6.999 999 999 999 999 555 910 848 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 6.999 999 999 999 999 555 910 848(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
6.999 999 999 999 999 555 910 848(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. First, convert to binary (in base 2) the integer part: 6.
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;
  • 6 ÷ 2 = 3 + 0;
  • 3 ÷ 2 = 1 + 1;
  • 1 ÷ 2 = 0 + 1;

2. 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.

6(10) =

110(2)


3. Convert to binary (base 2) the fractional part: 0.999 999 999 999 999 555 910 848.

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) 0.999 999 999 999 999 555 910 848 × 2 = 1 + 0.999 999 999 999 999 111 821 696;
  • 2) 0.999 999 999 999 999 111 821 696 × 2 = 1 + 0.999 999 999 999 998 223 643 392;
  • 3) 0.999 999 999 999 998 223 643 392 × 2 = 1 + 0.999 999 999 999 996 447 286 784;
  • 4) 0.999 999 999 999 996 447 286 784 × 2 = 1 + 0.999 999 999 999 992 894 573 568;
  • 5) 0.999 999 999 999 992 894 573 568 × 2 = 1 + 0.999 999 999 999 985 789 147 136;
  • 6) 0.999 999 999 999 985 789 147 136 × 2 = 1 + 0.999 999 999 999 971 578 294 272;
  • 7) 0.999 999 999 999 971 578 294 272 × 2 = 1 + 0.999 999 999 999 943 156 588 544;
  • 8) 0.999 999 999 999 943 156 588 544 × 2 = 1 + 0.999 999 999 999 886 313 177 088;
  • 9) 0.999 999 999 999 886 313 177 088 × 2 = 1 + 0.999 999 999 999 772 626 354 176;
  • 10) 0.999 999 999 999 772 626 354 176 × 2 = 1 + 0.999 999 999 999 545 252 708 352;
  • 11) 0.999 999 999 999 545 252 708 352 × 2 = 1 + 0.999 999 999 999 090 505 416 704;
  • 12) 0.999 999 999 999 090 505 416 704 × 2 = 1 + 0.999 999 999 998 181 010 833 408;
  • 13) 0.999 999 999 998 181 010 833 408 × 2 = 1 + 0.999 999 999 996 362 021 666 816;
  • 14) 0.999 999 999 996 362 021 666 816 × 2 = 1 + 0.999 999 999 992 724 043 333 632;
  • 15) 0.999 999 999 992 724 043 333 632 × 2 = 1 + 0.999 999 999 985 448 086 667 264;
  • 16) 0.999 999 999 985 448 086 667 264 × 2 = 1 + 0.999 999 999 970 896 173 334 528;
  • 17) 0.999 999 999 970 896 173 334 528 × 2 = 1 + 0.999 999 999 941 792 346 669 056;
  • 18) 0.999 999 999 941 792 346 669 056 × 2 = 1 + 0.999 999 999 883 584 693 338 112;
  • 19) 0.999 999 999 883 584 693 338 112 × 2 = 1 + 0.999 999 999 767 169 386 676 224;
  • 20) 0.999 999 999 767 169 386 676 224 × 2 = 1 + 0.999 999 999 534 338 773 352 448;
  • 21) 0.999 999 999 534 338 773 352 448 × 2 = 1 + 0.999 999 999 068 677 546 704 896;
  • 22) 0.999 999 999 068 677 546 704 896 × 2 = 1 + 0.999 999 998 137 355 093 409 792;
  • 23) 0.999 999 998 137 355 093 409 792 × 2 = 1 + 0.999 999 996 274 710 186 819 584;
  • 24) 0.999 999 996 274 710 186 819 584 × 2 = 1 + 0.999 999 992 549 420 373 639 168;
  • 25) 0.999 999 992 549 420 373 639 168 × 2 = 1 + 0.999 999 985 098 840 747 278 336;
  • 26) 0.999 999 985 098 840 747 278 336 × 2 = 1 + 0.999 999 970 197 681 494 556 672;
  • 27) 0.999 999 970 197 681 494 556 672 × 2 = 1 + 0.999 999 940 395 362 989 113 344;
  • 28) 0.999 999 940 395 362 989 113 344 × 2 = 1 + 0.999 999 880 790 725 978 226 688;
  • 29) 0.999 999 880 790 725 978 226 688 × 2 = 1 + 0.999 999 761 581 451 956 453 376;
  • 30) 0.999 999 761 581 451 956 453 376 × 2 = 1 + 0.999 999 523 162 903 912 906 752;
  • 31) 0.999 999 523 162 903 912 906 752 × 2 = 1 + 0.999 999 046 325 807 825 813 504;
  • 32) 0.999 999 046 325 807 825 813 504 × 2 = 1 + 0.999 998 092 651 615 651 627 008;
  • 33) 0.999 998 092 651 615 651 627 008 × 2 = 1 + 0.999 996 185 303 231 303 254 016;
  • 34) 0.999 996 185 303 231 303 254 016 × 2 = 1 + 0.999 992 370 606 462 606 508 032;
  • 35) 0.999 992 370 606 462 606 508 032 × 2 = 1 + 0.999 984 741 212 925 213 016 064;
  • 36) 0.999 984 741 212 925 213 016 064 × 2 = 1 + 0.999 969 482 425 850 426 032 128;
  • 37) 0.999 969 482 425 850 426 032 128 × 2 = 1 + 0.999 938 964 851 700 852 064 256;
  • 38) 0.999 938 964 851 700 852 064 256 × 2 = 1 + 0.999 877 929 703 401 704 128 512;
  • 39) 0.999 877 929 703 401 704 128 512 × 2 = 1 + 0.999 755 859 406 803 408 257 024;
  • 40) 0.999 755 859 406 803 408 257 024 × 2 = 1 + 0.999 511 718 813 606 816 514 048;
  • 41) 0.999 511 718 813 606 816 514 048 × 2 = 1 + 0.999 023 437 627 213 633 028 096;
  • 42) 0.999 023 437 627 213 633 028 096 × 2 = 1 + 0.998 046 875 254 427 266 056 192;
  • 43) 0.998 046 875 254 427 266 056 192 × 2 = 1 + 0.996 093 750 508 854 532 112 384;
  • 44) 0.996 093 750 508 854 532 112 384 × 2 = 1 + 0.992 187 501 017 709 064 224 768;
  • 45) 0.992 187 501 017 709 064 224 768 × 2 = 1 + 0.984 375 002 035 418 128 449 536;
  • 46) 0.984 375 002 035 418 128 449 536 × 2 = 1 + 0.968 750 004 070 836 256 899 072;
  • 47) 0.968 750 004 070 836 256 899 072 × 2 = 1 + 0.937 500 008 141 672 513 798 144;
  • 48) 0.937 500 008 141 672 513 798 144 × 2 = 1 + 0.875 000 016 283 345 027 596 288;
  • 49) 0.875 000 016 283 345 027 596 288 × 2 = 1 + 0.750 000 032 566 690 055 192 576;
  • 50) 0.750 000 032 566 690 055 192 576 × 2 = 1 + 0.500 000 065 133 380 110 385 152;
  • 51) 0.500 000 065 133 380 110 385 152 × 2 = 1 + 0.000 000 130 266 760 220 770 304;
  • 52) 0.000 000 130 266 760 220 770 304 × 2 = 0 + 0.000 000 260 533 520 441 540 608;
  • 53) 0.000 000 260 533 520 441 540 608 × 2 = 0 + 0.000 000 521 067 040 883 081 216;

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).


4. 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:


0.999 999 999 999 999 555 910 848(10) =

0.1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1110 0(2)

5. Positive number before normalization:

6.999 999 999 999 999 555 910 848(10) =

110.1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1110 0(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 2 positions to the left, so that only one non zero digit remains to the left of it:


6.999 999 999 999 999 555 910 848(10) =

110.1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1110 0(2) =

110.1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1110 0(2) × 20 =

1.1011 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 100(2) × 22


7. 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 0 (a positive number)

Exponent (unadjusted): 2

Mantissa (not normalized):
1.1011 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 100


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

2 + 2(11-1) - 1 =

(2 + 1 023)(10) =

1 025(10)


9. 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;
  • 1 025 ÷ 2 = 512 + 1;
  • 512 ÷ 2 = 256 + 0;
  • 256 ÷ 2 = 128 + 0;
  • 128 ÷ 2 = 64 + 0;
  • 64 ÷ 2 = 32 + 0;
  • 32 ÷ 2 = 16 + 0;
  • 16 ÷ 2 = 8 + 0;
  • 8 ÷ 2 = 4 + 0;
  • 4 ÷ 2 = 2 + 0;
  • 2 ÷ 2 = 1 + 0;
  • 1 ÷ 2 = 0 + 1;

10. Construct the base 2 representation of the adjusted exponent.

Take all the remainders starting from the bottom of the list constructed above.


Exponent (adjusted) =

1025(10) =

100 0000 0001(2)


11. 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, by removing the excess bits, from the right (if any of the excess bits is set on 1, we are losing precision...).


Mantissa (normalized) =

1. 1011 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 100 =

1011 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111


12. The three elements that make up the number's 64 bit double precision IEEE 754 binary floating point representation:

Sign (1 bit) =
0 (a positive number)

Exponent (11 bits) =
100 0000 0001

Mantissa (52 bits) =
1011 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111


Decimal number 6.999 999 999 999 999 555 910 848 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 100 0000 0001 - 1011 1111 1111 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