-0.016 738 891 601 562 496 530 553 22 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.016 738 891 601 562 496 530 553 22(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 22(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 22| = 0.016 738 891 601 562 496 530 553 22


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 22.

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 22 × 2 = 0 + 0.033 477 783 203 124 993 061 106 44;
  • 2) 0.033 477 783 203 124 993 061 106 44 × 2 = 0 + 0.066 955 566 406 249 986 122 212 88;
  • 3) 0.066 955 566 406 249 986 122 212 88 × 2 = 0 + 0.133 911 132 812 499 972 244 425 76;
  • 4) 0.133 911 132 812 499 972 244 425 76 × 2 = 0 + 0.267 822 265 624 999 944 488 851 52;
  • 5) 0.267 822 265 624 999 944 488 851 52 × 2 = 0 + 0.535 644 531 249 999 888 977 703 04;
  • 6) 0.535 644 531 249 999 888 977 703 04 × 2 = 1 + 0.071 289 062 499 999 777 955 406 08;
  • 7) 0.071 289 062 499 999 777 955 406 08 × 2 = 0 + 0.142 578 124 999 999 555 910 812 16;
  • 8) 0.142 578 124 999 999 555 910 812 16 × 2 = 0 + 0.285 156 249 999 999 111 821 624 32;
  • 9) 0.285 156 249 999 999 111 821 624 32 × 2 = 0 + 0.570 312 499 999 998 223 643 248 64;
  • 10) 0.570 312 499 999 998 223 643 248 64 × 2 = 1 + 0.140 624 999 999 996 447 286 497 28;
  • 11) 0.140 624 999 999 996 447 286 497 28 × 2 = 0 + 0.281 249 999 999 992 894 572 994 56;
  • 12) 0.281 249 999 999 992 894 572 994 56 × 2 = 0 + 0.562 499 999 999 985 789 145 989 12;
  • 13) 0.562 499 999 999 985 789 145 989 12 × 2 = 1 + 0.124 999 999 999 971 578 291 978 24;
  • 14) 0.124 999 999 999 971 578 291 978 24 × 2 = 0 + 0.249 999 999 999 943 156 583 956 48;
  • 15) 0.249 999 999 999 943 156 583 956 48 × 2 = 0 + 0.499 999 999 999 886 313 167 912 96;
  • 16) 0.499 999 999 999 886 313 167 912 96 × 2 = 0 + 0.999 999 999 999 772 626 335 825 92;
  • 17) 0.999 999 999 999 772 626 335 825 92 × 2 = 1 + 0.999 999 999 999 545 252 671 651 84;
  • 18) 0.999 999 999 999 545 252 671 651 84 × 2 = 1 + 0.999 999 999 999 090 505 343 303 68;
  • 19) 0.999 999 999 999 090 505 343 303 68 × 2 = 1 + 0.999 999 999 998 181 010 686 607 36;
  • 20) 0.999 999 999 998 181 010 686 607 36 × 2 = 1 + 0.999 999 999 996 362 021 373 214 72;
  • 21) 0.999 999 999 996 362 021 373 214 72 × 2 = 1 + 0.999 999 999 992 724 042 746 429 44;
  • 22) 0.999 999 999 992 724 042 746 429 44 × 2 = 1 + 0.999 999 999 985 448 085 492 858 88;
  • 23) 0.999 999 999 985 448 085 492 858 88 × 2 = 1 + 0.999 999 999 970 896 170 985 717 76;
  • 24) 0.999 999 999 970 896 170 985 717 76 × 2 = 1 + 0.999 999 999 941 792 341 971 435 52;
  • 25) 0.999 999 999 941 792 341 971 435 52 × 2 = 1 + 0.999 999 999 883 584 683 942 871 04;
  • 26) 0.999 999 999 883 584 683 942 871 04 × 2 = 1 + 0.999 999 999 767 169 367 885 742 08;
  • 27) 0.999 999 999 767 169 367 885 742 08 × 2 = 1 + 0.999 999 999 534 338 735 771 484 16;
  • 28) 0.999 999 999 534 338 735 771 484 16 × 2 = 1 + 0.999 999 999 068 677 471 542 968 32;
  • 29) 0.999 999 999 068 677 471 542 968 32 × 2 = 1 + 0.999 999 998 137 354 943 085 936 64;
  • 30) 0.999 999 998 137 354 943 085 936 64 × 2 = 1 + 0.999 999 996 274 709 886 171 873 28;
  • 31) 0.999 999 996 274 709 886 171 873 28 × 2 = 1 + 0.999 999 992 549 419 772 343 746 56;
  • 32) 0.999 999 992 549 419 772 343 746 56 × 2 = 1 + 0.999 999 985 098 839 544 687 493 12;
  • 33) 0.999 999 985 098 839 544 687 493 12 × 2 = 1 + 0.999 999 970 197 679 089 374 986 24;
  • 34) 0.999 999 970 197 679 089 374 986 24 × 2 = 1 + 0.999 999 940 395 358 178 749 972 48;
  • 35) 0.999 999 940 395 358 178 749 972 48 × 2 = 1 + 0.999 999 880 790 716 357 499 944 96;
  • 36) 0.999 999 880 790 716 357 499 944 96 × 2 = 1 + 0.999 999 761 581 432 714 999 889 92;
  • 37) 0.999 999 761 581 432 714 999 889 92 × 2 = 1 + 0.999 999 523 162 865 429 999 779 84;
  • 38) 0.999 999 523 162 865 429 999 779 84 × 2 = 1 + 0.999 999 046 325 730 859 999 559 68;
  • 39) 0.999 999 046 325 730 859 999 559 68 × 2 = 1 + 0.999 998 092 651 461 719 999 119 36;
  • 40) 0.999 998 092 651 461 719 999 119 36 × 2 = 1 + 0.999 996 185 302 923 439 998 238 72;
  • 41) 0.999 996 185 302 923 439 998 238 72 × 2 = 1 + 0.999 992 370 605 846 879 996 477 44;
  • 42) 0.999 992 370 605 846 879 996 477 44 × 2 = 1 + 0.999 984 741 211 693 759 992 954 88;
  • 43) 0.999 984 741 211 693 759 992 954 88 × 2 = 1 + 0.999 969 482 423 387 519 985 909 76;
  • 44) 0.999 969 482 423 387 519 985 909 76 × 2 = 1 + 0.999 938 964 846 775 039 971 819 52;
  • 45) 0.999 938 964 846 775 039 971 819 52 × 2 = 1 + 0.999 877 929 693 550 079 943 639 04;
  • 46) 0.999 877 929 693 550 079 943 639 04 × 2 = 1 + 0.999 755 859 387 100 159 887 278 08;
  • 47) 0.999 755 859 387 100 159 887 278 08 × 2 = 1 + 0.999 511 718 774 200 319 774 556 16;
  • 48) 0.999 511 718 774 200 319 774 556 16 × 2 = 1 + 0.999 023 437 548 400 639 549 112 32;
  • 49) 0.999 023 437 548 400 639 549 112 32 × 2 = 1 + 0.998 046 875 096 801 279 098 224 64;
  • 50) 0.998 046 875 096 801 279 098 224 64 × 2 = 1 + 0.996 093 750 193 602 558 196 449 28;
  • 51) 0.996 093 750 193 602 558 196 449 28 × 2 = 1 + 0.992 187 500 387 205 116 392 898 56;
  • 52) 0.992 187 500 387 205 116 392 898 56 × 2 = 1 + 0.984 375 000 774 410 232 785 797 12;
  • 53) 0.984 375 000 774 410 232 785 797 12 × 2 = 1 + 0.968 750 001 548 820 465 571 594 24;
  • 54) 0.968 750 001 548 820 465 571 594 24 × 2 = 1 + 0.937 500 003 097 640 931 143 188 48;
  • 55) 0.937 500 003 097 640 931 143 188 48 × 2 = 1 + 0.875 000 006 195 281 862 286 376 96;
  • 56) 0.875 000 006 195 281 862 286 376 96 × 2 = 1 + 0.750 000 012 390 563 724 572 753 92;
  • 57) 0.750 000 012 390 563 724 572 753 92 × 2 = 1 + 0.500 000 024 781 127 449 145 507 84;
  • 58) 0.500 000 024 781 127 449 145 507 84 × 2 = 1 + 0.000 000 049 562 254 898 291 015 68;

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 22(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 22(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 22(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 22 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