-0.016 738 891 601 562 496 530 657 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.016 738 891 601 562 496 530 657(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
-0.016 738 891 601 562 496 530 657(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:

|-0.016 738 891 601 562 496 530 657| = 0.016 738 891 601 562 496 530 657


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: 0.016 738 891 601 562 496 530 657.

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) 0.016 738 891 601 562 496 530 657 × 2 = 0 + 0.033 477 783 203 124 993 061 314;
  • 2) 0.033 477 783 203 124 993 061 314 × 2 = 0 + 0.066 955 566 406 249 986 122 628;
  • 3) 0.066 955 566 406 249 986 122 628 × 2 = 0 + 0.133 911 132 812 499 972 245 256;
  • 4) 0.133 911 132 812 499 972 245 256 × 2 = 0 + 0.267 822 265 624 999 944 490 512;
  • 5) 0.267 822 265 624 999 944 490 512 × 2 = 0 + 0.535 644 531 249 999 888 981 024;
  • 6) 0.535 644 531 249 999 888 981 024 × 2 = 1 + 0.071 289 062 499 999 777 962 048;
  • 7) 0.071 289 062 499 999 777 962 048 × 2 = 0 + 0.142 578 124 999 999 555 924 096;
  • 8) 0.142 578 124 999 999 555 924 096 × 2 = 0 + 0.285 156 249 999 999 111 848 192;
  • 9) 0.285 156 249 999 999 111 848 192 × 2 = 0 + 0.570 312 499 999 998 223 696 384;
  • 10) 0.570 312 499 999 998 223 696 384 × 2 = 1 + 0.140 624 999 999 996 447 392 768;
  • 11) 0.140 624 999 999 996 447 392 768 × 2 = 0 + 0.281 249 999 999 992 894 785 536;
  • 12) 0.281 249 999 999 992 894 785 536 × 2 = 0 + 0.562 499 999 999 985 789 571 072;
  • 13) 0.562 499 999 999 985 789 571 072 × 2 = 1 + 0.124 999 999 999 971 579 142 144;
  • 14) 0.124 999 999 999 971 579 142 144 × 2 = 0 + 0.249 999 999 999 943 158 284 288;
  • 15) 0.249 999 999 999 943 158 284 288 × 2 = 0 + 0.499 999 999 999 886 316 568 576;
  • 16) 0.499 999 999 999 886 316 568 576 × 2 = 0 + 0.999 999 999 999 772 633 137 152;
  • 17) 0.999 999 999 999 772 633 137 152 × 2 = 1 + 0.999 999 999 999 545 266 274 304;
  • 18) 0.999 999 999 999 545 266 274 304 × 2 = 1 + 0.999 999 999 999 090 532 548 608;
  • 19) 0.999 999 999 999 090 532 548 608 × 2 = 1 + 0.999 999 999 998 181 065 097 216;
  • 20) 0.999 999 999 998 181 065 097 216 × 2 = 1 + 0.999 999 999 996 362 130 194 432;
  • 21) 0.999 999 999 996 362 130 194 432 × 2 = 1 + 0.999 999 999 992 724 260 388 864;
  • 22) 0.999 999 999 992 724 260 388 864 × 2 = 1 + 0.999 999 999 985 448 520 777 728;
  • 23) 0.999 999 999 985 448 520 777 728 × 2 = 1 + 0.999 999 999 970 897 041 555 456;
  • 24) 0.999 999 999 970 897 041 555 456 × 2 = 1 + 0.999 999 999 941 794 083 110 912;
  • 25) 0.999 999 999 941 794 083 110 912 × 2 = 1 + 0.999 999 999 883 588 166 221 824;
  • 26) 0.999 999 999 883 588 166 221 824 × 2 = 1 + 0.999 999 999 767 176 332 443 648;
  • 27) 0.999 999 999 767 176 332 443 648 × 2 = 1 + 0.999 999 999 534 352 664 887 296;
  • 28) 0.999 999 999 534 352 664 887 296 × 2 = 1 + 0.999 999 999 068 705 329 774 592;
  • 29) 0.999 999 999 068 705 329 774 592 × 2 = 1 + 0.999 999 998 137 410 659 549 184;
  • 30) 0.999 999 998 137 410 659 549 184 × 2 = 1 + 0.999 999 996 274 821 319 098 368;
  • 31) 0.999 999 996 274 821 319 098 368 × 2 = 1 + 0.999 999 992 549 642 638 196 736;
  • 32) 0.999 999 992 549 642 638 196 736 × 2 = 1 + 0.999 999 985 099 285 276 393 472;
  • 33) 0.999 999 985 099 285 276 393 472 × 2 = 1 + 0.999 999 970 198 570 552 786 944;
  • 34) 0.999 999 970 198 570 552 786 944 × 2 = 1 + 0.999 999 940 397 141 105 573 888;
  • 35) 0.999 999 940 397 141 105 573 888 × 2 = 1 + 0.999 999 880 794 282 211 147 776;
  • 36) 0.999 999 880 794 282 211 147 776 × 2 = 1 + 0.999 999 761 588 564 422 295 552;
  • 37) 0.999 999 761 588 564 422 295 552 × 2 = 1 + 0.999 999 523 177 128 844 591 104;
  • 38) 0.999 999 523 177 128 844 591 104 × 2 = 1 + 0.999 999 046 354 257 689 182 208;
  • 39) 0.999 999 046 354 257 689 182 208 × 2 = 1 + 0.999 998 092 708 515 378 364 416;
  • 40) 0.999 998 092 708 515 378 364 416 × 2 = 1 + 0.999 996 185 417 030 756 728 832;
  • 41) 0.999 996 185 417 030 756 728 832 × 2 = 1 + 0.999 992 370 834 061 513 457 664;
  • 42) 0.999 992 370 834 061 513 457 664 × 2 = 1 + 0.999 984 741 668 123 026 915 328;
  • 43) 0.999 984 741 668 123 026 915 328 × 2 = 1 + 0.999 969 483 336 246 053 830 656;
  • 44) 0.999 969 483 336 246 053 830 656 × 2 = 1 + 0.999 938 966 672 492 107 661 312;
  • 45) 0.999 938 966 672 492 107 661 312 × 2 = 1 + 0.999 877 933 344 984 215 322 624;
  • 46) 0.999 877 933 344 984 215 322 624 × 2 = 1 + 0.999 755 866 689 968 430 645 248;
  • 47) 0.999 755 866 689 968 430 645 248 × 2 = 1 + 0.999 511 733 379 936 861 290 496;
  • 48) 0.999 511 733 379 936 861 290 496 × 2 = 1 + 0.999 023 466 759 873 722 580 992;
  • 49) 0.999 023 466 759 873 722 580 992 × 2 = 1 + 0.998 046 933 519 747 445 161 984;
  • 50) 0.998 046 933 519 747 445 161 984 × 2 = 1 + 0.996 093 867 039 494 890 323 968;
  • 51) 0.996 093 867 039 494 890 323 968 × 2 = 1 + 0.992 187 734 078 989 780 647 936;
  • 52) 0.992 187 734 078 989 780 647 936 × 2 = 1 + 0.984 375 468 157 979 561 295 872;
  • 53) 0.984 375 468 157 979 561 295 872 × 2 = 1 + 0.968 750 936 315 959 122 591 744;
  • 54) 0.968 750 936 315 959 122 591 744 × 2 = 1 + 0.937 501 872 631 918 245 183 488;
  • 55) 0.937 501 872 631 918 245 183 488 × 2 = 1 + 0.875 003 745 263 836 490 366 976;
  • 56) 0.875 003 745 263 836 490 366 976 × 2 = 1 + 0.750 007 490 527 672 980 733 952;
  • 57) 0.750 007 490 527 672 980 733 952 × 2 = 1 + 0.500 014 981 055 345 961 467 904;
  • 58) 0.500 014 981 055 345 961 467 904 × 2 = 1 + 0.000 029 962 110 691 922 935 808;

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:


0.016 738 891 601 562 496 530 657(10) =

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

6. Positive number before normalization:

0.016 738 891 601 562 496 530 657(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:


0.016 738 891 601 562 496 530 657(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 -0.016 738 891 601 562 496 530 657 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