-0.016 738 891 601 562 496 877 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.016 738 891 601 562 496 877(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
-0.016 738 891 601 562 496 877(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:

|-0.016 738 891 601 562 496 877| = 0.016 738 891 601 562 496 877


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: 0.016 738 891 601 562 496 877.

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) 0.016 738 891 601 562 496 877 × 2 = 0 + 0.033 477 783 203 124 993 754;
  • 2) 0.033 477 783 203 124 993 754 × 2 = 0 + 0.066 955 566 406 249 987 508;
  • 3) 0.066 955 566 406 249 987 508 × 2 = 0 + 0.133 911 132 812 499 975 016;
  • 4) 0.133 911 132 812 499 975 016 × 2 = 0 + 0.267 822 265 624 999 950 032;
  • 5) 0.267 822 265 624 999 950 032 × 2 = 0 + 0.535 644 531 249 999 900 064;
  • 6) 0.535 644 531 249 999 900 064 × 2 = 1 + 0.071 289 062 499 999 800 128;
  • 7) 0.071 289 062 499 999 800 128 × 2 = 0 + 0.142 578 124 999 999 600 256;
  • 8) 0.142 578 124 999 999 600 256 × 2 = 0 + 0.285 156 249 999 999 200 512;
  • 9) 0.285 156 249 999 999 200 512 × 2 = 0 + 0.570 312 499 999 998 401 024;
  • 10) 0.570 312 499 999 998 401 024 × 2 = 1 + 0.140 624 999 999 996 802 048;
  • 11) 0.140 624 999 999 996 802 048 × 2 = 0 + 0.281 249 999 999 993 604 096;
  • 12) 0.281 249 999 999 993 604 096 × 2 = 0 + 0.562 499 999 999 987 208 192;
  • 13) 0.562 499 999 999 987 208 192 × 2 = 1 + 0.124 999 999 999 974 416 384;
  • 14) 0.124 999 999 999 974 416 384 × 2 = 0 + 0.249 999 999 999 948 832 768;
  • 15) 0.249 999 999 999 948 832 768 × 2 = 0 + 0.499 999 999 999 897 665 536;
  • 16) 0.499 999 999 999 897 665 536 × 2 = 0 + 0.999 999 999 999 795 331 072;
  • 17) 0.999 999 999 999 795 331 072 × 2 = 1 + 0.999 999 999 999 590 662 144;
  • 18) 0.999 999 999 999 590 662 144 × 2 = 1 + 0.999 999 999 999 181 324 288;
  • 19) 0.999 999 999 999 181 324 288 × 2 = 1 + 0.999 999 999 998 362 648 576;
  • 20) 0.999 999 999 998 362 648 576 × 2 = 1 + 0.999 999 999 996 725 297 152;
  • 21) 0.999 999 999 996 725 297 152 × 2 = 1 + 0.999 999 999 993 450 594 304;
  • 22) 0.999 999 999 993 450 594 304 × 2 = 1 + 0.999 999 999 986 901 188 608;
  • 23) 0.999 999 999 986 901 188 608 × 2 = 1 + 0.999 999 999 973 802 377 216;
  • 24) 0.999 999 999 973 802 377 216 × 2 = 1 + 0.999 999 999 947 604 754 432;
  • 25) 0.999 999 999 947 604 754 432 × 2 = 1 + 0.999 999 999 895 209 508 864;
  • 26) 0.999 999 999 895 209 508 864 × 2 = 1 + 0.999 999 999 790 419 017 728;
  • 27) 0.999 999 999 790 419 017 728 × 2 = 1 + 0.999 999 999 580 838 035 456;
  • 28) 0.999 999 999 580 838 035 456 × 2 = 1 + 0.999 999 999 161 676 070 912;
  • 29) 0.999 999 999 161 676 070 912 × 2 = 1 + 0.999 999 998 323 352 141 824;
  • 30) 0.999 999 998 323 352 141 824 × 2 = 1 + 0.999 999 996 646 704 283 648;
  • 31) 0.999 999 996 646 704 283 648 × 2 = 1 + 0.999 999 993 293 408 567 296;
  • 32) 0.999 999 993 293 408 567 296 × 2 = 1 + 0.999 999 986 586 817 134 592;
  • 33) 0.999 999 986 586 817 134 592 × 2 = 1 + 0.999 999 973 173 634 269 184;
  • 34) 0.999 999 973 173 634 269 184 × 2 = 1 + 0.999 999 946 347 268 538 368;
  • 35) 0.999 999 946 347 268 538 368 × 2 = 1 + 0.999 999 892 694 537 076 736;
  • 36) 0.999 999 892 694 537 076 736 × 2 = 1 + 0.999 999 785 389 074 153 472;
  • 37) 0.999 999 785 389 074 153 472 × 2 = 1 + 0.999 999 570 778 148 306 944;
  • 38) 0.999 999 570 778 148 306 944 × 2 = 1 + 0.999 999 141 556 296 613 888;
  • 39) 0.999 999 141 556 296 613 888 × 2 = 1 + 0.999 998 283 112 593 227 776;
  • 40) 0.999 998 283 112 593 227 776 × 2 = 1 + 0.999 996 566 225 186 455 552;
  • 41) 0.999 996 566 225 186 455 552 × 2 = 1 + 0.999 993 132 450 372 911 104;
  • 42) 0.999 993 132 450 372 911 104 × 2 = 1 + 0.999 986 264 900 745 822 208;
  • 43) 0.999 986 264 900 745 822 208 × 2 = 1 + 0.999 972 529 801 491 644 416;
  • 44) 0.999 972 529 801 491 644 416 × 2 = 1 + 0.999 945 059 602 983 288 832;
  • 45) 0.999 945 059 602 983 288 832 × 2 = 1 + 0.999 890 119 205 966 577 664;
  • 46) 0.999 890 119 205 966 577 664 × 2 = 1 + 0.999 780 238 411 933 155 328;
  • 47) 0.999 780 238 411 933 155 328 × 2 = 1 + 0.999 560 476 823 866 310 656;
  • 48) 0.999 560 476 823 866 310 656 × 2 = 1 + 0.999 120 953 647 732 621 312;
  • 49) 0.999 120 953 647 732 621 312 × 2 = 1 + 0.998 241 907 295 465 242 624;
  • 50) 0.998 241 907 295 465 242 624 × 2 = 1 + 0.996 483 814 590 930 485 248;
  • 51) 0.996 483 814 590 930 485 248 × 2 = 1 + 0.992 967 629 181 860 970 496;
  • 52) 0.992 967 629 181 860 970 496 × 2 = 1 + 0.985 935 258 363 721 940 992;
  • 53) 0.985 935 258 363 721 940 992 × 2 = 1 + 0.971 870 516 727 443 881 984;
  • 54) 0.971 870 516 727 443 881 984 × 2 = 1 + 0.943 741 033 454 887 763 968;
  • 55) 0.943 741 033 454 887 763 968 × 2 = 1 + 0.887 482 066 909 775 527 936;
  • 56) 0.887 482 066 909 775 527 936 × 2 = 1 + 0.774 964 133 819 551 055 872;
  • 57) 0.774 964 133 819 551 055 872 × 2 = 1 + 0.549 928 267 639 102 111 744;
  • 58) 0.549 928 267 639 102 111 744 × 2 = 1 + 0.099 856 535 278 204 223 488;

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:


0.016 738 891 601 562 496 877(10) =

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

6. Positive number before normalization:

0.016 738 891 601 562 496 877(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:


0.016 738 891 601 562 496 877(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 -0.016 738 891 601 562 496 877 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