-0.016 738 891 601 562 496 530 553 096 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.016 738 891 601 562 496 530 553 096(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
-0.016 738 891 601 562 496 530 553 096(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:

|-0.016 738 891 601 562 496 530 553 096| = 0.016 738 891 601 562 496 530 553 096


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: 0.016 738 891 601 562 496 530 553 096.

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) 0.016 738 891 601 562 496 530 553 096 × 2 = 0 + 0.033 477 783 203 124 993 061 106 192;
  • 2) 0.033 477 783 203 124 993 061 106 192 × 2 = 0 + 0.066 955 566 406 249 986 122 212 384;
  • 3) 0.066 955 566 406 249 986 122 212 384 × 2 = 0 + 0.133 911 132 812 499 972 244 424 768;
  • 4) 0.133 911 132 812 499 972 244 424 768 × 2 = 0 + 0.267 822 265 624 999 944 488 849 536;
  • 5) 0.267 822 265 624 999 944 488 849 536 × 2 = 0 + 0.535 644 531 249 999 888 977 699 072;
  • 6) 0.535 644 531 249 999 888 977 699 072 × 2 = 1 + 0.071 289 062 499 999 777 955 398 144;
  • 7) 0.071 289 062 499 999 777 955 398 144 × 2 = 0 + 0.142 578 124 999 999 555 910 796 288;
  • 8) 0.142 578 124 999 999 555 910 796 288 × 2 = 0 + 0.285 156 249 999 999 111 821 592 576;
  • 9) 0.285 156 249 999 999 111 821 592 576 × 2 = 0 + 0.570 312 499 999 998 223 643 185 152;
  • 10) 0.570 312 499 999 998 223 643 185 152 × 2 = 1 + 0.140 624 999 999 996 447 286 370 304;
  • 11) 0.140 624 999 999 996 447 286 370 304 × 2 = 0 + 0.281 249 999 999 992 894 572 740 608;
  • 12) 0.281 249 999 999 992 894 572 740 608 × 2 = 0 + 0.562 499 999 999 985 789 145 481 216;
  • 13) 0.562 499 999 999 985 789 145 481 216 × 2 = 1 + 0.124 999 999 999 971 578 290 962 432;
  • 14) 0.124 999 999 999 971 578 290 962 432 × 2 = 0 + 0.249 999 999 999 943 156 581 924 864;
  • 15) 0.249 999 999 999 943 156 581 924 864 × 2 = 0 + 0.499 999 999 999 886 313 163 849 728;
  • 16) 0.499 999 999 999 886 313 163 849 728 × 2 = 0 + 0.999 999 999 999 772 626 327 699 456;
  • 17) 0.999 999 999 999 772 626 327 699 456 × 2 = 1 + 0.999 999 999 999 545 252 655 398 912;
  • 18) 0.999 999 999 999 545 252 655 398 912 × 2 = 1 + 0.999 999 999 999 090 505 310 797 824;
  • 19) 0.999 999 999 999 090 505 310 797 824 × 2 = 1 + 0.999 999 999 998 181 010 621 595 648;
  • 20) 0.999 999 999 998 181 010 621 595 648 × 2 = 1 + 0.999 999 999 996 362 021 243 191 296;
  • 21) 0.999 999 999 996 362 021 243 191 296 × 2 = 1 + 0.999 999 999 992 724 042 486 382 592;
  • 22) 0.999 999 999 992 724 042 486 382 592 × 2 = 1 + 0.999 999 999 985 448 084 972 765 184;
  • 23) 0.999 999 999 985 448 084 972 765 184 × 2 = 1 + 0.999 999 999 970 896 169 945 530 368;
  • 24) 0.999 999 999 970 896 169 945 530 368 × 2 = 1 + 0.999 999 999 941 792 339 891 060 736;
  • 25) 0.999 999 999 941 792 339 891 060 736 × 2 = 1 + 0.999 999 999 883 584 679 782 121 472;
  • 26) 0.999 999 999 883 584 679 782 121 472 × 2 = 1 + 0.999 999 999 767 169 359 564 242 944;
  • 27) 0.999 999 999 767 169 359 564 242 944 × 2 = 1 + 0.999 999 999 534 338 719 128 485 888;
  • 28) 0.999 999 999 534 338 719 128 485 888 × 2 = 1 + 0.999 999 999 068 677 438 256 971 776;
  • 29) 0.999 999 999 068 677 438 256 971 776 × 2 = 1 + 0.999 999 998 137 354 876 513 943 552;
  • 30) 0.999 999 998 137 354 876 513 943 552 × 2 = 1 + 0.999 999 996 274 709 753 027 887 104;
  • 31) 0.999 999 996 274 709 753 027 887 104 × 2 = 1 + 0.999 999 992 549 419 506 055 774 208;
  • 32) 0.999 999 992 549 419 506 055 774 208 × 2 = 1 + 0.999 999 985 098 839 012 111 548 416;
  • 33) 0.999 999 985 098 839 012 111 548 416 × 2 = 1 + 0.999 999 970 197 678 024 223 096 832;
  • 34) 0.999 999 970 197 678 024 223 096 832 × 2 = 1 + 0.999 999 940 395 356 048 446 193 664;
  • 35) 0.999 999 940 395 356 048 446 193 664 × 2 = 1 + 0.999 999 880 790 712 096 892 387 328;
  • 36) 0.999 999 880 790 712 096 892 387 328 × 2 = 1 + 0.999 999 761 581 424 193 784 774 656;
  • 37) 0.999 999 761 581 424 193 784 774 656 × 2 = 1 + 0.999 999 523 162 848 387 569 549 312;
  • 38) 0.999 999 523 162 848 387 569 549 312 × 2 = 1 + 0.999 999 046 325 696 775 139 098 624;
  • 39) 0.999 999 046 325 696 775 139 098 624 × 2 = 1 + 0.999 998 092 651 393 550 278 197 248;
  • 40) 0.999 998 092 651 393 550 278 197 248 × 2 = 1 + 0.999 996 185 302 787 100 556 394 496;
  • 41) 0.999 996 185 302 787 100 556 394 496 × 2 = 1 + 0.999 992 370 605 574 201 112 788 992;
  • 42) 0.999 992 370 605 574 201 112 788 992 × 2 = 1 + 0.999 984 741 211 148 402 225 577 984;
  • 43) 0.999 984 741 211 148 402 225 577 984 × 2 = 1 + 0.999 969 482 422 296 804 451 155 968;
  • 44) 0.999 969 482 422 296 804 451 155 968 × 2 = 1 + 0.999 938 964 844 593 608 902 311 936;
  • 45) 0.999 938 964 844 593 608 902 311 936 × 2 = 1 + 0.999 877 929 689 187 217 804 623 872;
  • 46) 0.999 877 929 689 187 217 804 623 872 × 2 = 1 + 0.999 755 859 378 374 435 609 247 744;
  • 47) 0.999 755 859 378 374 435 609 247 744 × 2 = 1 + 0.999 511 718 756 748 871 218 495 488;
  • 48) 0.999 511 718 756 748 871 218 495 488 × 2 = 1 + 0.999 023 437 513 497 742 436 990 976;
  • 49) 0.999 023 437 513 497 742 436 990 976 × 2 = 1 + 0.998 046 875 026 995 484 873 981 952;
  • 50) 0.998 046 875 026 995 484 873 981 952 × 2 = 1 + 0.996 093 750 053 990 969 747 963 904;
  • 51) 0.996 093 750 053 990 969 747 963 904 × 2 = 1 + 0.992 187 500 107 981 939 495 927 808;
  • 52) 0.992 187 500 107 981 939 495 927 808 × 2 = 1 + 0.984 375 000 215 963 878 991 855 616;
  • 53) 0.984 375 000 215 963 878 991 855 616 × 2 = 1 + 0.968 750 000 431 927 757 983 711 232;
  • 54) 0.968 750 000 431 927 757 983 711 232 × 2 = 1 + 0.937 500 000 863 855 515 967 422 464;
  • 55) 0.937 500 000 863 855 515 967 422 464 × 2 = 1 + 0.875 000 001 727 711 031 934 844 928;
  • 56) 0.875 000 001 727 711 031 934 844 928 × 2 = 1 + 0.750 000 003 455 422 063 869 689 856;
  • 57) 0.750 000 003 455 422 063 869 689 856 × 2 = 1 + 0.500 000 006 910 844 127 739 379 712;
  • 58) 0.500 000 006 910 844 127 739 379 712 × 2 = 1 + 0.000 000 013 821 688 255 478 759 424;

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:


0.016 738 891 601 562 496 530 553 096(10) =

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

6. Positive number before normalization:

0.016 738 891 601 562 496 530 553 096(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:


0.016 738 891 601 562 496 530 553 096(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 -0.016 738 891 601 562 496 530 553 096 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