-0.016 738 891 601 562 531 225 022 567 582 5 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.016 738 891 601 562 531 225 022 567 582 5(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
-0.016 738 891 601 562 531 225 022 567 582 5(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:

|-0.016 738 891 601 562 531 225 022 567 582 5| = 0.016 738 891 601 562 531 225 022 567 582 5


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: 0.016 738 891 601 562 531 225 022 567 582 5.

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) 0.016 738 891 601 562 531 225 022 567 582 5 × 2 = 0 + 0.033 477 783 203 125 062 450 045 135 165;
  • 2) 0.033 477 783 203 125 062 450 045 135 165 × 2 = 0 + 0.066 955 566 406 250 124 900 090 270 33;
  • 3) 0.066 955 566 406 250 124 900 090 270 33 × 2 = 0 + 0.133 911 132 812 500 249 800 180 540 66;
  • 4) 0.133 911 132 812 500 249 800 180 540 66 × 2 = 0 + 0.267 822 265 625 000 499 600 361 081 32;
  • 5) 0.267 822 265 625 000 499 600 361 081 32 × 2 = 0 + 0.535 644 531 250 000 999 200 722 162 64;
  • 6) 0.535 644 531 250 000 999 200 722 162 64 × 2 = 1 + 0.071 289 062 500 001 998 401 444 325 28;
  • 7) 0.071 289 062 500 001 998 401 444 325 28 × 2 = 0 + 0.142 578 125 000 003 996 802 888 650 56;
  • 8) 0.142 578 125 000 003 996 802 888 650 56 × 2 = 0 + 0.285 156 250 000 007 993 605 777 301 12;
  • 9) 0.285 156 250 000 007 993 605 777 301 12 × 2 = 0 + 0.570 312 500 000 015 987 211 554 602 24;
  • 10) 0.570 312 500 000 015 987 211 554 602 24 × 2 = 1 + 0.140 625 000 000 031 974 423 109 204 48;
  • 11) 0.140 625 000 000 031 974 423 109 204 48 × 2 = 0 + 0.281 250 000 000 063 948 846 218 408 96;
  • 12) 0.281 250 000 000 063 948 846 218 408 96 × 2 = 0 + 0.562 500 000 000 127 897 692 436 817 92;
  • 13) 0.562 500 000 000 127 897 692 436 817 92 × 2 = 1 + 0.125 000 000 000 255 795 384 873 635 84;
  • 14) 0.125 000 000 000 255 795 384 873 635 84 × 2 = 0 + 0.250 000 000 000 511 590 769 747 271 68;
  • 15) 0.250 000 000 000 511 590 769 747 271 68 × 2 = 0 + 0.500 000 000 001 023 181 539 494 543 36;
  • 16) 0.500 000 000 001 023 181 539 494 543 36 × 2 = 1 + 0.000 000 000 002 046 363 078 989 086 72;
  • 17) 0.000 000 000 002 046 363 078 989 086 72 × 2 = 0 + 0.000 000 000 004 092 726 157 978 173 44;
  • 18) 0.000 000 000 004 092 726 157 978 173 44 × 2 = 0 + 0.000 000 000 008 185 452 315 956 346 88;
  • 19) 0.000 000 000 008 185 452 315 956 346 88 × 2 = 0 + 0.000 000 000 016 370 904 631 912 693 76;
  • 20) 0.000 000 000 016 370 904 631 912 693 76 × 2 = 0 + 0.000 000 000 032 741 809 263 825 387 52;
  • 21) 0.000 000 000 032 741 809 263 825 387 52 × 2 = 0 + 0.000 000 000 065 483 618 527 650 775 04;
  • 22) 0.000 000 000 065 483 618 527 650 775 04 × 2 = 0 + 0.000 000 000 130 967 237 055 301 550 08;
  • 23) 0.000 000 000 130 967 237 055 301 550 08 × 2 = 0 + 0.000 000 000 261 934 474 110 603 100 16;
  • 24) 0.000 000 000 261 934 474 110 603 100 16 × 2 = 0 + 0.000 000 000 523 868 948 221 206 200 32;
  • 25) 0.000 000 000 523 868 948 221 206 200 32 × 2 = 0 + 0.000 000 001 047 737 896 442 412 400 64;
  • 26) 0.000 000 001 047 737 896 442 412 400 64 × 2 = 0 + 0.000 000 002 095 475 792 884 824 801 28;
  • 27) 0.000 000 002 095 475 792 884 824 801 28 × 2 = 0 + 0.000 000 004 190 951 585 769 649 602 56;
  • 28) 0.000 000 004 190 951 585 769 649 602 56 × 2 = 0 + 0.000 000 008 381 903 171 539 299 205 12;
  • 29) 0.000 000 008 381 903 171 539 299 205 12 × 2 = 0 + 0.000 000 016 763 806 343 078 598 410 24;
  • 30) 0.000 000 016 763 806 343 078 598 410 24 × 2 = 0 + 0.000 000 033 527 612 686 157 196 820 48;
  • 31) 0.000 000 033 527 612 686 157 196 820 48 × 2 = 0 + 0.000 000 067 055 225 372 314 393 640 96;
  • 32) 0.000 000 067 055 225 372 314 393 640 96 × 2 = 0 + 0.000 000 134 110 450 744 628 787 281 92;
  • 33) 0.000 000 134 110 450 744 628 787 281 92 × 2 = 0 + 0.000 000 268 220 901 489 257 574 563 84;
  • 34) 0.000 000 268 220 901 489 257 574 563 84 × 2 = 0 + 0.000 000 536 441 802 978 515 149 127 68;
  • 35) 0.000 000 536 441 802 978 515 149 127 68 × 2 = 0 + 0.000 001 072 883 605 957 030 298 255 36;
  • 36) 0.000 001 072 883 605 957 030 298 255 36 × 2 = 0 + 0.000 002 145 767 211 914 060 596 510 72;
  • 37) 0.000 002 145 767 211 914 060 596 510 72 × 2 = 0 + 0.000 004 291 534 423 828 121 193 021 44;
  • 38) 0.000 004 291 534 423 828 121 193 021 44 × 2 = 0 + 0.000 008 583 068 847 656 242 386 042 88;
  • 39) 0.000 008 583 068 847 656 242 386 042 88 × 2 = 0 + 0.000 017 166 137 695 312 484 772 085 76;
  • 40) 0.000 017 166 137 695 312 484 772 085 76 × 2 = 0 + 0.000 034 332 275 390 624 969 544 171 52;
  • 41) 0.000 034 332 275 390 624 969 544 171 52 × 2 = 0 + 0.000 068 664 550 781 249 939 088 343 04;
  • 42) 0.000 068 664 550 781 249 939 088 343 04 × 2 = 0 + 0.000 137 329 101 562 499 878 176 686 08;
  • 43) 0.000 137 329 101 562 499 878 176 686 08 × 2 = 0 + 0.000 274 658 203 124 999 756 353 372 16;
  • 44) 0.000 274 658 203 124 999 756 353 372 16 × 2 = 0 + 0.000 549 316 406 249 999 512 706 744 32;
  • 45) 0.000 549 316 406 249 999 512 706 744 32 × 2 = 0 + 0.001 098 632 812 499 999 025 413 488 64;
  • 46) 0.001 098 632 812 499 999 025 413 488 64 × 2 = 0 + 0.002 197 265 624 999 998 050 826 977 28;
  • 47) 0.002 197 265 624 999 998 050 826 977 28 × 2 = 0 + 0.004 394 531 249 999 996 101 653 954 56;
  • 48) 0.004 394 531 249 999 996 101 653 954 56 × 2 = 0 + 0.008 789 062 499 999 992 203 307 909 12;
  • 49) 0.008 789 062 499 999 992 203 307 909 12 × 2 = 0 + 0.017 578 124 999 999 984 406 615 818 24;
  • 50) 0.017 578 124 999 999 984 406 615 818 24 × 2 = 0 + 0.035 156 249 999 999 968 813 231 636 48;
  • 51) 0.035 156 249 999 999 968 813 231 636 48 × 2 = 0 + 0.070 312 499 999 999 937 626 463 272 96;
  • 52) 0.070 312 499 999 999 937 626 463 272 96 × 2 = 0 + 0.140 624 999 999 999 875 252 926 545 92;
  • 53) 0.140 624 999 999 999 875 252 926 545 92 × 2 = 0 + 0.281 249 999 999 999 750 505 853 091 84;
  • 54) 0.281 249 999 999 999 750 505 853 091 84 × 2 = 0 + 0.562 499 999 999 999 501 011 706 183 68;
  • 55) 0.562 499 999 999 999 501 011 706 183 68 × 2 = 1 + 0.124 999 999 999 999 002 023 412 367 36;
  • 56) 0.124 999 999 999 999 002 023 412 367 36 × 2 = 0 + 0.249 999 999 999 998 004 046 824 734 72;
  • 57) 0.249 999 999 999 998 004 046 824 734 72 × 2 = 0 + 0.499 999 999 999 996 008 093 649 469 44;
  • 58) 0.499 999 999 999 996 008 093 649 469 44 × 2 = 0 + 0.999 999 999 999 992 016 187 298 938 88;

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:


0.016 738 891 601 562 531 225 022 567 582 5(10) =

0.0000 0100 0100 1001 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 00(2)

6. Positive number before normalization:

0.016 738 891 601 562 531 225 022 567 582 5(10) =

0.0000 0100 0100 1001 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 00(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:


0.016 738 891 601 562 531 225 022 567 582 5(10) =

0.0000 0100 0100 1001 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 00(2) =

0.0000 0100 0100 1001 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 00(2) × 20 =

1.0001 0010 0100 0000 0000 0000 0000 0000 0000 0000 0000 0000 1000(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 0100 0000 0000 0000 0000 0000 0000 0000 0000 0000 1000


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 0100 0000 0000 0000 0000 0000 0000 0000 0000 0000 1000 =

0001 0010 0100 0000 0000 0000 0000 0000 0000 0000 0000 0000 1000


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 0100 0000 0000 0000 0000 0000 0000 0000 0000 0000 1000


Decimal number -0.016 738 891 601 562 531 225 022 567 582 5 converted to 64 bit double precision IEEE 754 binary floating point representation:

1 - 011 1111 1001 - 0001 0010 0100 0000 0000 0000 0000 0000 0000 0000 0000 0000 1000

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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