-0.016 738 891 601 562 496 706 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.016 738 891 601 562 496 706(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
-0.016 738 891 601 562 496 706(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:

|-0.016 738 891 601 562 496 706| = 0.016 738 891 601 562 496 706


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: 0.016 738 891 601 562 496 706.

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) 0.016 738 891 601 562 496 706 × 2 = 0 + 0.033 477 783 203 124 993 412;
  • 2) 0.033 477 783 203 124 993 412 × 2 = 0 + 0.066 955 566 406 249 986 824;
  • 3) 0.066 955 566 406 249 986 824 × 2 = 0 + 0.133 911 132 812 499 973 648;
  • 4) 0.133 911 132 812 499 973 648 × 2 = 0 + 0.267 822 265 624 999 947 296;
  • 5) 0.267 822 265 624 999 947 296 × 2 = 0 + 0.535 644 531 249 999 894 592;
  • 6) 0.535 644 531 249 999 894 592 × 2 = 1 + 0.071 289 062 499 999 789 184;
  • 7) 0.071 289 062 499 999 789 184 × 2 = 0 + 0.142 578 124 999 999 578 368;
  • 8) 0.142 578 124 999 999 578 368 × 2 = 0 + 0.285 156 249 999 999 156 736;
  • 9) 0.285 156 249 999 999 156 736 × 2 = 0 + 0.570 312 499 999 998 313 472;
  • 10) 0.570 312 499 999 998 313 472 × 2 = 1 + 0.140 624 999 999 996 626 944;
  • 11) 0.140 624 999 999 996 626 944 × 2 = 0 + 0.281 249 999 999 993 253 888;
  • 12) 0.281 249 999 999 993 253 888 × 2 = 0 + 0.562 499 999 999 986 507 776;
  • 13) 0.562 499 999 999 986 507 776 × 2 = 1 + 0.124 999 999 999 973 015 552;
  • 14) 0.124 999 999 999 973 015 552 × 2 = 0 + 0.249 999 999 999 946 031 104;
  • 15) 0.249 999 999 999 946 031 104 × 2 = 0 + 0.499 999 999 999 892 062 208;
  • 16) 0.499 999 999 999 892 062 208 × 2 = 0 + 0.999 999 999 999 784 124 416;
  • 17) 0.999 999 999 999 784 124 416 × 2 = 1 + 0.999 999 999 999 568 248 832;
  • 18) 0.999 999 999 999 568 248 832 × 2 = 1 + 0.999 999 999 999 136 497 664;
  • 19) 0.999 999 999 999 136 497 664 × 2 = 1 + 0.999 999 999 998 272 995 328;
  • 20) 0.999 999 999 998 272 995 328 × 2 = 1 + 0.999 999 999 996 545 990 656;
  • 21) 0.999 999 999 996 545 990 656 × 2 = 1 + 0.999 999 999 993 091 981 312;
  • 22) 0.999 999 999 993 091 981 312 × 2 = 1 + 0.999 999 999 986 183 962 624;
  • 23) 0.999 999 999 986 183 962 624 × 2 = 1 + 0.999 999 999 972 367 925 248;
  • 24) 0.999 999 999 972 367 925 248 × 2 = 1 + 0.999 999 999 944 735 850 496;
  • 25) 0.999 999 999 944 735 850 496 × 2 = 1 + 0.999 999 999 889 471 700 992;
  • 26) 0.999 999 999 889 471 700 992 × 2 = 1 + 0.999 999 999 778 943 401 984;
  • 27) 0.999 999 999 778 943 401 984 × 2 = 1 + 0.999 999 999 557 886 803 968;
  • 28) 0.999 999 999 557 886 803 968 × 2 = 1 + 0.999 999 999 115 773 607 936;
  • 29) 0.999 999 999 115 773 607 936 × 2 = 1 + 0.999 999 998 231 547 215 872;
  • 30) 0.999 999 998 231 547 215 872 × 2 = 1 + 0.999 999 996 463 094 431 744;
  • 31) 0.999 999 996 463 094 431 744 × 2 = 1 + 0.999 999 992 926 188 863 488;
  • 32) 0.999 999 992 926 188 863 488 × 2 = 1 + 0.999 999 985 852 377 726 976;
  • 33) 0.999 999 985 852 377 726 976 × 2 = 1 + 0.999 999 971 704 755 453 952;
  • 34) 0.999 999 971 704 755 453 952 × 2 = 1 + 0.999 999 943 409 510 907 904;
  • 35) 0.999 999 943 409 510 907 904 × 2 = 1 + 0.999 999 886 819 021 815 808;
  • 36) 0.999 999 886 819 021 815 808 × 2 = 1 + 0.999 999 773 638 043 631 616;
  • 37) 0.999 999 773 638 043 631 616 × 2 = 1 + 0.999 999 547 276 087 263 232;
  • 38) 0.999 999 547 276 087 263 232 × 2 = 1 + 0.999 999 094 552 174 526 464;
  • 39) 0.999 999 094 552 174 526 464 × 2 = 1 + 0.999 998 189 104 349 052 928;
  • 40) 0.999 998 189 104 349 052 928 × 2 = 1 + 0.999 996 378 208 698 105 856;
  • 41) 0.999 996 378 208 698 105 856 × 2 = 1 + 0.999 992 756 417 396 211 712;
  • 42) 0.999 992 756 417 396 211 712 × 2 = 1 + 0.999 985 512 834 792 423 424;
  • 43) 0.999 985 512 834 792 423 424 × 2 = 1 + 0.999 971 025 669 584 846 848;
  • 44) 0.999 971 025 669 584 846 848 × 2 = 1 + 0.999 942 051 339 169 693 696;
  • 45) 0.999 942 051 339 169 693 696 × 2 = 1 + 0.999 884 102 678 339 387 392;
  • 46) 0.999 884 102 678 339 387 392 × 2 = 1 + 0.999 768 205 356 678 774 784;
  • 47) 0.999 768 205 356 678 774 784 × 2 = 1 + 0.999 536 410 713 357 549 568;
  • 48) 0.999 536 410 713 357 549 568 × 2 = 1 + 0.999 072 821 426 715 099 136;
  • 49) 0.999 072 821 426 715 099 136 × 2 = 1 + 0.998 145 642 853 430 198 272;
  • 50) 0.998 145 642 853 430 198 272 × 2 = 1 + 0.996 291 285 706 860 396 544;
  • 51) 0.996 291 285 706 860 396 544 × 2 = 1 + 0.992 582 571 413 720 793 088;
  • 52) 0.992 582 571 413 720 793 088 × 2 = 1 + 0.985 165 142 827 441 586 176;
  • 53) 0.985 165 142 827 441 586 176 × 2 = 1 + 0.970 330 285 654 883 172 352;
  • 54) 0.970 330 285 654 883 172 352 × 2 = 1 + 0.940 660 571 309 766 344 704;
  • 55) 0.940 660 571 309 766 344 704 × 2 = 1 + 0.881 321 142 619 532 689 408;
  • 56) 0.881 321 142 619 532 689 408 × 2 = 1 + 0.762 642 285 239 065 378 816;
  • 57) 0.762 642 285 239 065 378 816 × 2 = 1 + 0.525 284 570 478 130 757 632;
  • 58) 0.525 284 570 478 130 757 632 × 2 = 1 + 0.050 569 140 956 261 515 264;

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:


0.016 738 891 601 562 496 706(10) =

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

6. Positive number before normalization:

0.016 738 891 601 562 496 706(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:


0.016 738 891 601 562 496 706(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 -0.016 738 891 601 562 496 706 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