-0.016 738 891 601 562 496 548 3 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.016 738 891 601 562 496 548 3(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
-0.016 738 891 601 562 496 548 3(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:

|-0.016 738 891 601 562 496 548 3| = 0.016 738 891 601 562 496 548 3


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: 0.016 738 891 601 562 496 548 3.

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) 0.016 738 891 601 562 496 548 3 × 2 = 0 + 0.033 477 783 203 124 993 096 6;
  • 2) 0.033 477 783 203 124 993 096 6 × 2 = 0 + 0.066 955 566 406 249 986 193 2;
  • 3) 0.066 955 566 406 249 986 193 2 × 2 = 0 + 0.133 911 132 812 499 972 386 4;
  • 4) 0.133 911 132 812 499 972 386 4 × 2 = 0 + 0.267 822 265 624 999 944 772 8;
  • 5) 0.267 822 265 624 999 944 772 8 × 2 = 0 + 0.535 644 531 249 999 889 545 6;
  • 6) 0.535 644 531 249 999 889 545 6 × 2 = 1 + 0.071 289 062 499 999 779 091 2;
  • 7) 0.071 289 062 499 999 779 091 2 × 2 = 0 + 0.142 578 124 999 999 558 182 4;
  • 8) 0.142 578 124 999 999 558 182 4 × 2 = 0 + 0.285 156 249 999 999 116 364 8;
  • 9) 0.285 156 249 999 999 116 364 8 × 2 = 0 + 0.570 312 499 999 998 232 729 6;
  • 10) 0.570 312 499 999 998 232 729 6 × 2 = 1 + 0.140 624 999 999 996 465 459 2;
  • 11) 0.140 624 999 999 996 465 459 2 × 2 = 0 + 0.281 249 999 999 992 930 918 4;
  • 12) 0.281 249 999 999 992 930 918 4 × 2 = 0 + 0.562 499 999 999 985 861 836 8;
  • 13) 0.562 499 999 999 985 861 836 8 × 2 = 1 + 0.124 999 999 999 971 723 673 6;
  • 14) 0.124 999 999 999 971 723 673 6 × 2 = 0 + 0.249 999 999 999 943 447 347 2;
  • 15) 0.249 999 999 999 943 447 347 2 × 2 = 0 + 0.499 999 999 999 886 894 694 4;
  • 16) 0.499 999 999 999 886 894 694 4 × 2 = 0 + 0.999 999 999 999 773 789 388 8;
  • 17) 0.999 999 999 999 773 789 388 8 × 2 = 1 + 0.999 999 999 999 547 578 777 6;
  • 18) 0.999 999 999 999 547 578 777 6 × 2 = 1 + 0.999 999 999 999 095 157 555 2;
  • 19) 0.999 999 999 999 095 157 555 2 × 2 = 1 + 0.999 999 999 998 190 315 110 4;
  • 20) 0.999 999 999 998 190 315 110 4 × 2 = 1 + 0.999 999 999 996 380 630 220 8;
  • 21) 0.999 999 999 996 380 630 220 8 × 2 = 1 + 0.999 999 999 992 761 260 441 6;
  • 22) 0.999 999 999 992 761 260 441 6 × 2 = 1 + 0.999 999 999 985 522 520 883 2;
  • 23) 0.999 999 999 985 522 520 883 2 × 2 = 1 + 0.999 999 999 971 045 041 766 4;
  • 24) 0.999 999 999 971 045 041 766 4 × 2 = 1 + 0.999 999 999 942 090 083 532 8;
  • 25) 0.999 999 999 942 090 083 532 8 × 2 = 1 + 0.999 999 999 884 180 167 065 6;
  • 26) 0.999 999 999 884 180 167 065 6 × 2 = 1 + 0.999 999 999 768 360 334 131 2;
  • 27) 0.999 999 999 768 360 334 131 2 × 2 = 1 + 0.999 999 999 536 720 668 262 4;
  • 28) 0.999 999 999 536 720 668 262 4 × 2 = 1 + 0.999 999 999 073 441 336 524 8;
  • 29) 0.999 999 999 073 441 336 524 8 × 2 = 1 + 0.999 999 998 146 882 673 049 6;
  • 30) 0.999 999 998 146 882 673 049 6 × 2 = 1 + 0.999 999 996 293 765 346 099 2;
  • 31) 0.999 999 996 293 765 346 099 2 × 2 = 1 + 0.999 999 992 587 530 692 198 4;
  • 32) 0.999 999 992 587 530 692 198 4 × 2 = 1 + 0.999 999 985 175 061 384 396 8;
  • 33) 0.999 999 985 175 061 384 396 8 × 2 = 1 + 0.999 999 970 350 122 768 793 6;
  • 34) 0.999 999 970 350 122 768 793 6 × 2 = 1 + 0.999 999 940 700 245 537 587 2;
  • 35) 0.999 999 940 700 245 537 587 2 × 2 = 1 + 0.999 999 881 400 491 075 174 4;
  • 36) 0.999 999 881 400 491 075 174 4 × 2 = 1 + 0.999 999 762 800 982 150 348 8;
  • 37) 0.999 999 762 800 982 150 348 8 × 2 = 1 + 0.999 999 525 601 964 300 697 6;
  • 38) 0.999 999 525 601 964 300 697 6 × 2 = 1 + 0.999 999 051 203 928 601 395 2;
  • 39) 0.999 999 051 203 928 601 395 2 × 2 = 1 + 0.999 998 102 407 857 202 790 4;
  • 40) 0.999 998 102 407 857 202 790 4 × 2 = 1 + 0.999 996 204 815 714 405 580 8;
  • 41) 0.999 996 204 815 714 405 580 8 × 2 = 1 + 0.999 992 409 631 428 811 161 6;
  • 42) 0.999 992 409 631 428 811 161 6 × 2 = 1 + 0.999 984 819 262 857 622 323 2;
  • 43) 0.999 984 819 262 857 622 323 2 × 2 = 1 + 0.999 969 638 525 715 244 646 4;
  • 44) 0.999 969 638 525 715 244 646 4 × 2 = 1 + 0.999 939 277 051 430 489 292 8;
  • 45) 0.999 939 277 051 430 489 292 8 × 2 = 1 + 0.999 878 554 102 860 978 585 6;
  • 46) 0.999 878 554 102 860 978 585 6 × 2 = 1 + 0.999 757 108 205 721 957 171 2;
  • 47) 0.999 757 108 205 721 957 171 2 × 2 = 1 + 0.999 514 216 411 443 914 342 4;
  • 48) 0.999 514 216 411 443 914 342 4 × 2 = 1 + 0.999 028 432 822 887 828 684 8;
  • 49) 0.999 028 432 822 887 828 684 8 × 2 = 1 + 0.998 056 865 645 775 657 369 6;
  • 50) 0.998 056 865 645 775 657 369 6 × 2 = 1 + 0.996 113 731 291 551 314 739 2;
  • 51) 0.996 113 731 291 551 314 739 2 × 2 = 1 + 0.992 227 462 583 102 629 478 4;
  • 52) 0.992 227 462 583 102 629 478 4 × 2 = 1 + 0.984 454 925 166 205 258 956 8;
  • 53) 0.984 454 925 166 205 258 956 8 × 2 = 1 + 0.968 909 850 332 410 517 913 6;
  • 54) 0.968 909 850 332 410 517 913 6 × 2 = 1 + 0.937 819 700 664 821 035 827 2;
  • 55) 0.937 819 700 664 821 035 827 2 × 2 = 1 + 0.875 639 401 329 642 071 654 4;
  • 56) 0.875 639 401 329 642 071 654 4 × 2 = 1 + 0.751 278 802 659 284 143 308 8;
  • 57) 0.751 278 802 659 284 143 308 8 × 2 = 1 + 0.502 557 605 318 568 286 617 6;
  • 58) 0.502 557 605 318 568 286 617 6 × 2 = 1 + 0.005 115 210 637 136 573 235 2;

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:


0.016 738 891 601 562 496 548 3(10) =

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

6. Positive number before normalization:

0.016 738 891 601 562 496 548 3(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:


0.016 738 891 601 562 496 548 3(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 -0.016 738 891 601 562 496 548 3 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