-0.016 738 891 601 562 496 530 438 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.016 738 891 601 562 496 530 438(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
-0.016 738 891 601 562 496 530 438(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:

|-0.016 738 891 601 562 496 530 438| = 0.016 738 891 601 562 496 530 438


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: 0.016 738 891 601 562 496 530 438.

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) 0.016 738 891 601 562 496 530 438 × 2 = 0 + 0.033 477 783 203 124 993 060 876;
  • 2) 0.033 477 783 203 124 993 060 876 × 2 = 0 + 0.066 955 566 406 249 986 121 752;
  • 3) 0.066 955 566 406 249 986 121 752 × 2 = 0 + 0.133 911 132 812 499 972 243 504;
  • 4) 0.133 911 132 812 499 972 243 504 × 2 = 0 + 0.267 822 265 624 999 944 487 008;
  • 5) 0.267 822 265 624 999 944 487 008 × 2 = 0 + 0.535 644 531 249 999 888 974 016;
  • 6) 0.535 644 531 249 999 888 974 016 × 2 = 1 + 0.071 289 062 499 999 777 948 032;
  • 7) 0.071 289 062 499 999 777 948 032 × 2 = 0 + 0.142 578 124 999 999 555 896 064;
  • 8) 0.142 578 124 999 999 555 896 064 × 2 = 0 + 0.285 156 249 999 999 111 792 128;
  • 9) 0.285 156 249 999 999 111 792 128 × 2 = 0 + 0.570 312 499 999 998 223 584 256;
  • 10) 0.570 312 499 999 998 223 584 256 × 2 = 1 + 0.140 624 999 999 996 447 168 512;
  • 11) 0.140 624 999 999 996 447 168 512 × 2 = 0 + 0.281 249 999 999 992 894 337 024;
  • 12) 0.281 249 999 999 992 894 337 024 × 2 = 0 + 0.562 499 999 999 985 788 674 048;
  • 13) 0.562 499 999 999 985 788 674 048 × 2 = 1 + 0.124 999 999 999 971 577 348 096;
  • 14) 0.124 999 999 999 971 577 348 096 × 2 = 0 + 0.249 999 999 999 943 154 696 192;
  • 15) 0.249 999 999 999 943 154 696 192 × 2 = 0 + 0.499 999 999 999 886 309 392 384;
  • 16) 0.499 999 999 999 886 309 392 384 × 2 = 0 + 0.999 999 999 999 772 618 784 768;
  • 17) 0.999 999 999 999 772 618 784 768 × 2 = 1 + 0.999 999 999 999 545 237 569 536;
  • 18) 0.999 999 999 999 545 237 569 536 × 2 = 1 + 0.999 999 999 999 090 475 139 072;
  • 19) 0.999 999 999 999 090 475 139 072 × 2 = 1 + 0.999 999 999 998 180 950 278 144;
  • 20) 0.999 999 999 998 180 950 278 144 × 2 = 1 + 0.999 999 999 996 361 900 556 288;
  • 21) 0.999 999 999 996 361 900 556 288 × 2 = 1 + 0.999 999 999 992 723 801 112 576;
  • 22) 0.999 999 999 992 723 801 112 576 × 2 = 1 + 0.999 999 999 985 447 602 225 152;
  • 23) 0.999 999 999 985 447 602 225 152 × 2 = 1 + 0.999 999 999 970 895 204 450 304;
  • 24) 0.999 999 999 970 895 204 450 304 × 2 = 1 + 0.999 999 999 941 790 408 900 608;
  • 25) 0.999 999 999 941 790 408 900 608 × 2 = 1 + 0.999 999 999 883 580 817 801 216;
  • 26) 0.999 999 999 883 580 817 801 216 × 2 = 1 + 0.999 999 999 767 161 635 602 432;
  • 27) 0.999 999 999 767 161 635 602 432 × 2 = 1 + 0.999 999 999 534 323 271 204 864;
  • 28) 0.999 999 999 534 323 271 204 864 × 2 = 1 + 0.999 999 999 068 646 542 409 728;
  • 29) 0.999 999 999 068 646 542 409 728 × 2 = 1 + 0.999 999 998 137 293 084 819 456;
  • 30) 0.999 999 998 137 293 084 819 456 × 2 = 1 + 0.999 999 996 274 586 169 638 912;
  • 31) 0.999 999 996 274 586 169 638 912 × 2 = 1 + 0.999 999 992 549 172 339 277 824;
  • 32) 0.999 999 992 549 172 339 277 824 × 2 = 1 + 0.999 999 985 098 344 678 555 648;
  • 33) 0.999 999 985 098 344 678 555 648 × 2 = 1 + 0.999 999 970 196 689 357 111 296;
  • 34) 0.999 999 970 196 689 357 111 296 × 2 = 1 + 0.999 999 940 393 378 714 222 592;
  • 35) 0.999 999 940 393 378 714 222 592 × 2 = 1 + 0.999 999 880 786 757 428 445 184;
  • 36) 0.999 999 880 786 757 428 445 184 × 2 = 1 + 0.999 999 761 573 514 856 890 368;
  • 37) 0.999 999 761 573 514 856 890 368 × 2 = 1 + 0.999 999 523 147 029 713 780 736;
  • 38) 0.999 999 523 147 029 713 780 736 × 2 = 1 + 0.999 999 046 294 059 427 561 472;
  • 39) 0.999 999 046 294 059 427 561 472 × 2 = 1 + 0.999 998 092 588 118 855 122 944;
  • 40) 0.999 998 092 588 118 855 122 944 × 2 = 1 + 0.999 996 185 176 237 710 245 888;
  • 41) 0.999 996 185 176 237 710 245 888 × 2 = 1 + 0.999 992 370 352 475 420 491 776;
  • 42) 0.999 992 370 352 475 420 491 776 × 2 = 1 + 0.999 984 740 704 950 840 983 552;
  • 43) 0.999 984 740 704 950 840 983 552 × 2 = 1 + 0.999 969 481 409 901 681 967 104;
  • 44) 0.999 969 481 409 901 681 967 104 × 2 = 1 + 0.999 938 962 819 803 363 934 208;
  • 45) 0.999 938 962 819 803 363 934 208 × 2 = 1 + 0.999 877 925 639 606 727 868 416;
  • 46) 0.999 877 925 639 606 727 868 416 × 2 = 1 + 0.999 755 851 279 213 455 736 832;
  • 47) 0.999 755 851 279 213 455 736 832 × 2 = 1 + 0.999 511 702 558 426 911 473 664;
  • 48) 0.999 511 702 558 426 911 473 664 × 2 = 1 + 0.999 023 405 116 853 822 947 328;
  • 49) 0.999 023 405 116 853 822 947 328 × 2 = 1 + 0.998 046 810 233 707 645 894 656;
  • 50) 0.998 046 810 233 707 645 894 656 × 2 = 1 + 0.996 093 620 467 415 291 789 312;
  • 51) 0.996 093 620 467 415 291 789 312 × 2 = 1 + 0.992 187 240 934 830 583 578 624;
  • 52) 0.992 187 240 934 830 583 578 624 × 2 = 1 + 0.984 374 481 869 661 167 157 248;
  • 53) 0.984 374 481 869 661 167 157 248 × 2 = 1 + 0.968 748 963 739 322 334 314 496;
  • 54) 0.968 748 963 739 322 334 314 496 × 2 = 1 + 0.937 497 927 478 644 668 628 992;
  • 55) 0.937 497 927 478 644 668 628 992 × 2 = 1 + 0.874 995 854 957 289 337 257 984;
  • 56) 0.874 995 854 957 289 337 257 984 × 2 = 1 + 0.749 991 709 914 578 674 515 968;
  • 57) 0.749 991 709 914 578 674 515 968 × 2 = 1 + 0.499 983 419 829 157 349 031 936;
  • 58) 0.499 983 419 829 157 349 031 936 × 2 = 0 + 0.999 966 839 658 314 698 063 872;

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:


0.016 738 891 601 562 496 530 438(10) =

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

6. Positive number before normalization:

0.016 738 891 601 562 496 530 438(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:


0.016 738 891 601 562 496 530 438(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 -0.016 738 891 601 562 496 530 438 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