64bit IEEE 754: Decimal ↗ Double Precision Floating Point Binary: -6.745 579 999 999 999 465 387 645 614 100 620 150 566 18 Convert the Number to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard, From a Base Ten Decimal System Number

Number -6.745 579 999 999 999 465 387 645 614 100 620 150 566 18(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:

|-6.745 579 999 999 999 465 387 645 614 100 620 150 566 18| = 6.745 579 999 999 999 465 387 645 614 100 620 150 566 18

2. First, convert to binary (in base 2) the integer part: 6.
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;
  • 6 ÷ 2 = 3 + 0;
  • 3 ÷ 2 = 1 + 1;
  • 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.


6(10) =


110(2)


4. Convert to binary (base 2) the fractional part: 0.745 579 999 999 999 465 387 645 614 100 620 150 566 18.

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) 0.745 579 999 999 999 465 387 645 614 100 620 150 566 18 × 2 = 1 + 0.491 159 999 999 998 930 775 291 228 201 240 301 132 36;
  • 2) 0.491 159 999 999 998 930 775 291 228 201 240 301 132 36 × 2 = 0 + 0.982 319 999 999 997 861 550 582 456 402 480 602 264 72;
  • 3) 0.982 319 999 999 997 861 550 582 456 402 480 602 264 72 × 2 = 1 + 0.964 639 999 999 995 723 101 164 912 804 961 204 529 44;
  • 4) 0.964 639 999 999 995 723 101 164 912 804 961 204 529 44 × 2 = 1 + 0.929 279 999 999 991 446 202 329 825 609 922 409 058 88;
  • 5) 0.929 279 999 999 991 446 202 329 825 609 922 409 058 88 × 2 = 1 + 0.858 559 999 999 982 892 404 659 651 219 844 818 117 76;
  • 6) 0.858 559 999 999 982 892 404 659 651 219 844 818 117 76 × 2 = 1 + 0.717 119 999 999 965 784 809 319 302 439 689 636 235 52;
  • 7) 0.717 119 999 999 965 784 809 319 302 439 689 636 235 52 × 2 = 1 + 0.434 239 999 999 931 569 618 638 604 879 379 272 471 04;
  • 8) 0.434 239 999 999 931 569 618 638 604 879 379 272 471 04 × 2 = 0 + 0.868 479 999 999 863 139 237 277 209 758 758 544 942 08;
  • 9) 0.868 479 999 999 863 139 237 277 209 758 758 544 942 08 × 2 = 1 + 0.736 959 999 999 726 278 474 554 419 517 517 089 884 16;
  • 10) 0.736 959 999 999 726 278 474 554 419 517 517 089 884 16 × 2 = 1 + 0.473 919 999 999 452 556 949 108 839 035 034 179 768 32;
  • 11) 0.473 919 999 999 452 556 949 108 839 035 034 179 768 32 × 2 = 0 + 0.947 839 999 998 905 113 898 217 678 070 068 359 536 64;
  • 12) 0.947 839 999 998 905 113 898 217 678 070 068 359 536 64 × 2 = 1 + 0.895 679 999 997 810 227 796 435 356 140 136 719 073 28;
  • 13) 0.895 679 999 997 810 227 796 435 356 140 136 719 073 28 × 2 = 1 + 0.791 359 999 995 620 455 592 870 712 280 273 438 146 56;
  • 14) 0.791 359 999 995 620 455 592 870 712 280 273 438 146 56 × 2 = 1 + 0.582 719 999 991 240 911 185 741 424 560 546 876 293 12;
  • 15) 0.582 719 999 991 240 911 185 741 424 560 546 876 293 12 × 2 = 1 + 0.165 439 999 982 481 822 371 482 849 121 093 752 586 24;
  • 16) 0.165 439 999 982 481 822 371 482 849 121 093 752 586 24 × 2 = 0 + 0.330 879 999 964 963 644 742 965 698 242 187 505 172 48;
  • 17) 0.330 879 999 964 963 644 742 965 698 242 187 505 172 48 × 2 = 0 + 0.661 759 999 929 927 289 485 931 396 484 375 010 344 96;
  • 18) 0.661 759 999 929 927 289 485 931 396 484 375 010 344 96 × 2 = 1 + 0.323 519 999 859 854 578 971 862 792 968 750 020 689 92;
  • 19) 0.323 519 999 859 854 578 971 862 792 968 750 020 689 92 × 2 = 0 + 0.647 039 999 719 709 157 943 725 585 937 500 041 379 84;
  • 20) 0.647 039 999 719 709 157 943 725 585 937 500 041 379 84 × 2 = 1 + 0.294 079 999 439 418 315 887 451 171 875 000 082 759 68;
  • 21) 0.294 079 999 439 418 315 887 451 171 875 000 082 759 68 × 2 = 0 + 0.588 159 998 878 836 631 774 902 343 750 000 165 519 36;
  • 22) 0.588 159 998 878 836 631 774 902 343 750 000 165 519 36 × 2 = 1 + 0.176 319 997 757 673 263 549 804 687 500 000 331 038 72;
  • 23) 0.176 319 997 757 673 263 549 804 687 500 000 331 038 72 × 2 = 0 + 0.352 639 995 515 346 527 099 609 375 000 000 662 077 44;
  • 24) 0.352 639 995 515 346 527 099 609 375 000 000 662 077 44 × 2 = 0 + 0.705 279 991 030 693 054 199 218 750 000 001 324 154 88;
  • 25) 0.705 279 991 030 693 054 199 218 750 000 001 324 154 88 × 2 = 1 + 0.410 559 982 061 386 108 398 437 500 000 002 648 309 76;
  • 26) 0.410 559 982 061 386 108 398 437 500 000 002 648 309 76 × 2 = 0 + 0.821 119 964 122 772 216 796 875 000 000 005 296 619 52;
  • 27) 0.821 119 964 122 772 216 796 875 000 000 005 296 619 52 × 2 = 1 + 0.642 239 928 245 544 433 593 750 000 000 010 593 239 04;
  • 28) 0.642 239 928 245 544 433 593 750 000 000 010 593 239 04 × 2 = 1 + 0.284 479 856 491 088 867 187 500 000 000 021 186 478 08;
  • 29) 0.284 479 856 491 088 867 187 500 000 000 021 186 478 08 × 2 = 0 + 0.568 959 712 982 177 734 375 000 000 000 042 372 956 16;
  • 30) 0.568 959 712 982 177 734 375 000 000 000 042 372 956 16 × 2 = 1 + 0.137 919 425 964 355 468 750 000 000 000 084 745 912 32;
  • 31) 0.137 919 425 964 355 468 750 000 000 000 084 745 912 32 × 2 = 0 + 0.275 838 851 928 710 937 500 000 000 000 169 491 824 64;
  • 32) 0.275 838 851 928 710 937 500 000 000 000 169 491 824 64 × 2 = 0 + 0.551 677 703 857 421 875 000 000 000 000 338 983 649 28;
  • 33) 0.551 677 703 857 421 875 000 000 000 000 338 983 649 28 × 2 = 1 + 0.103 355 407 714 843 750 000 000 000 000 677 967 298 56;
  • 34) 0.103 355 407 714 843 750 000 000 000 000 677 967 298 56 × 2 = 0 + 0.206 710 815 429 687 500 000 000 000 001 355 934 597 12;
  • 35) 0.206 710 815 429 687 500 000 000 000 001 355 934 597 12 × 2 = 0 + 0.413 421 630 859 375 000 000 000 000 002 711 869 194 24;
  • 36) 0.413 421 630 859 375 000 000 000 000 002 711 869 194 24 × 2 = 0 + 0.826 843 261 718 750 000 000 000 000 005 423 738 388 48;
  • 37) 0.826 843 261 718 750 000 000 000 000 005 423 738 388 48 × 2 = 1 + 0.653 686 523 437 500 000 000 000 000 010 847 476 776 96;
  • 38) 0.653 686 523 437 500 000 000 000 000 010 847 476 776 96 × 2 = 1 + 0.307 373 046 875 000 000 000 000 000 021 694 953 553 92;
  • 39) 0.307 373 046 875 000 000 000 000 000 021 694 953 553 92 × 2 = 0 + 0.614 746 093 750 000 000 000 000 000 043 389 907 107 84;
  • 40) 0.614 746 093 750 000 000 000 000 000 043 389 907 107 84 × 2 = 1 + 0.229 492 187 500 000 000 000 000 000 086 779 814 215 68;
  • 41) 0.229 492 187 500 000 000 000 000 000 086 779 814 215 68 × 2 = 0 + 0.458 984 375 000 000 000 000 000 000 173 559 628 431 36;
  • 42) 0.458 984 375 000 000 000 000 000 000 173 559 628 431 36 × 2 = 0 + 0.917 968 750 000 000 000 000 000 000 347 119 256 862 72;
  • 43) 0.917 968 750 000 000 000 000 000 000 347 119 256 862 72 × 2 = 1 + 0.835 937 500 000 000 000 000 000 000 694 238 513 725 44;
  • 44) 0.835 937 500 000 000 000 000 000 000 694 238 513 725 44 × 2 = 1 + 0.671 875 000 000 000 000 000 000 001 388 477 027 450 88;
  • 45) 0.671 875 000 000 000 000 000 000 001 388 477 027 450 88 × 2 = 1 + 0.343 750 000 000 000 000 000 000 002 776 954 054 901 76;
  • 46) 0.343 750 000 000 000 000 000 000 002 776 954 054 901 76 × 2 = 0 + 0.687 500 000 000 000 000 000 000 005 553 908 109 803 52;
  • 47) 0.687 500 000 000 000 000 000 000 005 553 908 109 803 52 × 2 = 1 + 0.375 000 000 000 000 000 000 000 011 107 816 219 607 04;
  • 48) 0.375 000 000 000 000 000 000 000 011 107 816 219 607 04 × 2 = 0 + 0.750 000 000 000 000 000 000 000 022 215 632 439 214 08;
  • 49) 0.750 000 000 000 000 000 000 000 022 215 632 439 214 08 × 2 = 1 + 0.500 000 000 000 000 000 000 000 044 431 264 878 428 16;
  • 50) 0.500 000 000 000 000 000 000 000 044 431 264 878 428 16 × 2 = 1 + 0.000 000 000 000 000 000 000 000 088 862 529 756 856 32;
  • 51) 0.000 000 000 000 000 000 000 000 088 862 529 756 856 32 × 2 = 0 + 0.000 000 000 000 000 000 000 000 177 725 059 513 712 64;
  • 52) 0.000 000 000 000 000 000 000 000 177 725 059 513 712 64 × 2 = 0 + 0.000 000 000 000 000 000 000 000 355 450 119 027 425 28;
  • 53) 0.000 000 000 000 000 000 000 000 355 450 119 027 425 28 × 2 = 0 + 0.000 000 000 000 000 000 000 000 710 900 238 054 850 56;

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:


0.745 579 999 999 999 465 387 645 614 100 620 150 566 18(10) =


0.1011 1110 1101 1110 0101 0100 1011 0100 1000 1101 0011 1010 1100 0(2)


6. Positive number before normalization:

6.745 579 999 999 999 465 387 645 614 100 620 150 566 18(10) =


110.1011 1110 1101 1110 0101 0100 1011 0100 1000 1101 0011 1010 1100 0(2)

7. Normalize the binary representation of the number.

Shift the decimal mark 2 positions to the left, so that only one non zero digit remains to the left of it:


6.745 579 999 999 999 465 387 645 614 100 620 150 566 18(10) =


110.1011 1110 1101 1110 0101 0100 1011 0100 1000 1101 0011 1010 1100 0(2) =


110.1011 1110 1101 1110 0101 0100 1011 0100 1000 1101 0011 1010 1100 0(2) × 20 =


1.1010 1111 1011 0111 1001 0101 0010 1101 0010 0011 0100 1110 1011 000(2) × 22


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): 2


Mantissa (not normalized):
1.1010 1111 1011 0111 1001 0101 0010 1101 0010 0011 0100 1110 1011 000


9. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =


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


2 + 2(11-1) - 1 =


(2 + 1 023)(10) =


1 025(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;
  • 1 025 ÷ 2 = 512 + 1;
  • 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) =


1025(10) =


100 0000 0001(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. 1010 1111 1011 0111 1001 0101 0010 1101 0010 0011 0100 1110 1011 000 =


1010 1111 1011 0111 1001 0101 0010 1101 0010 0011 0100 1110 1011


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 0001


Mantissa (52 bits) =
1010 1111 1011 0111 1001 0101 0010 1101 0010 0011 0100 1110 1011


The base ten decimal number -6.745 579 999 999 999 465 387 645 614 100 620 150 566 18 converted and written in 64 bit double precision IEEE 754 binary floating point representation:
1 - 100 0000 0001 - 1010 1111 1011 0111 1001 0101 0010 1101 0010 0011 0100 1110 1011

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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