-0.048 632 677 916 771 838 155 44 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.048 632 677 916 771 838 155 44(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
-0.048 632 677 916 771 838 155 44(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:

|-0.048 632 677 916 771 838 155 44| = 0.048 632 677 916 771 838 155 44


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: 0.048 632 677 916 771 838 155 44.

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) 0.048 632 677 916 771 838 155 44 × 2 = 0 + 0.097 265 355 833 543 676 310 88;
  • 2) 0.097 265 355 833 543 676 310 88 × 2 = 0 + 0.194 530 711 667 087 352 621 76;
  • 3) 0.194 530 711 667 087 352 621 76 × 2 = 0 + 0.389 061 423 334 174 705 243 52;
  • 4) 0.389 061 423 334 174 705 243 52 × 2 = 0 + 0.778 122 846 668 349 410 487 04;
  • 5) 0.778 122 846 668 349 410 487 04 × 2 = 1 + 0.556 245 693 336 698 820 974 08;
  • 6) 0.556 245 693 336 698 820 974 08 × 2 = 1 + 0.112 491 386 673 397 641 948 16;
  • 7) 0.112 491 386 673 397 641 948 16 × 2 = 0 + 0.224 982 773 346 795 283 896 32;
  • 8) 0.224 982 773 346 795 283 896 32 × 2 = 0 + 0.449 965 546 693 590 567 792 64;
  • 9) 0.449 965 546 693 590 567 792 64 × 2 = 0 + 0.899 931 093 387 181 135 585 28;
  • 10) 0.899 931 093 387 181 135 585 28 × 2 = 1 + 0.799 862 186 774 362 271 170 56;
  • 11) 0.799 862 186 774 362 271 170 56 × 2 = 1 + 0.599 724 373 548 724 542 341 12;
  • 12) 0.599 724 373 548 724 542 341 12 × 2 = 1 + 0.199 448 747 097 449 084 682 24;
  • 13) 0.199 448 747 097 449 084 682 24 × 2 = 0 + 0.398 897 494 194 898 169 364 48;
  • 14) 0.398 897 494 194 898 169 364 48 × 2 = 0 + 0.797 794 988 389 796 338 728 96;
  • 15) 0.797 794 988 389 796 338 728 96 × 2 = 1 + 0.595 589 976 779 592 677 457 92;
  • 16) 0.595 589 976 779 592 677 457 92 × 2 = 1 + 0.191 179 953 559 185 354 915 84;
  • 17) 0.191 179 953 559 185 354 915 84 × 2 = 0 + 0.382 359 907 118 370 709 831 68;
  • 18) 0.382 359 907 118 370 709 831 68 × 2 = 0 + 0.764 719 814 236 741 419 663 36;
  • 19) 0.764 719 814 236 741 419 663 36 × 2 = 1 + 0.529 439 628 473 482 839 326 72;
  • 20) 0.529 439 628 473 482 839 326 72 × 2 = 1 + 0.058 879 256 946 965 678 653 44;
  • 21) 0.058 879 256 946 965 678 653 44 × 2 = 0 + 0.117 758 513 893 931 357 306 88;
  • 22) 0.117 758 513 893 931 357 306 88 × 2 = 0 + 0.235 517 027 787 862 714 613 76;
  • 23) 0.235 517 027 787 862 714 613 76 × 2 = 0 + 0.471 034 055 575 725 429 227 52;
  • 24) 0.471 034 055 575 725 429 227 52 × 2 = 0 + 0.942 068 111 151 450 858 455 04;
  • 25) 0.942 068 111 151 450 858 455 04 × 2 = 1 + 0.884 136 222 302 901 716 910 08;
  • 26) 0.884 136 222 302 901 716 910 08 × 2 = 1 + 0.768 272 444 605 803 433 820 16;
  • 27) 0.768 272 444 605 803 433 820 16 × 2 = 1 + 0.536 544 889 211 606 867 640 32;
  • 28) 0.536 544 889 211 606 867 640 32 × 2 = 1 + 0.073 089 778 423 213 735 280 64;
  • 29) 0.073 089 778 423 213 735 280 64 × 2 = 0 + 0.146 179 556 846 427 470 561 28;
  • 30) 0.146 179 556 846 427 470 561 28 × 2 = 0 + 0.292 359 113 692 854 941 122 56;
  • 31) 0.292 359 113 692 854 941 122 56 × 2 = 0 + 0.584 718 227 385 709 882 245 12;
  • 32) 0.584 718 227 385 709 882 245 12 × 2 = 1 + 0.169 436 454 771 419 764 490 24;
  • 33) 0.169 436 454 771 419 764 490 24 × 2 = 0 + 0.338 872 909 542 839 528 980 48;
  • 34) 0.338 872 909 542 839 528 980 48 × 2 = 0 + 0.677 745 819 085 679 057 960 96;
  • 35) 0.677 745 819 085 679 057 960 96 × 2 = 1 + 0.355 491 638 171 358 115 921 92;
  • 36) 0.355 491 638 171 358 115 921 92 × 2 = 0 + 0.710 983 276 342 716 231 843 84;
  • 37) 0.710 983 276 342 716 231 843 84 × 2 = 1 + 0.421 966 552 685 432 463 687 68;
  • 38) 0.421 966 552 685 432 463 687 68 × 2 = 0 + 0.843 933 105 370 864 927 375 36;
  • 39) 0.843 933 105 370 864 927 375 36 × 2 = 1 + 0.687 866 210 741 729 854 750 72;
  • 40) 0.687 866 210 741 729 854 750 72 × 2 = 1 + 0.375 732 421 483 459 709 501 44;
  • 41) 0.375 732 421 483 459 709 501 44 × 2 = 0 + 0.751 464 842 966 919 419 002 88;
  • 42) 0.751 464 842 966 919 419 002 88 × 2 = 1 + 0.502 929 685 933 838 838 005 76;
  • 43) 0.502 929 685 933 838 838 005 76 × 2 = 1 + 0.005 859 371 867 677 676 011 52;
  • 44) 0.005 859 371 867 677 676 011 52 × 2 = 0 + 0.011 718 743 735 355 352 023 04;
  • 45) 0.011 718 743 735 355 352 023 04 × 2 = 0 + 0.023 437 487 470 710 704 046 08;
  • 46) 0.023 437 487 470 710 704 046 08 × 2 = 0 + 0.046 874 974 941 421 408 092 16;
  • 47) 0.046 874 974 941 421 408 092 16 × 2 = 0 + 0.093 749 949 882 842 816 184 32;
  • 48) 0.093 749 949 882 842 816 184 32 × 2 = 0 + 0.187 499 899 765 685 632 368 64;
  • 49) 0.187 499 899 765 685 632 368 64 × 2 = 0 + 0.374 999 799 531 371 264 737 28;
  • 50) 0.374 999 799 531 371 264 737 28 × 2 = 0 + 0.749 999 599 062 742 529 474 56;
  • 51) 0.749 999 599 062 742 529 474 56 × 2 = 1 + 0.499 999 198 125 485 058 949 12;
  • 52) 0.499 999 198 125 485 058 949 12 × 2 = 0 + 0.999 998 396 250 970 117 898 24;
  • 53) 0.999 998 396 250 970 117 898 24 × 2 = 1 + 0.999 996 792 501 940 235 796 48;
  • 54) 0.999 996 792 501 940 235 796 48 × 2 = 1 + 0.999 993 585 003 880 471 592 96;
  • 55) 0.999 993 585 003 880 471 592 96 × 2 = 1 + 0.999 987 170 007 760 943 185 92;
  • 56) 0.999 987 170 007 760 943 185 92 × 2 = 1 + 0.999 974 340 015 521 886 371 84;
  • 57) 0.999 974 340 015 521 886 371 84 × 2 = 1 + 0.999 948 680 031 043 772 743 68;

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:


0.048 632 677 916 771 838 155 44(10) =

0.0000 1100 0111 0011 0011 0000 1111 0001 0010 1011 0110 0000 0010 1111 1(2)

6. Positive number before normalization:

0.048 632 677 916 771 838 155 44(10) =

0.0000 1100 0111 0011 0011 0000 1111 0001 0010 1011 0110 0000 0010 1111 1(2)

7. Normalize the binary representation of the number.

Shift the decimal mark 5 positions to the right, so that only one non zero digit remains to the left of it:


0.048 632 677 916 771 838 155 44(10) =

0.0000 1100 0111 0011 0011 0000 1111 0001 0010 1011 0110 0000 0010 1111 1(2) =

0.0000 1100 0111 0011 0011 0000 1111 0001 0010 1011 0110 0000 0010 1111 1(2) × 20 =

1.1000 1110 0110 0110 0001 1110 0010 0101 0110 1100 0000 0101 1111(2) × 2-5


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): -5

Mantissa (not normalized):
1.1000 1110 0110 0110 0001 1110 0010 0101 0110 1100 0000 0101 1111


9. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

-5 + 2(11-1) - 1 =

(-5 + 1 023)(10) =

1 018(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;
  • 1 018 ÷ 2 = 509 + 0;
  • 509 ÷ 2 = 254 + 1;
  • 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) =

1018(10) =

011 1111 1010(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. 1000 1110 0110 0110 0001 1110 0010 0101 0110 1100 0000 0101 1111 =

1000 1110 0110 0110 0001 1110 0010 0101 0110 1100 0000 0101 1111


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 1010

Mantissa (52 bits) =
1000 1110 0110 0110 0001 1110 0010 0101 0110 1100 0000 0101 1111


Decimal number -0.048 632 677 916 771 838 155 44 converted to 64 bit double precision IEEE 754 binary floating point representation:

1 - 011 1111 1010 - 1000 1110 0110 0110 0001 1110 0010 0101 0110 1100 0000 0101 1111

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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