-0.016 738 891 601 562 496 530 554 42 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.016 738 891 601 562 496 530 554 42(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
-0.016 738 891 601 562 496 530 554 42(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:

|-0.016 738 891 601 562 496 530 554 42| = 0.016 738 891 601 562 496 530 554 42


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: 0.016 738 891 601 562 496 530 554 42.

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) 0.016 738 891 601 562 496 530 554 42 × 2 = 0 + 0.033 477 783 203 124 993 061 108 84;
  • 2) 0.033 477 783 203 124 993 061 108 84 × 2 = 0 + 0.066 955 566 406 249 986 122 217 68;
  • 3) 0.066 955 566 406 249 986 122 217 68 × 2 = 0 + 0.133 911 132 812 499 972 244 435 36;
  • 4) 0.133 911 132 812 499 972 244 435 36 × 2 = 0 + 0.267 822 265 624 999 944 488 870 72;
  • 5) 0.267 822 265 624 999 944 488 870 72 × 2 = 0 + 0.535 644 531 249 999 888 977 741 44;
  • 6) 0.535 644 531 249 999 888 977 741 44 × 2 = 1 + 0.071 289 062 499 999 777 955 482 88;
  • 7) 0.071 289 062 499 999 777 955 482 88 × 2 = 0 + 0.142 578 124 999 999 555 910 965 76;
  • 8) 0.142 578 124 999 999 555 910 965 76 × 2 = 0 + 0.285 156 249 999 999 111 821 931 52;
  • 9) 0.285 156 249 999 999 111 821 931 52 × 2 = 0 + 0.570 312 499 999 998 223 643 863 04;
  • 10) 0.570 312 499 999 998 223 643 863 04 × 2 = 1 + 0.140 624 999 999 996 447 287 726 08;
  • 11) 0.140 624 999 999 996 447 287 726 08 × 2 = 0 + 0.281 249 999 999 992 894 575 452 16;
  • 12) 0.281 249 999 999 992 894 575 452 16 × 2 = 0 + 0.562 499 999 999 985 789 150 904 32;
  • 13) 0.562 499 999 999 985 789 150 904 32 × 2 = 1 + 0.124 999 999 999 971 578 301 808 64;
  • 14) 0.124 999 999 999 971 578 301 808 64 × 2 = 0 + 0.249 999 999 999 943 156 603 617 28;
  • 15) 0.249 999 999 999 943 156 603 617 28 × 2 = 0 + 0.499 999 999 999 886 313 207 234 56;
  • 16) 0.499 999 999 999 886 313 207 234 56 × 2 = 0 + 0.999 999 999 999 772 626 414 469 12;
  • 17) 0.999 999 999 999 772 626 414 469 12 × 2 = 1 + 0.999 999 999 999 545 252 828 938 24;
  • 18) 0.999 999 999 999 545 252 828 938 24 × 2 = 1 + 0.999 999 999 999 090 505 657 876 48;
  • 19) 0.999 999 999 999 090 505 657 876 48 × 2 = 1 + 0.999 999 999 998 181 011 315 752 96;
  • 20) 0.999 999 999 998 181 011 315 752 96 × 2 = 1 + 0.999 999 999 996 362 022 631 505 92;
  • 21) 0.999 999 999 996 362 022 631 505 92 × 2 = 1 + 0.999 999 999 992 724 045 263 011 84;
  • 22) 0.999 999 999 992 724 045 263 011 84 × 2 = 1 + 0.999 999 999 985 448 090 526 023 68;
  • 23) 0.999 999 999 985 448 090 526 023 68 × 2 = 1 + 0.999 999 999 970 896 181 052 047 36;
  • 24) 0.999 999 999 970 896 181 052 047 36 × 2 = 1 + 0.999 999 999 941 792 362 104 094 72;
  • 25) 0.999 999 999 941 792 362 104 094 72 × 2 = 1 + 0.999 999 999 883 584 724 208 189 44;
  • 26) 0.999 999 999 883 584 724 208 189 44 × 2 = 1 + 0.999 999 999 767 169 448 416 378 88;
  • 27) 0.999 999 999 767 169 448 416 378 88 × 2 = 1 + 0.999 999 999 534 338 896 832 757 76;
  • 28) 0.999 999 999 534 338 896 832 757 76 × 2 = 1 + 0.999 999 999 068 677 793 665 515 52;
  • 29) 0.999 999 999 068 677 793 665 515 52 × 2 = 1 + 0.999 999 998 137 355 587 331 031 04;
  • 30) 0.999 999 998 137 355 587 331 031 04 × 2 = 1 + 0.999 999 996 274 711 174 662 062 08;
  • 31) 0.999 999 996 274 711 174 662 062 08 × 2 = 1 + 0.999 999 992 549 422 349 324 124 16;
  • 32) 0.999 999 992 549 422 349 324 124 16 × 2 = 1 + 0.999 999 985 098 844 698 648 248 32;
  • 33) 0.999 999 985 098 844 698 648 248 32 × 2 = 1 + 0.999 999 970 197 689 397 296 496 64;
  • 34) 0.999 999 970 197 689 397 296 496 64 × 2 = 1 + 0.999 999 940 395 378 794 592 993 28;
  • 35) 0.999 999 940 395 378 794 592 993 28 × 2 = 1 + 0.999 999 880 790 757 589 185 986 56;
  • 36) 0.999 999 880 790 757 589 185 986 56 × 2 = 1 + 0.999 999 761 581 515 178 371 973 12;
  • 37) 0.999 999 761 581 515 178 371 973 12 × 2 = 1 + 0.999 999 523 163 030 356 743 946 24;
  • 38) 0.999 999 523 163 030 356 743 946 24 × 2 = 1 + 0.999 999 046 326 060 713 487 892 48;
  • 39) 0.999 999 046 326 060 713 487 892 48 × 2 = 1 + 0.999 998 092 652 121 426 975 784 96;
  • 40) 0.999 998 092 652 121 426 975 784 96 × 2 = 1 + 0.999 996 185 304 242 853 951 569 92;
  • 41) 0.999 996 185 304 242 853 951 569 92 × 2 = 1 + 0.999 992 370 608 485 707 903 139 84;
  • 42) 0.999 992 370 608 485 707 903 139 84 × 2 = 1 + 0.999 984 741 216 971 415 806 279 68;
  • 43) 0.999 984 741 216 971 415 806 279 68 × 2 = 1 + 0.999 969 482 433 942 831 612 559 36;
  • 44) 0.999 969 482 433 942 831 612 559 36 × 2 = 1 + 0.999 938 964 867 885 663 225 118 72;
  • 45) 0.999 938 964 867 885 663 225 118 72 × 2 = 1 + 0.999 877 929 735 771 326 450 237 44;
  • 46) 0.999 877 929 735 771 326 450 237 44 × 2 = 1 + 0.999 755 859 471 542 652 900 474 88;
  • 47) 0.999 755 859 471 542 652 900 474 88 × 2 = 1 + 0.999 511 718 943 085 305 800 949 76;
  • 48) 0.999 511 718 943 085 305 800 949 76 × 2 = 1 + 0.999 023 437 886 170 611 601 899 52;
  • 49) 0.999 023 437 886 170 611 601 899 52 × 2 = 1 + 0.998 046 875 772 341 223 203 799 04;
  • 50) 0.998 046 875 772 341 223 203 799 04 × 2 = 1 + 0.996 093 751 544 682 446 407 598 08;
  • 51) 0.996 093 751 544 682 446 407 598 08 × 2 = 1 + 0.992 187 503 089 364 892 815 196 16;
  • 52) 0.992 187 503 089 364 892 815 196 16 × 2 = 1 + 0.984 375 006 178 729 785 630 392 32;
  • 53) 0.984 375 006 178 729 785 630 392 32 × 2 = 1 + 0.968 750 012 357 459 571 260 784 64;
  • 54) 0.968 750 012 357 459 571 260 784 64 × 2 = 1 + 0.937 500 024 714 919 142 521 569 28;
  • 55) 0.937 500 024 714 919 142 521 569 28 × 2 = 1 + 0.875 000 049 429 838 285 043 138 56;
  • 56) 0.875 000 049 429 838 285 043 138 56 × 2 = 1 + 0.750 000 098 859 676 570 086 277 12;
  • 57) 0.750 000 098 859 676 570 086 277 12 × 2 = 1 + 0.500 000 197 719 353 140 172 554 24;
  • 58) 0.500 000 197 719 353 140 172 554 24 × 2 = 1 + 0.000 000 395 438 706 280 345 108 48;

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:


0.016 738 891 601 562 496 530 554 42(10) =

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

6. Positive number before normalization:

0.016 738 891 601 562 496 530 554 42(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:


0.016 738 891 601 562 496 530 554 42(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 -0.016 738 891 601 562 496 530 554 42 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