-0.145 067 813 487 901 050 859 9 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.145 067 813 487 901 050 859 9(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
-0.145 067 813 487 901 050 859 9(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:

|-0.145 067 813 487 901 050 859 9| = 0.145 067 813 487 901 050 859 9


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: 0.145 067 813 487 901 050 859 9.

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) 0.145 067 813 487 901 050 859 9 × 2 = 0 + 0.290 135 626 975 802 101 719 8;
  • 2) 0.290 135 626 975 802 101 719 8 × 2 = 0 + 0.580 271 253 951 604 203 439 6;
  • 3) 0.580 271 253 951 604 203 439 6 × 2 = 1 + 0.160 542 507 903 208 406 879 2;
  • 4) 0.160 542 507 903 208 406 879 2 × 2 = 0 + 0.321 085 015 806 416 813 758 4;
  • 5) 0.321 085 015 806 416 813 758 4 × 2 = 0 + 0.642 170 031 612 833 627 516 8;
  • 6) 0.642 170 031 612 833 627 516 8 × 2 = 1 + 0.284 340 063 225 667 255 033 6;
  • 7) 0.284 340 063 225 667 255 033 6 × 2 = 0 + 0.568 680 126 451 334 510 067 2;
  • 8) 0.568 680 126 451 334 510 067 2 × 2 = 1 + 0.137 360 252 902 669 020 134 4;
  • 9) 0.137 360 252 902 669 020 134 4 × 2 = 0 + 0.274 720 505 805 338 040 268 8;
  • 10) 0.274 720 505 805 338 040 268 8 × 2 = 0 + 0.549 441 011 610 676 080 537 6;
  • 11) 0.549 441 011 610 676 080 537 6 × 2 = 1 + 0.098 882 023 221 352 161 075 2;
  • 12) 0.098 882 023 221 352 161 075 2 × 2 = 0 + 0.197 764 046 442 704 322 150 4;
  • 13) 0.197 764 046 442 704 322 150 4 × 2 = 0 + 0.395 528 092 885 408 644 300 8;
  • 14) 0.395 528 092 885 408 644 300 8 × 2 = 0 + 0.791 056 185 770 817 288 601 6;
  • 15) 0.791 056 185 770 817 288 601 6 × 2 = 1 + 0.582 112 371 541 634 577 203 2;
  • 16) 0.582 112 371 541 634 577 203 2 × 2 = 1 + 0.164 224 743 083 269 154 406 4;
  • 17) 0.164 224 743 083 269 154 406 4 × 2 = 0 + 0.328 449 486 166 538 308 812 8;
  • 18) 0.328 449 486 166 538 308 812 8 × 2 = 0 + 0.656 898 972 333 076 617 625 6;
  • 19) 0.656 898 972 333 076 617 625 6 × 2 = 1 + 0.313 797 944 666 153 235 251 2;
  • 20) 0.313 797 944 666 153 235 251 2 × 2 = 0 + 0.627 595 889 332 306 470 502 4;
  • 21) 0.627 595 889 332 306 470 502 4 × 2 = 1 + 0.255 191 778 664 612 941 004 8;
  • 22) 0.255 191 778 664 612 941 004 8 × 2 = 0 + 0.510 383 557 329 225 882 009 6;
  • 23) 0.510 383 557 329 225 882 009 6 × 2 = 1 + 0.020 767 114 658 451 764 019 2;
  • 24) 0.020 767 114 658 451 764 019 2 × 2 = 0 + 0.041 534 229 316 903 528 038 4;
  • 25) 0.041 534 229 316 903 528 038 4 × 2 = 0 + 0.083 068 458 633 807 056 076 8;
  • 26) 0.083 068 458 633 807 056 076 8 × 2 = 0 + 0.166 136 917 267 614 112 153 6;
  • 27) 0.166 136 917 267 614 112 153 6 × 2 = 0 + 0.332 273 834 535 228 224 307 2;
  • 28) 0.332 273 834 535 228 224 307 2 × 2 = 0 + 0.664 547 669 070 456 448 614 4;
  • 29) 0.664 547 669 070 456 448 614 4 × 2 = 1 + 0.329 095 338 140 912 897 228 8;
  • 30) 0.329 095 338 140 912 897 228 8 × 2 = 0 + 0.658 190 676 281 825 794 457 6;
  • 31) 0.658 190 676 281 825 794 457 6 × 2 = 1 + 0.316 381 352 563 651 588 915 2;
  • 32) 0.316 381 352 563 651 588 915 2 × 2 = 0 + 0.632 762 705 127 303 177 830 4;
  • 33) 0.632 762 705 127 303 177 830 4 × 2 = 1 + 0.265 525 410 254 606 355 660 8;
  • 34) 0.265 525 410 254 606 355 660 8 × 2 = 0 + 0.531 050 820 509 212 711 321 6;
  • 35) 0.531 050 820 509 212 711 321 6 × 2 = 1 + 0.062 101 641 018 425 422 643 2;
  • 36) 0.062 101 641 018 425 422 643 2 × 2 = 0 + 0.124 203 282 036 850 845 286 4;
  • 37) 0.124 203 282 036 850 845 286 4 × 2 = 0 + 0.248 406 564 073 701 690 572 8;
  • 38) 0.248 406 564 073 701 690 572 8 × 2 = 0 + 0.496 813 128 147 403 381 145 6;
  • 39) 0.496 813 128 147 403 381 145 6 × 2 = 0 + 0.993 626 256 294 806 762 291 2;
  • 40) 0.993 626 256 294 806 762 291 2 × 2 = 1 + 0.987 252 512 589 613 524 582 4;
  • 41) 0.987 252 512 589 613 524 582 4 × 2 = 1 + 0.974 505 025 179 227 049 164 8;
  • 42) 0.974 505 025 179 227 049 164 8 × 2 = 1 + 0.949 010 050 358 454 098 329 6;
  • 43) 0.949 010 050 358 454 098 329 6 × 2 = 1 + 0.898 020 100 716 908 196 659 2;
  • 44) 0.898 020 100 716 908 196 659 2 × 2 = 1 + 0.796 040 201 433 816 393 318 4;
  • 45) 0.796 040 201 433 816 393 318 4 × 2 = 1 + 0.592 080 402 867 632 786 636 8;
  • 46) 0.592 080 402 867 632 786 636 8 × 2 = 1 + 0.184 160 805 735 265 573 273 6;
  • 47) 0.184 160 805 735 265 573 273 6 × 2 = 0 + 0.368 321 611 470 531 146 547 2;
  • 48) 0.368 321 611 470 531 146 547 2 × 2 = 0 + 0.736 643 222 941 062 293 094 4;
  • 49) 0.736 643 222 941 062 293 094 4 × 2 = 1 + 0.473 286 445 882 124 586 188 8;
  • 50) 0.473 286 445 882 124 586 188 8 × 2 = 0 + 0.946 572 891 764 249 172 377 6;
  • 51) 0.946 572 891 764 249 172 377 6 × 2 = 1 + 0.893 145 783 528 498 344 755 2;
  • 52) 0.893 145 783 528 498 344 755 2 × 2 = 1 + 0.786 291 567 056 996 689 510 4;
  • 53) 0.786 291 567 056 996 689 510 4 × 2 = 1 + 0.572 583 134 113 993 379 020 8;
  • 54) 0.572 583 134 113 993 379 020 8 × 2 = 1 + 0.145 166 268 227 986 758 041 6;
  • 55) 0.145 166 268 227 986 758 041 6 × 2 = 0 + 0.290 332 536 455 973 516 083 2;

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:


0.145 067 813 487 901 050 859 9(10) =

0.0010 0101 0010 0011 0010 1010 0000 1010 1010 0001 1111 1100 1011 110(2)

6. Positive number before normalization:

0.145 067 813 487 901 050 859 9(10) =

0.0010 0101 0010 0011 0010 1010 0000 1010 1010 0001 1111 1100 1011 110(2)

7. Normalize the binary representation of the number.

Shift the decimal mark 3 positions to the right, so that only one non zero digit remains to the left of it:


0.145 067 813 487 901 050 859 9(10) =

0.0010 0101 0010 0011 0010 1010 0000 1010 1010 0001 1111 1100 1011 110(2) =

0.0010 0101 0010 0011 0010 1010 0000 1010 1010 0001 1111 1100 1011 110(2) × 20 =

1.0010 1001 0001 1001 0101 0000 0101 0101 0000 1111 1110 0101 1110(2) × 2-3


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): -3

Mantissa (not normalized):
1.0010 1001 0001 1001 0101 0000 0101 0101 0000 1111 1110 0101 1110


9. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

Exponent (unadjusted) + 2(11-1) - 1 =

-3 + 2(11-1) - 1 =

(-3 + 1 023)(10) =

1 020(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;
  • 1 020 ÷ 2 = 510 + 0;
  • 510 ÷ 2 = 255 + 0;
  • 255 ÷ 2 = 127 + 1;
  • 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) =

1020(10) =

011 1111 1100(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. 0010 1001 0001 1001 0101 0000 0101 0101 0000 1111 1110 0101 1110 =

0010 1001 0001 1001 0101 0000 0101 0101 0000 1111 1110 0101 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 1100

Mantissa (52 bits) =
0010 1001 0001 1001 0101 0000 0101 0101 0000 1111 1110 0101 1110


Decimal number -0.145 067 813 487 901 050 859 9 converted to 64 bit double precision IEEE 754 binary floating point representation:

1 - 011 1111 1100 - 0010 1001 0001 1001 0101 0000 0101 0101 0000 1111 1110 0101 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