3.421 299 999 999 999 563 726 760 243 298 485 875 129 699 706 943 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 3.421 299 999 999 999 563 726 760 243 298 485 875 129 699 706 943(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
3.421 299 999 999 999 563 726 760 243 298 485 875 129 699 706 943(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: 3.
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;
  • 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.

3(10) =

11(2)


3. Convert to binary (base 2) the fractional part: 0.421 299 999 999 999 563 726 760 243 298 485 875 129 699 706 943.

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) 0.421 299 999 999 999 563 726 760 243 298 485 875 129 699 706 943 × 2 = 0 + 0.842 599 999 999 999 127 453 520 486 596 971 750 259 399 413 886;
  • 2) 0.842 599 999 999 999 127 453 520 486 596 971 750 259 399 413 886 × 2 = 1 + 0.685 199 999 999 998 254 907 040 973 193 943 500 518 798 827 772;
  • 3) 0.685 199 999 999 998 254 907 040 973 193 943 500 518 798 827 772 × 2 = 1 + 0.370 399 999 999 996 509 814 081 946 387 887 001 037 597 655 544;
  • 4) 0.370 399 999 999 996 509 814 081 946 387 887 001 037 597 655 544 × 2 = 0 + 0.740 799 999 999 993 019 628 163 892 775 774 002 075 195 311 088;
  • 5) 0.740 799 999 999 993 019 628 163 892 775 774 002 075 195 311 088 × 2 = 1 + 0.481 599 999 999 986 039 256 327 785 551 548 004 150 390 622 176;
  • 6) 0.481 599 999 999 986 039 256 327 785 551 548 004 150 390 622 176 × 2 = 0 + 0.963 199 999 999 972 078 512 655 571 103 096 008 300 781 244 352;
  • 7) 0.963 199 999 999 972 078 512 655 571 103 096 008 300 781 244 352 × 2 = 1 + 0.926 399 999 999 944 157 025 311 142 206 192 016 601 562 488 704;
  • 8) 0.926 399 999 999 944 157 025 311 142 206 192 016 601 562 488 704 × 2 = 1 + 0.852 799 999 999 888 314 050 622 284 412 384 033 203 124 977 408;
  • 9) 0.852 799 999 999 888 314 050 622 284 412 384 033 203 124 977 408 × 2 = 1 + 0.705 599 999 999 776 628 101 244 568 824 768 066 406 249 954 816;
  • 10) 0.705 599 999 999 776 628 101 244 568 824 768 066 406 249 954 816 × 2 = 1 + 0.411 199 999 999 553 256 202 489 137 649 536 132 812 499 909 632;
  • 11) 0.411 199 999 999 553 256 202 489 137 649 536 132 812 499 909 632 × 2 = 0 + 0.822 399 999 999 106 512 404 978 275 299 072 265 624 999 819 264;
  • 12) 0.822 399 999 999 106 512 404 978 275 299 072 265 624 999 819 264 × 2 = 1 + 0.644 799 999 998 213 024 809 956 550 598 144 531 249 999 638 528;
  • 13) 0.644 799 999 998 213 024 809 956 550 598 144 531 249 999 638 528 × 2 = 1 + 0.289 599 999 996 426 049 619 913 101 196 289 062 499 999 277 056;
  • 14) 0.289 599 999 996 426 049 619 913 101 196 289 062 499 999 277 056 × 2 = 0 + 0.579 199 999 992 852 099 239 826 202 392 578 124 999 998 554 112;
  • 15) 0.579 199 999 992 852 099 239 826 202 392 578 124 999 998 554 112 × 2 = 1 + 0.158 399 999 985 704 198 479 652 404 785 156 249 999 997 108 224;
  • 16) 0.158 399 999 985 704 198 479 652 404 785 156 249 999 997 108 224 × 2 = 0 + 0.316 799 999 971 408 396 959 304 809 570 312 499 999 994 216 448;
  • 17) 0.316 799 999 971 408 396 959 304 809 570 312 499 999 994 216 448 × 2 = 0 + 0.633 599 999 942 816 793 918 609 619 140 624 999 999 988 432 896;
  • 18) 0.633 599 999 942 816 793 918 609 619 140 624 999 999 988 432 896 × 2 = 1 + 0.267 199 999 885 633 587 837 219 238 281 249 999 999 976 865 792;
  • 19) 0.267 199 999 885 633 587 837 219 238 281 249 999 999 976 865 792 × 2 = 0 + 0.534 399 999 771 267 175 674 438 476 562 499 999 999 953 731 584;
  • 20) 0.534 399 999 771 267 175 674 438 476 562 499 999 999 953 731 584 × 2 = 1 + 0.068 799 999 542 534 351 348 876 953 124 999 999 999 907 463 168;
  • 21) 0.068 799 999 542 534 351 348 876 953 124 999 999 999 907 463 168 × 2 = 0 + 0.137 599 999 085 068 702 697 753 906 249 999 999 999 814 926 336;
  • 22) 0.137 599 999 085 068 702 697 753 906 249 999 999 999 814 926 336 × 2 = 0 + 0.275 199 998 170 137 405 395 507 812 499 999 999 999 629 852 672;
  • 23) 0.275 199 998 170 137 405 395 507 812 499 999 999 999 629 852 672 × 2 = 0 + 0.550 399 996 340 274 810 791 015 624 999 999 999 999 259 705 344;
  • 24) 0.550 399 996 340 274 810 791 015 624 999 999 999 999 259 705 344 × 2 = 1 + 0.100 799 992 680 549 621 582 031 249 999 999 999 998 519 410 688;
  • 25) 0.100 799 992 680 549 621 582 031 249 999 999 999 998 519 410 688 × 2 = 0 + 0.201 599 985 361 099 243 164 062 499 999 999 999 997 038 821 376;
  • 26) 0.201 599 985 361 099 243 164 062 499 999 999 999 997 038 821 376 × 2 = 0 + 0.403 199 970 722 198 486 328 124 999 999 999 999 994 077 642 752;
  • 27) 0.403 199 970 722 198 486 328 124 999 999 999 999 994 077 642 752 × 2 = 0 + 0.806 399 941 444 396 972 656 249 999 999 999 999 988 155 285 504;
  • 28) 0.806 399 941 444 396 972 656 249 999 999 999 999 988 155 285 504 × 2 = 1 + 0.612 799 882 888 793 945 312 499 999 999 999 999 976 310 571 008;
  • 29) 0.612 799 882 888 793 945 312 499 999 999 999 999 976 310 571 008 × 2 = 1 + 0.225 599 765 777 587 890 624 999 999 999 999 999 952 621 142 016;
  • 30) 0.225 599 765 777 587 890 624 999 999 999 999 999 952 621 142 016 × 2 = 0 + 0.451 199 531 555 175 781 249 999 999 999 999 999 905 242 284 032;
  • 31) 0.451 199 531 555 175 781 249 999 999 999 999 999 905 242 284 032 × 2 = 0 + 0.902 399 063 110 351 562 499 999 999 999 999 999 810 484 568 064;
  • 32) 0.902 399 063 110 351 562 499 999 999 999 999 999 810 484 568 064 × 2 = 1 + 0.804 798 126 220 703 124 999 999 999 999 999 999 620 969 136 128;
  • 33) 0.804 798 126 220 703 124 999 999 999 999 999 999 620 969 136 128 × 2 = 1 + 0.609 596 252 441 406 249 999 999 999 999 999 999 241 938 272 256;
  • 34) 0.609 596 252 441 406 249 999 999 999 999 999 999 241 938 272 256 × 2 = 1 + 0.219 192 504 882 812 499 999 999 999 999 999 998 483 876 544 512;
  • 35) 0.219 192 504 882 812 499 999 999 999 999 999 998 483 876 544 512 × 2 = 0 + 0.438 385 009 765 624 999 999 999 999 999 999 996 967 753 089 024;
  • 36) 0.438 385 009 765 624 999 999 999 999 999 999 996 967 753 089 024 × 2 = 0 + 0.876 770 019 531 249 999 999 999 999 999 999 993 935 506 178 048;
  • 37) 0.876 770 019 531 249 999 999 999 999 999 999 993 935 506 178 048 × 2 = 1 + 0.753 540 039 062 499 999 999 999 999 999 999 987 871 012 356 096;
  • 38) 0.753 540 039 062 499 999 999 999 999 999 999 987 871 012 356 096 × 2 = 1 + 0.507 080 078 124 999 999 999 999 999 999 999 975 742 024 712 192;
  • 39) 0.507 080 078 124 999 999 999 999 999 999 999 975 742 024 712 192 × 2 = 1 + 0.014 160 156 249 999 999 999 999 999 999 999 951 484 049 424 384;
  • 40) 0.014 160 156 249 999 999 999 999 999 999 999 951 484 049 424 384 × 2 = 0 + 0.028 320 312 499 999 999 999 999 999 999 999 902 968 098 848 768;
  • 41) 0.028 320 312 499 999 999 999 999 999 999 999 902 968 098 848 768 × 2 = 0 + 0.056 640 624 999 999 999 999 999 999 999 999 805 936 197 697 536;
  • 42) 0.056 640 624 999 999 999 999 999 999 999 999 805 936 197 697 536 × 2 = 0 + 0.113 281 249 999 999 999 999 999 999 999 999 611 872 395 395 072;
  • 43) 0.113 281 249 999 999 999 999 999 999 999 999 611 872 395 395 072 × 2 = 0 + 0.226 562 499 999 999 999 999 999 999 999 999 223 744 790 790 144;
  • 44) 0.226 562 499 999 999 999 999 999 999 999 999 223 744 790 790 144 × 2 = 0 + 0.453 124 999 999 999 999 999 999 999 999 998 447 489 581 580 288;
  • 45) 0.453 124 999 999 999 999 999 999 999 999 998 447 489 581 580 288 × 2 = 0 + 0.906 249 999 999 999 999 999 999 999 999 996 894 979 163 160 576;
  • 46) 0.906 249 999 999 999 999 999 999 999 999 996 894 979 163 160 576 × 2 = 1 + 0.812 499 999 999 999 999 999 999 999 999 993 789 958 326 321 152;
  • 47) 0.812 499 999 999 999 999 999 999 999 999 993 789 958 326 321 152 × 2 = 1 + 0.624 999 999 999 999 999 999 999 999 999 987 579 916 652 642 304;
  • 48) 0.624 999 999 999 999 999 999 999 999 999 987 579 916 652 642 304 × 2 = 1 + 0.249 999 999 999 999 999 999 999 999 999 975 159 833 305 284 608;
  • 49) 0.249 999 999 999 999 999 999 999 999 999 975 159 833 305 284 608 × 2 = 0 + 0.499 999 999 999 999 999 999 999 999 999 950 319 666 610 569 216;
  • 50) 0.499 999 999 999 999 999 999 999 999 999 950 319 666 610 569 216 × 2 = 0 + 0.999 999 999 999 999 999 999 999 999 999 900 639 333 221 138 432;
  • 51) 0.999 999 999 999 999 999 999 999 999 999 900 639 333 221 138 432 × 2 = 1 + 0.999 999 999 999 999 999 999 999 999 999 801 278 666 442 276 864;
  • 52) 0.999 999 999 999 999 999 999 999 999 999 801 278 666 442 276 864 × 2 = 1 + 0.999 999 999 999 999 999 999 999 999 999 602 557 332 884 553 728;
  • 53) 0.999 999 999 999 999 999 999 999 999 999 602 557 332 884 553 728 × 2 = 1 + 0.999 999 999 999 999 999 999 999 999 999 205 114 665 769 107 456;

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:


0.421 299 999 999 999 563 726 760 243 298 485 875 129 699 706 943(10) =

0.0110 1011 1101 1010 0101 0001 0001 1001 1100 1110 0000 0111 0011 1(2)

5. Positive number before normalization:

3.421 299 999 999 999 563 726 760 243 298 485 875 129 699 706 943(10) =

11.0110 1011 1101 1010 0101 0001 0001 1001 1100 1110 0000 0111 0011 1(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 1 positions to the left, so that only one non zero digit remains to the left of it:


3.421 299 999 999 999 563 726 760 243 298 485 875 129 699 706 943(10) =

11.0110 1011 1101 1010 0101 0001 0001 1001 1100 1110 0000 0111 0011 1(2) =

11.0110 1011 1101 1010 0101 0001 0001 1001 1100 1110 0000 0111 0011 1(2) × 20 =

1.1011 0101 1110 1101 0010 1000 1000 1100 1110 0111 0000 0011 1001 11(2) × 21


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): 1

Mantissa (not normalized):
1.1011 0101 1110 1101 0010 1000 1000 1100 1110 0111 0000 0011 1001 11


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

1 + 2(11-1) - 1 =

(1 + 1 023)(10) =

1 024(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;
  • 1 024 ÷ 2 = 512 + 0;
  • 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) =

1024(10) =

100 0000 0000(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 0101 1110 1101 0010 1000 1000 1100 1110 0111 0000 0011 1001 11 =

1011 0101 1110 1101 0010 1000 1000 1100 1110 0111 0000 0011 1001


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 0000

Mantissa (52 bits) =
1011 0101 1110 1101 0010 1000 1000 1100 1110 0111 0000 0011 1001


Decimal number 3.421 299 999 999 999 563 726 760 243 298 485 875 129 699 706 943 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 100 0000 0000 - 1011 0101 1110 1101 0010 1000 1000 1100 1110 0111 0000 0011 1001

Share

» Convert decimal 3.421 299 999 999 999 563 726 760 243 298 485 875 129 699 706 971 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 08, 2026 [August]: Decimal Numbers Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard. Calculations performed during the month of: August - by our visitors

» Improve your social skills: The Attention Economy - The Battlefield For Your Focus. Gemini Notebook YouTube Video of 7.50 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