-0.048 632 677 916 771 838 155 796 105 2 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.048 632 677 916 771 838 155 796 105 2(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
-0.048 632 677 916 771 838 155 796 105 2(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:

|-0.048 632 677 916 771 838 155 796 105 2| = 0.048 632 677 916 771 838 155 796 105 2


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: 0.048 632 677 916 771 838 155 796 105 2.

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) 0.048 632 677 916 771 838 155 796 105 2 × 2 = 0 + 0.097 265 355 833 543 676 311 592 210 4;
  • 2) 0.097 265 355 833 543 676 311 592 210 4 × 2 = 0 + 0.194 530 711 667 087 352 623 184 420 8;
  • 3) 0.194 530 711 667 087 352 623 184 420 8 × 2 = 0 + 0.389 061 423 334 174 705 246 368 841 6;
  • 4) 0.389 061 423 334 174 705 246 368 841 6 × 2 = 0 + 0.778 122 846 668 349 410 492 737 683 2;
  • 5) 0.778 122 846 668 349 410 492 737 683 2 × 2 = 1 + 0.556 245 693 336 698 820 985 475 366 4;
  • 6) 0.556 245 693 336 698 820 985 475 366 4 × 2 = 1 + 0.112 491 386 673 397 641 970 950 732 8;
  • 7) 0.112 491 386 673 397 641 970 950 732 8 × 2 = 0 + 0.224 982 773 346 795 283 941 901 465 6;
  • 8) 0.224 982 773 346 795 283 941 901 465 6 × 2 = 0 + 0.449 965 546 693 590 567 883 802 931 2;
  • 9) 0.449 965 546 693 590 567 883 802 931 2 × 2 = 0 + 0.899 931 093 387 181 135 767 605 862 4;
  • 10) 0.899 931 093 387 181 135 767 605 862 4 × 2 = 1 + 0.799 862 186 774 362 271 535 211 724 8;
  • 11) 0.799 862 186 774 362 271 535 211 724 8 × 2 = 1 + 0.599 724 373 548 724 543 070 423 449 6;
  • 12) 0.599 724 373 548 724 543 070 423 449 6 × 2 = 1 + 0.199 448 747 097 449 086 140 846 899 2;
  • 13) 0.199 448 747 097 449 086 140 846 899 2 × 2 = 0 + 0.398 897 494 194 898 172 281 693 798 4;
  • 14) 0.398 897 494 194 898 172 281 693 798 4 × 2 = 0 + 0.797 794 988 389 796 344 563 387 596 8;
  • 15) 0.797 794 988 389 796 344 563 387 596 8 × 2 = 1 + 0.595 589 976 779 592 689 126 775 193 6;
  • 16) 0.595 589 976 779 592 689 126 775 193 6 × 2 = 1 + 0.191 179 953 559 185 378 253 550 387 2;
  • 17) 0.191 179 953 559 185 378 253 550 387 2 × 2 = 0 + 0.382 359 907 118 370 756 507 100 774 4;
  • 18) 0.382 359 907 118 370 756 507 100 774 4 × 2 = 0 + 0.764 719 814 236 741 513 014 201 548 8;
  • 19) 0.764 719 814 236 741 513 014 201 548 8 × 2 = 1 + 0.529 439 628 473 483 026 028 403 097 6;
  • 20) 0.529 439 628 473 483 026 028 403 097 6 × 2 = 1 + 0.058 879 256 946 966 052 056 806 195 2;
  • 21) 0.058 879 256 946 966 052 056 806 195 2 × 2 = 0 + 0.117 758 513 893 932 104 113 612 390 4;
  • 22) 0.117 758 513 893 932 104 113 612 390 4 × 2 = 0 + 0.235 517 027 787 864 208 227 224 780 8;
  • 23) 0.235 517 027 787 864 208 227 224 780 8 × 2 = 0 + 0.471 034 055 575 728 416 454 449 561 6;
  • 24) 0.471 034 055 575 728 416 454 449 561 6 × 2 = 0 + 0.942 068 111 151 456 832 908 899 123 2;
  • 25) 0.942 068 111 151 456 832 908 899 123 2 × 2 = 1 + 0.884 136 222 302 913 665 817 798 246 4;
  • 26) 0.884 136 222 302 913 665 817 798 246 4 × 2 = 1 + 0.768 272 444 605 827 331 635 596 492 8;
  • 27) 0.768 272 444 605 827 331 635 596 492 8 × 2 = 1 + 0.536 544 889 211 654 663 271 192 985 6;
  • 28) 0.536 544 889 211 654 663 271 192 985 6 × 2 = 1 + 0.073 089 778 423 309 326 542 385 971 2;
  • 29) 0.073 089 778 423 309 326 542 385 971 2 × 2 = 0 + 0.146 179 556 846 618 653 084 771 942 4;
  • 30) 0.146 179 556 846 618 653 084 771 942 4 × 2 = 0 + 0.292 359 113 693 237 306 169 543 884 8;
  • 31) 0.292 359 113 693 237 306 169 543 884 8 × 2 = 0 + 0.584 718 227 386 474 612 339 087 769 6;
  • 32) 0.584 718 227 386 474 612 339 087 769 6 × 2 = 1 + 0.169 436 454 772 949 224 678 175 539 2;
  • 33) 0.169 436 454 772 949 224 678 175 539 2 × 2 = 0 + 0.338 872 909 545 898 449 356 351 078 4;
  • 34) 0.338 872 909 545 898 449 356 351 078 4 × 2 = 0 + 0.677 745 819 091 796 898 712 702 156 8;
  • 35) 0.677 745 819 091 796 898 712 702 156 8 × 2 = 1 + 0.355 491 638 183 593 797 425 404 313 6;
  • 36) 0.355 491 638 183 593 797 425 404 313 6 × 2 = 0 + 0.710 983 276 367 187 594 850 808 627 2;
  • 37) 0.710 983 276 367 187 594 850 808 627 2 × 2 = 1 + 0.421 966 552 734 375 189 701 617 254 4;
  • 38) 0.421 966 552 734 375 189 701 617 254 4 × 2 = 0 + 0.843 933 105 468 750 379 403 234 508 8;
  • 39) 0.843 933 105 468 750 379 403 234 508 8 × 2 = 1 + 0.687 866 210 937 500 758 806 469 017 6;
  • 40) 0.687 866 210 937 500 758 806 469 017 6 × 2 = 1 + 0.375 732 421 875 001 517 612 938 035 2;
  • 41) 0.375 732 421 875 001 517 612 938 035 2 × 2 = 0 + 0.751 464 843 750 003 035 225 876 070 4;
  • 42) 0.751 464 843 750 003 035 225 876 070 4 × 2 = 1 + 0.502 929 687 500 006 070 451 752 140 8;
  • 43) 0.502 929 687 500 006 070 451 752 140 8 × 2 = 1 + 0.005 859 375 000 012 140 903 504 281 6;
  • 44) 0.005 859 375 000 012 140 903 504 281 6 × 2 = 0 + 0.011 718 750 000 024 281 807 008 563 2;
  • 45) 0.011 718 750 000 024 281 807 008 563 2 × 2 = 0 + 0.023 437 500 000 048 563 614 017 126 4;
  • 46) 0.023 437 500 000 048 563 614 017 126 4 × 2 = 0 + 0.046 875 000 000 097 127 228 034 252 8;
  • 47) 0.046 875 000 000 097 127 228 034 252 8 × 2 = 0 + 0.093 750 000 000 194 254 456 068 505 6;
  • 48) 0.093 750 000 000 194 254 456 068 505 6 × 2 = 0 + 0.187 500 000 000 388 508 912 137 011 2;
  • 49) 0.187 500 000 000 388 508 912 137 011 2 × 2 = 0 + 0.375 000 000 000 777 017 824 274 022 4;
  • 50) 0.375 000 000 000 777 017 824 274 022 4 × 2 = 0 + 0.750 000 000 001 554 035 648 548 044 8;
  • 51) 0.750 000 000 001 554 035 648 548 044 8 × 2 = 1 + 0.500 000 000 003 108 071 297 096 089 6;
  • 52) 0.500 000 000 003 108 071 297 096 089 6 × 2 = 1 + 0.000 000 000 006 216 142 594 192 179 2;
  • 53) 0.000 000 000 006 216 142 594 192 179 2 × 2 = 0 + 0.000 000 000 012 432 285 188 384 358 4;
  • 54) 0.000 000 000 012 432 285 188 384 358 4 × 2 = 0 + 0.000 000 000 024 864 570 376 768 716 8;
  • 55) 0.000 000 000 024 864 570 376 768 716 8 × 2 = 0 + 0.000 000 000 049 729 140 753 537 433 6;
  • 56) 0.000 000 000 049 729 140 753 537 433 6 × 2 = 0 + 0.000 000 000 099 458 281 507 074 867 2;
  • 57) 0.000 000 000 099 458 281 507 074 867 2 × 2 = 0 + 0.000 000 000 198 916 563 014 149 734 4;

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:


0.048 632 677 916 771 838 155 796 105 2(10) =

0.0000 1100 0111 0011 0011 0000 1111 0001 0010 1011 0110 0000 0011 0000 0(2)

6. Positive number before normalization:

0.048 632 677 916 771 838 155 796 105 2(10) =

0.0000 1100 0111 0011 0011 0000 1111 0001 0010 1011 0110 0000 0011 0000 0(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:


0.048 632 677 916 771 838 155 796 105 2(10) =

0.0000 1100 0111 0011 0011 0000 1111 0001 0010 1011 0110 0000 0011 0000 0(2) =

0.0000 1100 0111 0011 0011 0000 1111 0001 0010 1011 0110 0000 0011 0000 0(2) × 20 =

1.1000 1110 0110 0110 0001 1110 0010 0101 0110 1100 0000 0110 0000(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 0110 0000


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 0110 0000 =

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

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


Decimal number -0.048 632 677 916 771 838 155 796 105 2 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 0110 0000

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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