-0.016 738 891 601 562 496 774 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.016 738 891 601 562 496 774(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
-0.016 738 891 601 562 496 774(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:

|-0.016 738 891 601 562 496 774| = 0.016 738 891 601 562 496 774


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: 0.016 738 891 601 562 496 774.

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) 0.016 738 891 601 562 496 774 × 2 = 0 + 0.033 477 783 203 124 993 548;
  • 2) 0.033 477 783 203 124 993 548 × 2 = 0 + 0.066 955 566 406 249 987 096;
  • 3) 0.066 955 566 406 249 987 096 × 2 = 0 + 0.133 911 132 812 499 974 192;
  • 4) 0.133 911 132 812 499 974 192 × 2 = 0 + 0.267 822 265 624 999 948 384;
  • 5) 0.267 822 265 624 999 948 384 × 2 = 0 + 0.535 644 531 249 999 896 768;
  • 6) 0.535 644 531 249 999 896 768 × 2 = 1 + 0.071 289 062 499 999 793 536;
  • 7) 0.071 289 062 499 999 793 536 × 2 = 0 + 0.142 578 124 999 999 587 072;
  • 8) 0.142 578 124 999 999 587 072 × 2 = 0 + 0.285 156 249 999 999 174 144;
  • 9) 0.285 156 249 999 999 174 144 × 2 = 0 + 0.570 312 499 999 998 348 288;
  • 10) 0.570 312 499 999 998 348 288 × 2 = 1 + 0.140 624 999 999 996 696 576;
  • 11) 0.140 624 999 999 996 696 576 × 2 = 0 + 0.281 249 999 999 993 393 152;
  • 12) 0.281 249 999 999 993 393 152 × 2 = 0 + 0.562 499 999 999 986 786 304;
  • 13) 0.562 499 999 999 986 786 304 × 2 = 1 + 0.124 999 999 999 973 572 608;
  • 14) 0.124 999 999 999 973 572 608 × 2 = 0 + 0.249 999 999 999 947 145 216;
  • 15) 0.249 999 999 999 947 145 216 × 2 = 0 + 0.499 999 999 999 894 290 432;
  • 16) 0.499 999 999 999 894 290 432 × 2 = 0 + 0.999 999 999 999 788 580 864;
  • 17) 0.999 999 999 999 788 580 864 × 2 = 1 + 0.999 999 999 999 577 161 728;
  • 18) 0.999 999 999 999 577 161 728 × 2 = 1 + 0.999 999 999 999 154 323 456;
  • 19) 0.999 999 999 999 154 323 456 × 2 = 1 + 0.999 999 999 998 308 646 912;
  • 20) 0.999 999 999 998 308 646 912 × 2 = 1 + 0.999 999 999 996 617 293 824;
  • 21) 0.999 999 999 996 617 293 824 × 2 = 1 + 0.999 999 999 993 234 587 648;
  • 22) 0.999 999 999 993 234 587 648 × 2 = 1 + 0.999 999 999 986 469 175 296;
  • 23) 0.999 999 999 986 469 175 296 × 2 = 1 + 0.999 999 999 972 938 350 592;
  • 24) 0.999 999 999 972 938 350 592 × 2 = 1 + 0.999 999 999 945 876 701 184;
  • 25) 0.999 999 999 945 876 701 184 × 2 = 1 + 0.999 999 999 891 753 402 368;
  • 26) 0.999 999 999 891 753 402 368 × 2 = 1 + 0.999 999 999 783 506 804 736;
  • 27) 0.999 999 999 783 506 804 736 × 2 = 1 + 0.999 999 999 567 013 609 472;
  • 28) 0.999 999 999 567 013 609 472 × 2 = 1 + 0.999 999 999 134 027 218 944;
  • 29) 0.999 999 999 134 027 218 944 × 2 = 1 + 0.999 999 998 268 054 437 888;
  • 30) 0.999 999 998 268 054 437 888 × 2 = 1 + 0.999 999 996 536 108 875 776;
  • 31) 0.999 999 996 536 108 875 776 × 2 = 1 + 0.999 999 993 072 217 751 552;
  • 32) 0.999 999 993 072 217 751 552 × 2 = 1 + 0.999 999 986 144 435 503 104;
  • 33) 0.999 999 986 144 435 503 104 × 2 = 1 + 0.999 999 972 288 871 006 208;
  • 34) 0.999 999 972 288 871 006 208 × 2 = 1 + 0.999 999 944 577 742 012 416;
  • 35) 0.999 999 944 577 742 012 416 × 2 = 1 + 0.999 999 889 155 484 024 832;
  • 36) 0.999 999 889 155 484 024 832 × 2 = 1 + 0.999 999 778 310 968 049 664;
  • 37) 0.999 999 778 310 968 049 664 × 2 = 1 + 0.999 999 556 621 936 099 328;
  • 38) 0.999 999 556 621 936 099 328 × 2 = 1 + 0.999 999 113 243 872 198 656;
  • 39) 0.999 999 113 243 872 198 656 × 2 = 1 + 0.999 998 226 487 744 397 312;
  • 40) 0.999 998 226 487 744 397 312 × 2 = 1 + 0.999 996 452 975 488 794 624;
  • 41) 0.999 996 452 975 488 794 624 × 2 = 1 + 0.999 992 905 950 977 589 248;
  • 42) 0.999 992 905 950 977 589 248 × 2 = 1 + 0.999 985 811 901 955 178 496;
  • 43) 0.999 985 811 901 955 178 496 × 2 = 1 + 0.999 971 623 803 910 356 992;
  • 44) 0.999 971 623 803 910 356 992 × 2 = 1 + 0.999 943 247 607 820 713 984;
  • 45) 0.999 943 247 607 820 713 984 × 2 = 1 + 0.999 886 495 215 641 427 968;
  • 46) 0.999 886 495 215 641 427 968 × 2 = 1 + 0.999 772 990 431 282 855 936;
  • 47) 0.999 772 990 431 282 855 936 × 2 = 1 + 0.999 545 980 862 565 711 872;
  • 48) 0.999 545 980 862 565 711 872 × 2 = 1 + 0.999 091 961 725 131 423 744;
  • 49) 0.999 091 961 725 131 423 744 × 2 = 1 + 0.998 183 923 450 262 847 488;
  • 50) 0.998 183 923 450 262 847 488 × 2 = 1 + 0.996 367 846 900 525 694 976;
  • 51) 0.996 367 846 900 525 694 976 × 2 = 1 + 0.992 735 693 801 051 389 952;
  • 52) 0.992 735 693 801 051 389 952 × 2 = 1 + 0.985 471 387 602 102 779 904;
  • 53) 0.985 471 387 602 102 779 904 × 2 = 1 + 0.970 942 775 204 205 559 808;
  • 54) 0.970 942 775 204 205 559 808 × 2 = 1 + 0.941 885 550 408 411 119 616;
  • 55) 0.941 885 550 408 411 119 616 × 2 = 1 + 0.883 771 100 816 822 239 232;
  • 56) 0.883 771 100 816 822 239 232 × 2 = 1 + 0.767 542 201 633 644 478 464;
  • 57) 0.767 542 201 633 644 478 464 × 2 = 1 + 0.535 084 403 267 288 956 928;
  • 58) 0.535 084 403 267 288 956 928 × 2 = 1 + 0.070 168 806 534 577 913 856;

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:


0.016 738 891 601 562 496 774(10) =

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

6. Positive number before normalization:

0.016 738 891 601 562 496 774(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:


0.016 738 891 601 562 496 774(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 -0.016 738 891 601 562 496 774 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