-0.016 738 891 601 562 496 530 547 7 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.016 738 891 601 562 496 530 547 7(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
-0.016 738 891 601 562 496 530 547 7(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:

|-0.016 738 891 601 562 496 530 547 7| = 0.016 738 891 601 562 496 530 547 7


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: 0.016 738 891 601 562 496 530 547 7.

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) 0.016 738 891 601 562 496 530 547 7 × 2 = 0 + 0.033 477 783 203 124 993 061 095 4;
  • 2) 0.033 477 783 203 124 993 061 095 4 × 2 = 0 + 0.066 955 566 406 249 986 122 190 8;
  • 3) 0.066 955 566 406 249 986 122 190 8 × 2 = 0 + 0.133 911 132 812 499 972 244 381 6;
  • 4) 0.133 911 132 812 499 972 244 381 6 × 2 = 0 + 0.267 822 265 624 999 944 488 763 2;
  • 5) 0.267 822 265 624 999 944 488 763 2 × 2 = 0 + 0.535 644 531 249 999 888 977 526 4;
  • 6) 0.535 644 531 249 999 888 977 526 4 × 2 = 1 + 0.071 289 062 499 999 777 955 052 8;
  • 7) 0.071 289 062 499 999 777 955 052 8 × 2 = 0 + 0.142 578 124 999 999 555 910 105 6;
  • 8) 0.142 578 124 999 999 555 910 105 6 × 2 = 0 + 0.285 156 249 999 999 111 820 211 2;
  • 9) 0.285 156 249 999 999 111 820 211 2 × 2 = 0 + 0.570 312 499 999 998 223 640 422 4;
  • 10) 0.570 312 499 999 998 223 640 422 4 × 2 = 1 + 0.140 624 999 999 996 447 280 844 8;
  • 11) 0.140 624 999 999 996 447 280 844 8 × 2 = 0 + 0.281 249 999 999 992 894 561 689 6;
  • 12) 0.281 249 999 999 992 894 561 689 6 × 2 = 0 + 0.562 499 999 999 985 789 123 379 2;
  • 13) 0.562 499 999 999 985 789 123 379 2 × 2 = 1 + 0.124 999 999 999 971 578 246 758 4;
  • 14) 0.124 999 999 999 971 578 246 758 4 × 2 = 0 + 0.249 999 999 999 943 156 493 516 8;
  • 15) 0.249 999 999 999 943 156 493 516 8 × 2 = 0 + 0.499 999 999 999 886 312 987 033 6;
  • 16) 0.499 999 999 999 886 312 987 033 6 × 2 = 0 + 0.999 999 999 999 772 625 974 067 2;
  • 17) 0.999 999 999 999 772 625 974 067 2 × 2 = 1 + 0.999 999 999 999 545 251 948 134 4;
  • 18) 0.999 999 999 999 545 251 948 134 4 × 2 = 1 + 0.999 999 999 999 090 503 896 268 8;
  • 19) 0.999 999 999 999 090 503 896 268 8 × 2 = 1 + 0.999 999 999 998 181 007 792 537 6;
  • 20) 0.999 999 999 998 181 007 792 537 6 × 2 = 1 + 0.999 999 999 996 362 015 585 075 2;
  • 21) 0.999 999 999 996 362 015 585 075 2 × 2 = 1 + 0.999 999 999 992 724 031 170 150 4;
  • 22) 0.999 999 999 992 724 031 170 150 4 × 2 = 1 + 0.999 999 999 985 448 062 340 300 8;
  • 23) 0.999 999 999 985 448 062 340 300 8 × 2 = 1 + 0.999 999 999 970 896 124 680 601 6;
  • 24) 0.999 999 999 970 896 124 680 601 6 × 2 = 1 + 0.999 999 999 941 792 249 361 203 2;
  • 25) 0.999 999 999 941 792 249 361 203 2 × 2 = 1 + 0.999 999 999 883 584 498 722 406 4;
  • 26) 0.999 999 999 883 584 498 722 406 4 × 2 = 1 + 0.999 999 999 767 168 997 444 812 8;
  • 27) 0.999 999 999 767 168 997 444 812 8 × 2 = 1 + 0.999 999 999 534 337 994 889 625 6;
  • 28) 0.999 999 999 534 337 994 889 625 6 × 2 = 1 + 0.999 999 999 068 675 989 779 251 2;
  • 29) 0.999 999 999 068 675 989 779 251 2 × 2 = 1 + 0.999 999 998 137 351 979 558 502 4;
  • 30) 0.999 999 998 137 351 979 558 502 4 × 2 = 1 + 0.999 999 996 274 703 959 117 004 8;
  • 31) 0.999 999 996 274 703 959 117 004 8 × 2 = 1 + 0.999 999 992 549 407 918 234 009 6;
  • 32) 0.999 999 992 549 407 918 234 009 6 × 2 = 1 + 0.999 999 985 098 815 836 468 019 2;
  • 33) 0.999 999 985 098 815 836 468 019 2 × 2 = 1 + 0.999 999 970 197 631 672 936 038 4;
  • 34) 0.999 999 970 197 631 672 936 038 4 × 2 = 1 + 0.999 999 940 395 263 345 872 076 8;
  • 35) 0.999 999 940 395 263 345 872 076 8 × 2 = 1 + 0.999 999 880 790 526 691 744 153 6;
  • 36) 0.999 999 880 790 526 691 744 153 6 × 2 = 1 + 0.999 999 761 581 053 383 488 307 2;
  • 37) 0.999 999 761 581 053 383 488 307 2 × 2 = 1 + 0.999 999 523 162 106 766 976 614 4;
  • 38) 0.999 999 523 162 106 766 976 614 4 × 2 = 1 + 0.999 999 046 324 213 533 953 228 8;
  • 39) 0.999 999 046 324 213 533 953 228 8 × 2 = 1 + 0.999 998 092 648 427 067 906 457 6;
  • 40) 0.999 998 092 648 427 067 906 457 6 × 2 = 1 + 0.999 996 185 296 854 135 812 915 2;
  • 41) 0.999 996 185 296 854 135 812 915 2 × 2 = 1 + 0.999 992 370 593 708 271 625 830 4;
  • 42) 0.999 992 370 593 708 271 625 830 4 × 2 = 1 + 0.999 984 741 187 416 543 251 660 8;
  • 43) 0.999 984 741 187 416 543 251 660 8 × 2 = 1 + 0.999 969 482 374 833 086 503 321 6;
  • 44) 0.999 969 482 374 833 086 503 321 6 × 2 = 1 + 0.999 938 964 749 666 173 006 643 2;
  • 45) 0.999 938 964 749 666 173 006 643 2 × 2 = 1 + 0.999 877 929 499 332 346 013 286 4;
  • 46) 0.999 877 929 499 332 346 013 286 4 × 2 = 1 + 0.999 755 858 998 664 692 026 572 8;
  • 47) 0.999 755 858 998 664 692 026 572 8 × 2 = 1 + 0.999 511 717 997 329 384 053 145 6;
  • 48) 0.999 511 717 997 329 384 053 145 6 × 2 = 1 + 0.999 023 435 994 658 768 106 291 2;
  • 49) 0.999 023 435 994 658 768 106 291 2 × 2 = 1 + 0.998 046 871 989 317 536 212 582 4;
  • 50) 0.998 046 871 989 317 536 212 582 4 × 2 = 1 + 0.996 093 743 978 635 072 425 164 8;
  • 51) 0.996 093 743 978 635 072 425 164 8 × 2 = 1 + 0.992 187 487 957 270 144 850 329 6;
  • 52) 0.992 187 487 957 270 144 850 329 6 × 2 = 1 + 0.984 374 975 914 540 289 700 659 2;
  • 53) 0.984 374 975 914 540 289 700 659 2 × 2 = 1 + 0.968 749 951 829 080 579 401 318 4;
  • 54) 0.968 749 951 829 080 579 401 318 4 × 2 = 1 + 0.937 499 903 658 161 158 802 636 8;
  • 55) 0.937 499 903 658 161 158 802 636 8 × 2 = 1 + 0.874 999 807 316 322 317 605 273 6;
  • 56) 0.874 999 807 316 322 317 605 273 6 × 2 = 1 + 0.749 999 614 632 644 635 210 547 2;
  • 57) 0.749 999 614 632 644 635 210 547 2 × 2 = 1 + 0.499 999 229 265 289 270 421 094 4;
  • 58) 0.499 999 229 265 289 270 421 094 4 × 2 = 0 + 0.999 998 458 530 578 540 842 188 8;

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:


0.016 738 891 601 562 496 530 547 7(10) =

0.0000 0100 0100 1000 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 10(2)

6. Positive number before normalization:

0.016 738 891 601 562 496 530 547 7(10) =

0.0000 0100 0100 1000 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 10(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:


0.016 738 891 601 562 496 530 547 7(10) =

0.0000 0100 0100 1000 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 10(2) =

0.0000 0100 0100 1000 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 10(2) × 20 =

1.0001 0010 0011 1111 1111 1111 1111 1111 1111 1111 1111 1111 1110(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 1110


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 1110 =

0001 0010 0011 1111 1111 1111 1111 1111 1111 1111 1111 1111 1110


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 1110


Decimal number -0.016 738 891 601 562 496 530 547 7 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 1110

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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