 2 539 475 043 084 457 647 041 264 127 + 1;
  • 2 539 475 043 084 457 647 041 264 127 ÷ 2 = 1 269 737 521 542 228 823 520 632 063 + 1;
  • 1 269 737 521 542 228 823 520 632 063 ÷ 2 = 634 868 760 771 114 411 760 316 031 + 1;
  • 634 868 760 771 114 411 760 316 031 ÷ 2 = 317 434 380 385 557 205 880 158 015 + 1;
  • 317 434 380 385 557 205 880 158 015 ÷ 2 = 158 717 190 192 778 602 940 079 007 + 1;
  • 158 717 190 192 778 602 940 079 007 ÷ 2 = 79 358 595 096 389 301 470 039 503 + 1;
  • 79 358 595 096 389 301 470 039 503 ÷ 2 = 39 679 297 548 194 650 735 019 751 + 1;
  • 39 679 297 548 194 650 735 019 751 ÷ 2 = 19 839 648 774 097 325 367 509 875 + 1;
  • 19 839 648 774 097 325 367 509 875 ÷ 2 = 9 919 824 387 048 662 683 754 937 + 1;
  • 9 919 824 387 048 662 683 754 937 ÷ 2 = 4 959 912 193 524 331 341 877 468 + 1;
  • 4 959 912 193 524 331 341 877 468 ÷ 2 = 2 479 956 096 762 165 670 938 734 + 0;
  • 2 479 956 096 762 165 670 938 734 ÷ 2 = 1 239 978 048 381 082 835 469 367 + 0;
  • 1 239 978 048 381 082 835 469 367 ÷ 2 = 619 989 024 190 541 417 734 683 + 1;
  • 619 989 024 190 541 417 734 683 ÷ 2 = 309 994 512 095 270 708 867 341 + 1;
  • 309 994 512 095 270 708 867 341 ÷ 2 = 154 997 256 047 635 354 433 670 + 1;
  • 154 997 256 047 635 354 433 670 ÷ 2 = 77 498 628 023 817 677 216 835 + 0;
  • 77 498 628 023 817 677 216 835 ÷ 2 = 38 749 314 011 908 838 608 417 + 1;
  • 38 749 314 011 908 838 608 417 ÷ 2 = 19 374 657 005 954 419 304 208 + 1;
  • 19 374 657 005 954 419 304 208 ÷ 2 = 9 687 328 502 977 209 652 104 + 0;
  • 9 687 328 502 977 209 652 104 ÷ 2 = 4 843 664 251 488 604 826 052 + 0;
  • 4 843 664 251 488 604 826 052 ÷ 2 = 2 421 832 125 744 302 413 026 + 0;
  • 2 421 832 125 744 302 413 026 ÷ 2 = 1 210 916 062 872 151 206 513 + 0;
  • 1 210 916 062 872 151 206 513 ÷ 2 = 605 458 031 436 075 603 256 + 1;
  • 605 458 031 436 075 603 256 ÷ 2 = 302 729 015 718 037 801 628 + 0;
  • 302 729 015 718 037 801 628 ÷ 2 = 151 364 507 859 018 900 814 + 0;
  • 151 364 507 859 018 900 814 ÷ 2 = 75 682 253 929 509 450 407 + 0;
  • 75 682 253 929 509 450 407 ÷ 2 = 37 841 126 964 754 725 203 + 1;
  • 37 841 126 964 754 725 203 ÷ 2 = 18 920 563 482 377 362 601 + 1;
  • 18 920 563 482 377 362 601 ÷ 2 = 9 460 281 741 188 681 300 + 1;
  • 9 460 281 741 188 681 300 ÷ 2 = 4 730 140 870 594 340 650 + 0;
  • 4 730 140 870 594 340 650 ÷ 2 = 2 365 070 435 297 170 325 + 0;
  • 2 365 070 435 297 170 325 ÷ 2 = 1 182 535 217 648 585 162 + 1;
  • 1 182 535 217 648 585 162 ÷ 2 = 591 267 608 824 292 581 + 0;
  • 591 267 608 824 292 581 ÷ 2 = 295 633 804 412 146 290 + 1;
  • 295 633 804 412 146 290 ÷ 2 = 147 816 902 206 073 145 + 0;
  • 147 816 902 206 073 145 ÷ 2 = 73 908 451 103 036 572 + 1;
  • 73 908 451 103 036 572 ÷ 2 = 36 954 225 551 518 286 + 0;
  • 36 954 225 551 518 286 ÷ 2 = 18 477 112 775 759 143 + 0;
  • 18 477 112 775 759 143 ÷ 2 = 9 238 556 387 879 571 + 1;
  • 9 238 556 387 879 571 ÷ 2 = 4 619 278 193 939 785 + 1;
  • 4 619 278 193 939 785 ÷ 2 = 2 309 639 096 969 892 + 1;
  • 2 309 639 096 969 892 ÷ 2 = 1 154 819 548 484 946 + 0;
  • 1 154 819 548 484 946 ÷ 2 = 577 409 774 242 473 + 0;
  • 577 409 774 242 473 ÷ 2 = 288 704 887 121 236 + 1;
  • 288 704 887 121 236 ÷ 2 = 144 352 443 560 618 + 0;
  • 144 352 443 560 618 ÷ 2 = 72 176 221 780 309 + 0;
  • 72 176 221 780 309 ÷ 2 = 36 088 110 890 154 + 1;
  • 36 088 110 890 154 ÷ 2 = 18 044 055 445 077 + 0;
  • 18 044 055 445 077 ÷ 2 = 9 022 027 722 538 + 1;
  • 9 022 027 722 538 ÷ 2 = 4 511 013 861 269 + 0;
  • 4 511 013 861 269 ÷ 2 = 2 255 506 930 634 + 1;
  • 2 255 506 930 634 ÷ 2 = 1 127 753 465 317 + 0;
  • 1 127 753 465 317 ÷ 2 = 563 876 732 658 + 1;
  • 563 876 732 658 ÷ 2 = 281 938 366 329 + 0;
  • 281 938 366 329 ÷ 2 = 140 969 183 164 + 1;
  • 140 969 183 164 ÷ 2 = 70 484 591 582 + 0;
  • 70 484 591 582 ÷ 2 = 35 242 295 791 + 0;
  • 35 242 295 791 ÷ 2 = 17 621 147 895 + 1;
  • 17 621 147 895 ÷ 2 = 8 810 573 947 + 1;
  • 8 810 573 947 ÷ 2 = 4 405 286 973 + 1;
  • 4 405 286 973 ÷ 2 = 2 202 643 486 + 1;
  • 2 202 643 486 ÷ 2 = 1 101 321 743 + 0;
  • 1 101 321 743 ÷ 2 = 550 660 871 + 1;
  • 550 660 871 ÷ 2 = 275 330 435 + 1;
  • 275 330 435 ÷ 2 = 137 665 217 + 1;
  • 137 665 217 ÷ 2 = 68 832 608 + 1;
  • 68 832 608 ÷ 2 = 34 416 304 + 0;
  • 34 416 304 ÷ 2 = 17 208 152 + 0;
  • 17 208 152 ÷ 2 = 8 604 076 + 0;
  • 8 604 076 ÷ 2 = 4 302 038 + 0;
  • 4 302 038 ÷ 2 = 2 151 019 + 0;
  • 2 151 019 ÷ 2 = 1 075 509 + 1;
  • 1 075 509 ÷ 2 = 537 754 + 1;
  • 537 754 ÷ 2 = 268 877 + 0;
  • 268 877 ÷ 2 = 134 438 + 1;
  • 134 438 ÷ 2 = 67 219 + 0;
  • 67 219 ÷ 2 = 33 609 + 1;
  • 33 609 ÷ 2 = 16 804 + 1;
  • 16 804 ÷ 2 = 8 402 + 0;
  • 8 402 ÷ 2 = 4 201 + 0;
  • 4 201 ÷ 2 = 2 100 + 1;
  • 2 100 ÷ 2 = 1 050 + 0;
  • 1 050 ÷ 2 = 525 + 0;
  • 525 ÷ 2 = 262 + 1;
  • 262 ÷ 2 = 131 + 0;
  • 131 ÷ 2 = 65 + 1;
  • 65 ÷ 2 = 32 + 1;
  • 32 ÷ 2 = 16 + 0;
  • 16 ÷ 2 = 8 + 0;
  • 8 ÷ 2 = 4 + 0;
  • 4 ÷ 2 = 2 + 0;
  • 2 ÷ 2 = 1 + 0;
  • 1 ÷ 2 = 0 + 1;

2. Construct the base 2 representation of the positive number.

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


43 627 849 036 223 746 239 757 362 364 923 236 237(10) =


10 0000 1101 0010 0110 1011 0000 0111 1011 1100 1010 1010 1001 0011 1001 0101 0011 1000 1000 0110 1110 0111 1111 1111 0000 1011 0001 1011 1100 1111 1000 1101(2)


3. Normalize the binary representation of the number.

Shift the decimal mark 125 positions to the left, so that only one non zero digit remains to the left of it:


43 627 849 036 223 746 239 757 362 364 923 236 237(10) =


10 0000 1101 0010 0110 1011 0000 0111 1011 1100 1010 1010 1001 0011 1001 0101 0011 1000 1000 0110 1110 0111 1111 1111 0000 1011 0001 1011 1100 1111 1000 1101(2) =


10 0000 1101 0010 0110 1011 0000 0111 1011 1100 1010 1010 1001 0011 1001 0101 0011 1000 1000 0110 1110 0111 1111 1111 0000 1011 0001 1011 1100 1111 1000 1101(2) × 20 =


1.0000 0110 1001 0011 0101 1000 0011 1101 1110 0101 0101 0100 1001 1100 1010 1001 1100 0100 0011 0111 0011 1111 1111 1000 0101 1000 1101 1110 0111 1100 0110 1(2) × 2125


4. 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): 125


Mantissa (not normalized):
1.0000 0110 1001 0011 0101 1000 0011 1101 1110 0101 0101 0100 1001 1100 1010 1001 1100 0100 0011 0111 0011 1111 1111 1000 0101 1000 1101 1110 0111 1100 0110 1


5. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =


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


125 + 2(11-1) - 1 =


(125 + 1 023)(10) =


1 148(10)


6. 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;
  • 1 148 ÷ 2 = 574 + 0;
  • 574 ÷ 2 = 287 + 0;
  • 287 ÷ 2 = 143 + 1;
  • 143 ÷ 2 = 71 + 1;
  • 71 ÷ 2 = 35 + 1;
  • 35 ÷ 2 = 17 + 1;
  • 17 ÷ 2 = 8 + 1;
  • 8 ÷ 2 = 4 + 0;
  • 4 ÷ 2 = 2 + 0;
  • 2 ÷ 2 = 1 + 0;
  • 1 ÷ 2 = 0 + 1;

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

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


Exponent (adjusted) =


1148(10) =


100 0111 1100(2)


8. 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. 0000 0110 1001 0011 0101 1000 0011 1101 1110 0101 0101 0100 1001 1 1001 0101 0011 1000 1000 0110 1110 0111 1111 1111 0000 1011 0001 1011 1100 1111 1000 1101 =


0000 0110 1001 0011 0101 1000 0011 1101 1110 0101 0101 0100 1001


9. 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) =
100 0111 1100


Mantissa (52 bits) =
0000 0110 1001 0011 0101 1000 0011 1101 1110 0101 0101 0100 1001


The base ten decimal number 43 627 849 036 223 746 239 757 362 364 923 236 237 converted and written in 64 bit double precision IEEE 754 binary floating point representation:
0 - 100 0111 1100 - 0000 0110 1001 0011 0101 1000 0011 1101 1110 0101 0101 0100 1001

The latest decimal numbers converted from base ten to 64 bit double precision IEEE 754 floating point binary standard representation

Number 43 627 849 036 223 746 239 757 362 364 923 236 237 converted from decimal system (written in base ten) to 64 bit double precision IEEE 754 binary floating point representation standard Mar 28 08:31 UTC (GMT)
Number 1 252 593 685 converted from decimal system (written in base ten) to 64 bit double precision IEEE 754 binary floating point representation standard Mar 28 08:31 UTC (GMT)
Number 19 649 converted from decimal system (written in base ten) to 64 bit double precision IEEE 754 binary floating point representation standard Mar 28 08:31 UTC (GMT)
Number 2.010 205 converted from decimal system (written in base ten) to 64 bit double precision IEEE 754 binary floating point representation standard Mar 28 08:31 UTC (GMT)
Number 1 280 converted from decimal system (written in base ten) to 64 bit double precision IEEE 754 binary floating point representation standard Mar 28 08:31 UTC (GMT)
Number 4 232 converted from decimal system (written in base ten) to 64 bit double precision IEEE 754 binary floating point representation standard Mar 28 08:31 UTC (GMT)
Number 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 94 converted from decimal system (written in base ten) to 64 bit double precision IEEE 754 binary floating point representation standard Mar 28 08:31 UTC (GMT)
All base ten decimal numbers converted to 64 bit double precision IEEE 754 binary floating point