-0.016 738 891 601 562 496 466 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.016 738 891 601 562 496 466(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
-0.016 738 891 601 562 496 466(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:

|-0.016 738 891 601 562 496 466| = 0.016 738 891 601 562 496 466


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: 0.016 738 891 601 562 496 466.

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) 0.016 738 891 601 562 496 466 × 2 = 0 + 0.033 477 783 203 124 992 932;
  • 2) 0.033 477 783 203 124 992 932 × 2 = 0 + 0.066 955 566 406 249 985 864;
  • 3) 0.066 955 566 406 249 985 864 × 2 = 0 + 0.133 911 132 812 499 971 728;
  • 4) 0.133 911 132 812 499 971 728 × 2 = 0 + 0.267 822 265 624 999 943 456;
  • 5) 0.267 822 265 624 999 943 456 × 2 = 0 + 0.535 644 531 249 999 886 912;
  • 6) 0.535 644 531 249 999 886 912 × 2 = 1 + 0.071 289 062 499 999 773 824;
  • 7) 0.071 289 062 499 999 773 824 × 2 = 0 + 0.142 578 124 999 999 547 648;
  • 8) 0.142 578 124 999 999 547 648 × 2 = 0 + 0.285 156 249 999 999 095 296;
  • 9) 0.285 156 249 999 999 095 296 × 2 = 0 + 0.570 312 499 999 998 190 592;
  • 10) 0.570 312 499 999 998 190 592 × 2 = 1 + 0.140 624 999 999 996 381 184;
  • 11) 0.140 624 999 999 996 381 184 × 2 = 0 + 0.281 249 999 999 992 762 368;
  • 12) 0.281 249 999 999 992 762 368 × 2 = 0 + 0.562 499 999 999 985 524 736;
  • 13) 0.562 499 999 999 985 524 736 × 2 = 1 + 0.124 999 999 999 971 049 472;
  • 14) 0.124 999 999 999 971 049 472 × 2 = 0 + 0.249 999 999 999 942 098 944;
  • 15) 0.249 999 999 999 942 098 944 × 2 = 0 + 0.499 999 999 999 884 197 888;
  • 16) 0.499 999 999 999 884 197 888 × 2 = 0 + 0.999 999 999 999 768 395 776;
  • 17) 0.999 999 999 999 768 395 776 × 2 = 1 + 0.999 999 999 999 536 791 552;
  • 18) 0.999 999 999 999 536 791 552 × 2 = 1 + 0.999 999 999 999 073 583 104;
  • 19) 0.999 999 999 999 073 583 104 × 2 = 1 + 0.999 999 999 998 147 166 208;
  • 20) 0.999 999 999 998 147 166 208 × 2 = 1 + 0.999 999 999 996 294 332 416;
  • 21) 0.999 999 999 996 294 332 416 × 2 = 1 + 0.999 999 999 992 588 664 832;
  • 22) 0.999 999 999 992 588 664 832 × 2 = 1 + 0.999 999 999 985 177 329 664;
  • 23) 0.999 999 999 985 177 329 664 × 2 = 1 + 0.999 999 999 970 354 659 328;
  • 24) 0.999 999 999 970 354 659 328 × 2 = 1 + 0.999 999 999 940 709 318 656;
  • 25) 0.999 999 999 940 709 318 656 × 2 = 1 + 0.999 999 999 881 418 637 312;
  • 26) 0.999 999 999 881 418 637 312 × 2 = 1 + 0.999 999 999 762 837 274 624;
  • 27) 0.999 999 999 762 837 274 624 × 2 = 1 + 0.999 999 999 525 674 549 248;
  • 28) 0.999 999 999 525 674 549 248 × 2 = 1 + 0.999 999 999 051 349 098 496;
  • 29) 0.999 999 999 051 349 098 496 × 2 = 1 + 0.999 999 998 102 698 196 992;
  • 30) 0.999 999 998 102 698 196 992 × 2 = 1 + 0.999 999 996 205 396 393 984;
  • 31) 0.999 999 996 205 396 393 984 × 2 = 1 + 0.999 999 992 410 792 787 968;
  • 32) 0.999 999 992 410 792 787 968 × 2 = 1 + 0.999 999 984 821 585 575 936;
  • 33) 0.999 999 984 821 585 575 936 × 2 = 1 + 0.999 999 969 643 171 151 872;
  • 34) 0.999 999 969 643 171 151 872 × 2 = 1 + 0.999 999 939 286 342 303 744;
  • 35) 0.999 999 939 286 342 303 744 × 2 = 1 + 0.999 999 878 572 684 607 488;
  • 36) 0.999 999 878 572 684 607 488 × 2 = 1 + 0.999 999 757 145 369 214 976;
  • 37) 0.999 999 757 145 369 214 976 × 2 = 1 + 0.999 999 514 290 738 429 952;
  • 38) 0.999 999 514 290 738 429 952 × 2 = 1 + 0.999 999 028 581 476 859 904;
  • 39) 0.999 999 028 581 476 859 904 × 2 = 1 + 0.999 998 057 162 953 719 808;
  • 40) 0.999 998 057 162 953 719 808 × 2 = 1 + 0.999 996 114 325 907 439 616;
  • 41) 0.999 996 114 325 907 439 616 × 2 = 1 + 0.999 992 228 651 814 879 232;
  • 42) 0.999 992 228 651 814 879 232 × 2 = 1 + 0.999 984 457 303 629 758 464;
  • 43) 0.999 984 457 303 629 758 464 × 2 = 1 + 0.999 968 914 607 259 516 928;
  • 44) 0.999 968 914 607 259 516 928 × 2 = 1 + 0.999 937 829 214 519 033 856;
  • 45) 0.999 937 829 214 519 033 856 × 2 = 1 + 0.999 875 658 429 038 067 712;
  • 46) 0.999 875 658 429 038 067 712 × 2 = 1 + 0.999 751 316 858 076 135 424;
  • 47) 0.999 751 316 858 076 135 424 × 2 = 1 + 0.999 502 633 716 152 270 848;
  • 48) 0.999 502 633 716 152 270 848 × 2 = 1 + 0.999 005 267 432 304 541 696;
  • 49) 0.999 005 267 432 304 541 696 × 2 = 1 + 0.998 010 534 864 609 083 392;
  • 50) 0.998 010 534 864 609 083 392 × 2 = 1 + 0.996 021 069 729 218 166 784;
  • 51) 0.996 021 069 729 218 166 784 × 2 = 1 + 0.992 042 139 458 436 333 568;
  • 52) 0.992 042 139 458 436 333 568 × 2 = 1 + 0.984 084 278 916 872 667 136;
  • 53) 0.984 084 278 916 872 667 136 × 2 = 1 + 0.968 168 557 833 745 334 272;
  • 54) 0.968 168 557 833 745 334 272 × 2 = 1 + 0.936 337 115 667 490 668 544;
  • 55) 0.936 337 115 667 490 668 544 × 2 = 1 + 0.872 674 231 334 981 337 088;
  • 56) 0.872 674 231 334 981 337 088 × 2 = 1 + 0.745 348 462 669 962 674 176;
  • 57) 0.745 348 462 669 962 674 176 × 2 = 1 + 0.490 696 925 339 925 348 352;
  • 58) 0.490 696 925 339 925 348 352 × 2 = 0 + 0.981 393 850 679 850 696 704;

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:


0.016 738 891 601 562 496 466(10) =

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

6. Positive number before normalization:

0.016 738 891 601 562 496 466(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:


0.016 738 891 601 562 496 466(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 -0.016 738 891 601 562 496 466 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