-0.016 738 891 601 562 496 530 553 97 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.016 738 891 601 562 496 530 553 97(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
-0.016 738 891 601 562 496 530 553 97(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:

|-0.016 738 891 601 562 496 530 553 97| = 0.016 738 891 601 562 496 530 553 97


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: 0.016 738 891 601 562 496 530 553 97.

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) 0.016 738 891 601 562 496 530 553 97 × 2 = 0 + 0.033 477 783 203 124 993 061 107 94;
  • 2) 0.033 477 783 203 124 993 061 107 94 × 2 = 0 + 0.066 955 566 406 249 986 122 215 88;
  • 3) 0.066 955 566 406 249 986 122 215 88 × 2 = 0 + 0.133 911 132 812 499 972 244 431 76;
  • 4) 0.133 911 132 812 499 972 244 431 76 × 2 = 0 + 0.267 822 265 624 999 944 488 863 52;
  • 5) 0.267 822 265 624 999 944 488 863 52 × 2 = 0 + 0.535 644 531 249 999 888 977 727 04;
  • 6) 0.535 644 531 249 999 888 977 727 04 × 2 = 1 + 0.071 289 062 499 999 777 955 454 08;
  • 7) 0.071 289 062 499 999 777 955 454 08 × 2 = 0 + 0.142 578 124 999 999 555 910 908 16;
  • 8) 0.142 578 124 999 999 555 910 908 16 × 2 = 0 + 0.285 156 249 999 999 111 821 816 32;
  • 9) 0.285 156 249 999 999 111 821 816 32 × 2 = 0 + 0.570 312 499 999 998 223 643 632 64;
  • 10) 0.570 312 499 999 998 223 643 632 64 × 2 = 1 + 0.140 624 999 999 996 447 287 265 28;
  • 11) 0.140 624 999 999 996 447 287 265 28 × 2 = 0 + 0.281 249 999 999 992 894 574 530 56;
  • 12) 0.281 249 999 999 992 894 574 530 56 × 2 = 0 + 0.562 499 999 999 985 789 149 061 12;
  • 13) 0.562 499 999 999 985 789 149 061 12 × 2 = 1 + 0.124 999 999 999 971 578 298 122 24;
  • 14) 0.124 999 999 999 971 578 298 122 24 × 2 = 0 + 0.249 999 999 999 943 156 596 244 48;
  • 15) 0.249 999 999 999 943 156 596 244 48 × 2 = 0 + 0.499 999 999 999 886 313 192 488 96;
  • 16) 0.499 999 999 999 886 313 192 488 96 × 2 = 0 + 0.999 999 999 999 772 626 384 977 92;
  • 17) 0.999 999 999 999 772 626 384 977 92 × 2 = 1 + 0.999 999 999 999 545 252 769 955 84;
  • 18) 0.999 999 999 999 545 252 769 955 84 × 2 = 1 + 0.999 999 999 999 090 505 539 911 68;
  • 19) 0.999 999 999 999 090 505 539 911 68 × 2 = 1 + 0.999 999 999 998 181 011 079 823 36;
  • 20) 0.999 999 999 998 181 011 079 823 36 × 2 = 1 + 0.999 999 999 996 362 022 159 646 72;
  • 21) 0.999 999 999 996 362 022 159 646 72 × 2 = 1 + 0.999 999 999 992 724 044 319 293 44;
  • 22) 0.999 999 999 992 724 044 319 293 44 × 2 = 1 + 0.999 999 999 985 448 088 638 586 88;
  • 23) 0.999 999 999 985 448 088 638 586 88 × 2 = 1 + 0.999 999 999 970 896 177 277 173 76;
  • 24) 0.999 999 999 970 896 177 277 173 76 × 2 = 1 + 0.999 999 999 941 792 354 554 347 52;
  • 25) 0.999 999 999 941 792 354 554 347 52 × 2 = 1 + 0.999 999 999 883 584 709 108 695 04;
  • 26) 0.999 999 999 883 584 709 108 695 04 × 2 = 1 + 0.999 999 999 767 169 418 217 390 08;
  • 27) 0.999 999 999 767 169 418 217 390 08 × 2 = 1 + 0.999 999 999 534 338 836 434 780 16;
  • 28) 0.999 999 999 534 338 836 434 780 16 × 2 = 1 + 0.999 999 999 068 677 672 869 560 32;
  • 29) 0.999 999 999 068 677 672 869 560 32 × 2 = 1 + 0.999 999 998 137 355 345 739 120 64;
  • 30) 0.999 999 998 137 355 345 739 120 64 × 2 = 1 + 0.999 999 996 274 710 691 478 241 28;
  • 31) 0.999 999 996 274 710 691 478 241 28 × 2 = 1 + 0.999 999 992 549 421 382 956 482 56;
  • 32) 0.999 999 992 549 421 382 956 482 56 × 2 = 1 + 0.999 999 985 098 842 765 912 965 12;
  • 33) 0.999 999 985 098 842 765 912 965 12 × 2 = 1 + 0.999 999 970 197 685 531 825 930 24;
  • 34) 0.999 999 970 197 685 531 825 930 24 × 2 = 1 + 0.999 999 940 395 371 063 651 860 48;
  • 35) 0.999 999 940 395 371 063 651 860 48 × 2 = 1 + 0.999 999 880 790 742 127 303 720 96;
  • 36) 0.999 999 880 790 742 127 303 720 96 × 2 = 1 + 0.999 999 761 581 484 254 607 441 92;
  • 37) 0.999 999 761 581 484 254 607 441 92 × 2 = 1 + 0.999 999 523 162 968 509 214 883 84;
  • 38) 0.999 999 523 162 968 509 214 883 84 × 2 = 1 + 0.999 999 046 325 937 018 429 767 68;
  • 39) 0.999 999 046 325 937 018 429 767 68 × 2 = 1 + 0.999 998 092 651 874 036 859 535 36;
  • 40) 0.999 998 092 651 874 036 859 535 36 × 2 = 1 + 0.999 996 185 303 748 073 719 070 72;
  • 41) 0.999 996 185 303 748 073 719 070 72 × 2 = 1 + 0.999 992 370 607 496 147 438 141 44;
  • 42) 0.999 992 370 607 496 147 438 141 44 × 2 = 1 + 0.999 984 741 214 992 294 876 282 88;
  • 43) 0.999 984 741 214 992 294 876 282 88 × 2 = 1 + 0.999 969 482 429 984 589 752 565 76;
  • 44) 0.999 969 482 429 984 589 752 565 76 × 2 = 1 + 0.999 938 964 859 969 179 505 131 52;
  • 45) 0.999 938 964 859 969 179 505 131 52 × 2 = 1 + 0.999 877 929 719 938 359 010 263 04;
  • 46) 0.999 877 929 719 938 359 010 263 04 × 2 = 1 + 0.999 755 859 439 876 718 020 526 08;
  • 47) 0.999 755 859 439 876 718 020 526 08 × 2 = 1 + 0.999 511 718 879 753 436 041 052 16;
  • 48) 0.999 511 718 879 753 436 041 052 16 × 2 = 1 + 0.999 023 437 759 506 872 082 104 32;
  • 49) 0.999 023 437 759 506 872 082 104 32 × 2 = 1 + 0.998 046 875 519 013 744 164 208 64;
  • 50) 0.998 046 875 519 013 744 164 208 64 × 2 = 1 + 0.996 093 751 038 027 488 328 417 28;
  • 51) 0.996 093 751 038 027 488 328 417 28 × 2 = 1 + 0.992 187 502 076 054 976 656 834 56;
  • 52) 0.992 187 502 076 054 976 656 834 56 × 2 = 1 + 0.984 375 004 152 109 953 313 669 12;
  • 53) 0.984 375 004 152 109 953 313 669 12 × 2 = 1 + 0.968 750 008 304 219 906 627 338 24;
  • 54) 0.968 750 008 304 219 906 627 338 24 × 2 = 1 + 0.937 500 016 608 439 813 254 676 48;
  • 55) 0.937 500 016 608 439 813 254 676 48 × 2 = 1 + 0.875 000 033 216 879 626 509 352 96;
  • 56) 0.875 000 033 216 879 626 509 352 96 × 2 = 1 + 0.750 000 066 433 759 253 018 705 92;
  • 57) 0.750 000 066 433 759 253 018 705 92 × 2 = 1 + 0.500 000 132 867 518 506 037 411 84;
  • 58) 0.500 000 132 867 518 506 037 411 84 × 2 = 1 + 0.000 000 265 735 037 012 074 823 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:


0.016 738 891 601 562 496 530 553 97(10) =

0.0000 0100 0100 1000 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 11(2)

6. Positive number before normalization:

0.016 738 891 601 562 496 530 553 97(10) =

0.0000 0100 0100 1000 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 11(2)

7. Normalize the binary representation of the number.

Shift the decimal mark 6 positions to the right, so that only one non zero digit remains to the left of it:


0.016 738 891 601 562 496 530 553 97(10) =

0.0000 0100 0100 1000 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 11(2) =

0.0000 0100 0100 1000 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 11(2) × 20 =

1.0001 0010 0011 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111(2) × 2-6


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): -6

Mantissa (not normalized):
1.0001 0010 0011 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111


9. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

-6 + 2(11-1) - 1 =

(-6 + 1 023)(10) =

1 017(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;
  • 1 017 ÷ 2 = 508 + 1;
  • 508 ÷ 2 = 254 + 0;
  • 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) =

1017(10) =

011 1111 1001(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. 0001 0010 0011 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 =

0001 0010 0011 1111 1111 1111 1111 1111 1111 1111 1111 1111 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 1001

Mantissa (52 bits) =
0001 0010 0011 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111


Decimal number -0.016 738 891 601 562 496 530 553 97 converted to 64 bit double precision IEEE 754 binary floating point representation:

1 - 011 1111 1001 - 0001 0010 0011 1111 1111 1111 1111 1111 1111 1111 1111 1111 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