0.000 999 999 999 999 999 803 976 247 214 620 798 56 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 999 999 999 999 999 803 976 247 214 620 798 56(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
0.000 999 999 999 999 999 803 976 247 214 620 798 56(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: 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;

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.

0(10) =

0(2)


3. Convert to binary (base 2) the fractional part: 0.000 999 999 999 999 999 803 976 247 214 620 798 56.

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) 0.000 999 999 999 999 999 803 976 247 214 620 798 56 × 2 = 0 + 0.001 999 999 999 999 999 607 952 494 429 241 597 12;
  • 2) 0.001 999 999 999 999 999 607 952 494 429 241 597 12 × 2 = 0 + 0.003 999 999 999 999 999 215 904 988 858 483 194 24;
  • 3) 0.003 999 999 999 999 999 215 904 988 858 483 194 24 × 2 = 0 + 0.007 999 999 999 999 998 431 809 977 716 966 388 48;
  • 4) 0.007 999 999 999 999 998 431 809 977 716 966 388 48 × 2 = 0 + 0.015 999 999 999 999 996 863 619 955 433 932 776 96;
  • 5) 0.015 999 999 999 999 996 863 619 955 433 932 776 96 × 2 = 0 + 0.031 999 999 999 999 993 727 239 910 867 865 553 92;
  • 6) 0.031 999 999 999 999 993 727 239 910 867 865 553 92 × 2 = 0 + 0.063 999 999 999 999 987 454 479 821 735 731 107 84;
  • 7) 0.063 999 999 999 999 987 454 479 821 735 731 107 84 × 2 = 0 + 0.127 999 999 999 999 974 908 959 643 471 462 215 68;
  • 8) 0.127 999 999 999 999 974 908 959 643 471 462 215 68 × 2 = 0 + 0.255 999 999 999 999 949 817 919 286 942 924 431 36;
  • 9) 0.255 999 999 999 999 949 817 919 286 942 924 431 36 × 2 = 0 + 0.511 999 999 999 999 899 635 838 573 885 848 862 72;
  • 10) 0.511 999 999 999 999 899 635 838 573 885 848 862 72 × 2 = 1 + 0.023 999 999 999 999 799 271 677 147 771 697 725 44;
  • 11) 0.023 999 999 999 999 799 271 677 147 771 697 725 44 × 2 = 0 + 0.047 999 999 999 999 598 543 354 295 543 395 450 88;
  • 12) 0.047 999 999 999 999 598 543 354 295 543 395 450 88 × 2 = 0 + 0.095 999 999 999 999 197 086 708 591 086 790 901 76;
  • 13) 0.095 999 999 999 999 197 086 708 591 086 790 901 76 × 2 = 0 + 0.191 999 999 999 998 394 173 417 182 173 581 803 52;
  • 14) 0.191 999 999 999 998 394 173 417 182 173 581 803 52 × 2 = 0 + 0.383 999 999 999 996 788 346 834 364 347 163 607 04;
  • 15) 0.383 999 999 999 996 788 346 834 364 347 163 607 04 × 2 = 0 + 0.767 999 999 999 993 576 693 668 728 694 327 214 08;
  • 16) 0.767 999 999 999 993 576 693 668 728 694 327 214 08 × 2 = 1 + 0.535 999 999 999 987 153 387 337 457 388 654 428 16;
  • 17) 0.535 999 999 999 987 153 387 337 457 388 654 428 16 × 2 = 1 + 0.071 999 999 999 974 306 774 674 914 777 308 856 32;
  • 18) 0.071 999 999 999 974 306 774 674 914 777 308 856 32 × 2 = 0 + 0.143 999 999 999 948 613 549 349 829 554 617 712 64;
  • 19) 0.143 999 999 999 948 613 549 349 829 554 617 712 64 × 2 = 0 + 0.287 999 999 999 897 227 098 699 659 109 235 425 28;
  • 20) 0.287 999 999 999 897 227 098 699 659 109 235 425 28 × 2 = 0 + 0.575 999 999 999 794 454 197 399 318 218 470 850 56;
  • 21) 0.575 999 999 999 794 454 197 399 318 218 470 850 56 × 2 = 1 + 0.151 999 999 999 588 908 394 798 636 436 941 701 12;
  • 22) 0.151 999 999 999 588 908 394 798 636 436 941 701 12 × 2 = 0 + 0.303 999 999 999 177 816 789 597 272 873 883 402 24;
  • 23) 0.303 999 999 999 177 816 789 597 272 873 883 402 24 × 2 = 0 + 0.607 999 999 998 355 633 579 194 545 747 766 804 48;
  • 24) 0.607 999 999 998 355 633 579 194 545 747 766 804 48 × 2 = 1 + 0.215 999 999 996 711 267 158 389 091 495 533 608 96;
  • 25) 0.215 999 999 996 711 267 158 389 091 495 533 608 96 × 2 = 0 + 0.431 999 999 993 422 534 316 778 182 991 067 217 92;
  • 26) 0.431 999 999 993 422 534 316 778 182 991 067 217 92 × 2 = 0 + 0.863 999 999 986 845 068 633 556 365 982 134 435 84;
  • 27) 0.863 999 999 986 845 068 633 556 365 982 134 435 84 × 2 = 1 + 0.727 999 999 973 690 137 267 112 731 964 268 871 68;
  • 28) 0.727 999 999 973 690 137 267 112 731 964 268 871 68 × 2 = 1 + 0.455 999 999 947 380 274 534 225 463 928 537 743 36;
  • 29) 0.455 999 999 947 380 274 534 225 463 928 537 743 36 × 2 = 0 + 0.911 999 999 894 760 549 068 450 927 857 075 486 72;
  • 30) 0.911 999 999 894 760 549 068 450 927 857 075 486 72 × 2 = 1 + 0.823 999 999 789 521 098 136 901 855 714 150 973 44;
  • 31) 0.823 999 999 789 521 098 136 901 855 714 150 973 44 × 2 = 1 + 0.647 999 999 579 042 196 273 803 711 428 301 946 88;
  • 32) 0.647 999 999 579 042 196 273 803 711 428 301 946 88 × 2 = 1 + 0.295 999 999 158 084 392 547 607 422 856 603 893 76;
  • 33) 0.295 999 999 158 084 392 547 607 422 856 603 893 76 × 2 = 0 + 0.591 999 998 316 168 785 095 214 845 713 207 787 52;
  • 34) 0.591 999 998 316 168 785 095 214 845 713 207 787 52 × 2 = 1 + 0.183 999 996 632 337 570 190 429 691 426 415 575 04;
  • 35) 0.183 999 996 632 337 570 190 429 691 426 415 575 04 × 2 = 0 + 0.367 999 993 264 675 140 380 859 382 852 831 150 08;
  • 36) 0.367 999 993 264 675 140 380 859 382 852 831 150 08 × 2 = 0 + 0.735 999 986 529 350 280 761 718 765 705 662 300 16;
  • 37) 0.735 999 986 529 350 280 761 718 765 705 662 300 16 × 2 = 1 + 0.471 999 973 058 700 561 523 437 531 411 324 600 32;
  • 38) 0.471 999 973 058 700 561 523 437 531 411 324 600 32 × 2 = 0 + 0.943 999 946 117 401 123 046 875 062 822 649 200 64;
  • 39) 0.943 999 946 117 401 123 046 875 062 822 649 200 64 × 2 = 1 + 0.887 999 892 234 802 246 093 750 125 645 298 401 28;
  • 40) 0.887 999 892 234 802 246 093 750 125 645 298 401 28 × 2 = 1 + 0.775 999 784 469 604 492 187 500 251 290 596 802 56;
  • 41) 0.775 999 784 469 604 492 187 500 251 290 596 802 56 × 2 = 1 + 0.551 999 568 939 208 984 375 000 502 581 193 605 12;
  • 42) 0.551 999 568 939 208 984 375 000 502 581 193 605 12 × 2 = 1 + 0.103 999 137 878 417 968 750 001 005 162 387 210 24;
  • 43) 0.103 999 137 878 417 968 750 001 005 162 387 210 24 × 2 = 0 + 0.207 998 275 756 835 937 500 002 010 324 774 420 48;
  • 44) 0.207 998 275 756 835 937 500 002 010 324 774 420 48 × 2 = 0 + 0.415 996 551 513 671 875 000 004 020 649 548 840 96;
  • 45) 0.415 996 551 513 671 875 000 004 020 649 548 840 96 × 2 = 0 + 0.831 993 103 027 343 750 000 008 041 299 097 681 92;
  • 46) 0.831 993 103 027 343 750 000 008 041 299 097 681 92 × 2 = 1 + 0.663 986 206 054 687 500 000 016 082 598 195 363 84;
  • 47) 0.663 986 206 054 687 500 000 016 082 598 195 363 84 × 2 = 1 + 0.327 972 412 109 375 000 000 032 165 196 390 727 68;
  • 48) 0.327 972 412 109 375 000 000 032 165 196 390 727 68 × 2 = 0 + 0.655 944 824 218 750 000 000 064 330 392 781 455 36;
  • 49) 0.655 944 824 218 750 000 000 064 330 392 781 455 36 × 2 = 1 + 0.311 889 648 437 500 000 000 128 660 785 562 910 72;
  • 50) 0.311 889 648 437 500 000 000 128 660 785 562 910 72 × 2 = 0 + 0.623 779 296 875 000 000 000 257 321 571 125 821 44;
  • 51) 0.623 779 296 875 000 000 000 257 321 571 125 821 44 × 2 = 1 + 0.247 558 593 750 000 000 000 514 643 142 251 642 88;
  • 52) 0.247 558 593 750 000 000 000 514 643 142 251 642 88 × 2 = 0 + 0.495 117 187 500 000 000 001 029 286 284 503 285 76;
  • 53) 0.495 117 187 500 000 000 001 029 286 284 503 285 76 × 2 = 0 + 0.990 234 375 000 000 000 002 058 572 569 006 571 52;
  • 54) 0.990 234 375 000 000 000 002 058 572 569 006 571 52 × 2 = 1 + 0.980 468 750 000 000 000 004 117 145 138 013 143 04;
  • 55) 0.980 468 750 000 000 000 004 117 145 138 013 143 04 × 2 = 1 + 0.960 937 500 000 000 000 008 234 290 276 026 286 08;
  • 56) 0.960 937 500 000 000 000 008 234 290 276 026 286 08 × 2 = 1 + 0.921 875 000 000 000 000 016 468 580 552 052 572 16;
  • 57) 0.921 875 000 000 000 000 016 468 580 552 052 572 16 × 2 = 1 + 0.843 750 000 000 000 000 032 937 161 104 105 144 32;
  • 58) 0.843 750 000 000 000 000 032 937 161 104 105 144 32 × 2 = 1 + 0.687 500 000 000 000 000 065 874 322 208 210 288 64;
  • 59) 0.687 500 000 000 000 000 065 874 322 208 210 288 64 × 2 = 1 + 0.375 000 000 000 000 000 131 748 644 416 420 577 28;
  • 60) 0.375 000 000 000 000 000 131 748 644 416 420 577 28 × 2 = 0 + 0.750 000 000 000 000 000 263 497 288 832 841 154 56;
  • 61) 0.750 000 000 000 000 000 263 497 288 832 841 154 56 × 2 = 1 + 0.500 000 000 000 000 000 526 994 577 665 682 309 12;
  • 62) 0.500 000 000 000 000 000 526 994 577 665 682 309 12 × 2 = 1 + 0.000 000 000 000 000 001 053 989 155 331 364 618 24;

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:


0.000 999 999 999 999 999 803 976 247 214 620 798 56(10) =

0.0000 0000 0100 0001 1000 1001 0011 0111 0100 1011 1100 0110 1010 0111 1110 11(2)

5. Positive number before normalization:

0.000 999 999 999 999 999 803 976 247 214 620 798 56(10) =

0.0000 0000 0100 0001 1000 1001 0011 0111 0100 1011 1100 0110 1010 0111 1110 11(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 10 positions to the right, so that only one non zero digit remains to the left of it:


0.000 999 999 999 999 999 803 976 247 214 620 798 56(10) =

0.0000 0000 0100 0001 1000 1001 0011 0111 0100 1011 1100 0110 1010 0111 1110 11(2) =

0.0000 0000 0100 0001 1000 1001 0011 0111 0100 1011 1100 0110 1010 0111 1110 11(2) × 20 =

1.0000 0110 0010 0100 1101 1101 0010 1111 0001 1010 1001 1111 1011(2) × 2-10


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): -10

Mantissa (not normalized):
1.0000 0110 0010 0100 1101 1101 0010 1111 0001 1010 1001 1111 1011


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

-10 + 2(11-1) - 1 =

(-10 + 1 023)(10) =

1 013(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;
  • 1 013 ÷ 2 = 506 + 1;
  • 506 ÷ 2 = 253 + 0;
  • 253 ÷ 2 = 126 + 1;
  • 126 ÷ 2 = 63 + 0;
  • 63 ÷ 2 = 31 + 1;
  • 31 ÷ 2 = 15 + 1;
  • 15 ÷ 2 = 7 + 1;
  • 7 ÷ 2 = 3 + 1;
  • 3 ÷ 2 = 1 + 1;
  • 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) =

1013(10) =

011 1111 0101(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, only if necessary (not the case here).


Mantissa (normalized) =

1. 0000 0110 0010 0100 1101 1101 0010 1111 0001 1010 1001 1111 1011 =

0000 0110 0010 0100 1101 1101 0010 1111 0001 1010 1001 1111 1011


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) =
011 1111 0101

Mantissa (52 bits) =
0000 0110 0010 0100 1101 1101 0010 1111 0001 1010 1001 1111 1011


Decimal number 0.000 999 999 999 999 999 803 976 247 214 620 798 56 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1111 0101 - 0000 0110 0010 0100 1101 1101 0010 1111 0001 1010 1001 1111 1011

Share

» Convert decimal 0.000 999 999 999 999 999 803 976 247 214 620 797 57 to 64 bit double precision IEEE 754 binary floating point representation standard

» Calculations Performed by Our Visitors: Decimal Numbers Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard. Data organized on a Monthly Basis

» Month 05, 2026 [May]: Decimal Numbers Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard. Calculations performed during the month of: May - by our visitors

» Improve your social skills: Your greatest rival, your most powerful ally. NotebookLM YouTube Video of 7.54 minutes

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