3.141 592 653 589 793 115 997 963 468 544 185 161 590 576 116 8 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 3.141 592 653 589 793 115 997 963 468 544 185 161 590 576 116 8(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
3.141 592 653 589 793 115 997 963 468 544 185 161 590 576 116 8(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: 0.141 592 653 589 793 115 997 963 468 544 185 161 590 576 116 8.

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) 0.141 592 653 589 793 115 997 963 468 544 185 161 590 576 116 8 × 2 = 0 + 0.283 185 307 179 586 231 995 926 937 088 370 323 181 152 233 6;
  • 2) 0.283 185 307 179 586 231 995 926 937 088 370 323 181 152 233 6 × 2 = 0 + 0.566 370 614 359 172 463 991 853 874 176 740 646 362 304 467 2;
  • 3) 0.566 370 614 359 172 463 991 853 874 176 740 646 362 304 467 2 × 2 = 1 + 0.132 741 228 718 344 927 983 707 748 353 481 292 724 608 934 4;
  • 4) 0.132 741 228 718 344 927 983 707 748 353 481 292 724 608 934 4 × 2 = 0 + 0.265 482 457 436 689 855 967 415 496 706 962 585 449 217 868 8;
  • 5) 0.265 482 457 436 689 855 967 415 496 706 962 585 449 217 868 8 × 2 = 0 + 0.530 964 914 873 379 711 934 830 993 413 925 170 898 435 737 6;
  • 6) 0.530 964 914 873 379 711 934 830 993 413 925 170 898 435 737 6 × 2 = 1 + 0.061 929 829 746 759 423 869 661 986 827 850 341 796 871 475 2;
  • 7) 0.061 929 829 746 759 423 869 661 986 827 850 341 796 871 475 2 × 2 = 0 + 0.123 859 659 493 518 847 739 323 973 655 700 683 593 742 950 4;
  • 8) 0.123 859 659 493 518 847 739 323 973 655 700 683 593 742 950 4 × 2 = 0 + 0.247 719 318 987 037 695 478 647 947 311 401 367 187 485 900 8;
  • 9) 0.247 719 318 987 037 695 478 647 947 311 401 367 187 485 900 8 × 2 = 0 + 0.495 438 637 974 075 390 957 295 894 622 802 734 374 971 801 6;
  • 10) 0.495 438 637 974 075 390 957 295 894 622 802 734 374 971 801 6 × 2 = 0 + 0.990 877 275 948 150 781 914 591 789 245 605 468 749 943 603 2;
  • 11) 0.990 877 275 948 150 781 914 591 789 245 605 468 749 943 603 2 × 2 = 1 + 0.981 754 551 896 301 563 829 183 578 491 210 937 499 887 206 4;
  • 12) 0.981 754 551 896 301 563 829 183 578 491 210 937 499 887 206 4 × 2 = 1 + 0.963 509 103 792 603 127 658 367 156 982 421 874 999 774 412 8;
  • 13) 0.963 509 103 792 603 127 658 367 156 982 421 874 999 774 412 8 × 2 = 1 + 0.927 018 207 585 206 255 316 734 313 964 843 749 999 548 825 6;
  • 14) 0.927 018 207 585 206 255 316 734 313 964 843 749 999 548 825 6 × 2 = 1 + 0.854 036 415 170 412 510 633 468 627 929 687 499 999 097 651 2;
  • 15) 0.854 036 415 170 412 510 633 468 627 929 687 499 999 097 651 2 × 2 = 1 + 0.708 072 830 340 825 021 266 937 255 859 374 999 998 195 302 4;
  • 16) 0.708 072 830 340 825 021 266 937 255 859 374 999 998 195 302 4 × 2 = 1 + 0.416 145 660 681 650 042 533 874 511 718 749 999 996 390 604 8;
  • 17) 0.416 145 660 681 650 042 533 874 511 718 749 999 996 390 604 8 × 2 = 0 + 0.832 291 321 363 300 085 067 749 023 437 499 999 992 781 209 6;
  • 18) 0.832 291 321 363 300 085 067 749 023 437 499 999 992 781 209 6 × 2 = 1 + 0.664 582 642 726 600 170 135 498 046 874 999 999 985 562 419 2;
  • 19) 0.664 582 642 726 600 170 135 498 046 874 999 999 985 562 419 2 × 2 = 1 + 0.329 165 285 453 200 340 270 996 093 749 999 999 971 124 838 4;
  • 20) 0.329 165 285 453 200 340 270 996 093 749 999 999 971 124 838 4 × 2 = 0 + 0.658 330 570 906 400 680 541 992 187 499 999 999 942 249 676 8;
  • 21) 0.658 330 570 906 400 680 541 992 187 499 999 999 942 249 676 8 × 2 = 1 + 0.316 661 141 812 801 361 083 984 374 999 999 999 884 499 353 6;
  • 22) 0.316 661 141 812 801 361 083 984 374 999 999 999 884 499 353 6 × 2 = 0 + 0.633 322 283 625 602 722 167 968 749 999 999 999 768 998 707 2;
  • 23) 0.633 322 283 625 602 722 167 968 749 999 999 999 768 998 707 2 × 2 = 1 + 0.266 644 567 251 205 444 335 937 499 999 999 999 537 997 414 4;
  • 24) 0.266 644 567 251 205 444 335 937 499 999 999 999 537 997 414 4 × 2 = 0 + 0.533 289 134 502 410 888 671 874 999 999 999 999 075 994 828 8;
  • 25) 0.533 289 134 502 410 888 671 874 999 999 999 999 075 994 828 8 × 2 = 1 + 0.066 578 269 004 821 777 343 749 999 999 999 998 151 989 657 6;
  • 26) 0.066 578 269 004 821 777 343 749 999 999 999 998 151 989 657 6 × 2 = 0 + 0.133 156 538 009 643 554 687 499 999 999 999 996 303 979 315 2;
  • 27) 0.133 156 538 009 643 554 687 499 999 999 999 996 303 979 315 2 × 2 = 0 + 0.266 313 076 019 287 109 374 999 999 999 999 992 607 958 630 4;
  • 28) 0.266 313 076 019 287 109 374 999 999 999 999 992 607 958 630 4 × 2 = 0 + 0.532 626 152 038 574 218 749 999 999 999 999 985 215 917 260 8;
  • 29) 0.532 626 152 038 574 218 749 999 999 999 999 985 215 917 260 8 × 2 = 1 + 0.065 252 304 077 148 437 499 999 999 999 999 970 431 834 521 6;
  • 30) 0.065 252 304 077 148 437 499 999 999 999 999 970 431 834 521 6 × 2 = 0 + 0.130 504 608 154 296 874 999 999 999 999 999 940 863 669 043 2;
  • 31) 0.130 504 608 154 296 874 999 999 999 999 999 940 863 669 043 2 × 2 = 0 + 0.261 009 216 308 593 749 999 999 999 999 999 881 727 338 086 4;
  • 32) 0.261 009 216 308 593 749 999 999 999 999 999 881 727 338 086 4 × 2 = 0 + 0.522 018 432 617 187 499 999 999 999 999 999 763 454 676 172 8;
  • 33) 0.522 018 432 617 187 499 999 999 999 999 999 763 454 676 172 8 × 2 = 1 + 0.044 036 865 234 374 999 999 999 999 999 999 526 909 352 345 6;
  • 34) 0.044 036 865 234 374 999 999 999 999 999 999 526 909 352 345 6 × 2 = 0 + 0.088 073 730 468 749 999 999 999 999 999 999 053 818 704 691 2;
  • 35) 0.088 073 730 468 749 999 999 999 999 999 999 053 818 704 691 2 × 2 = 0 + 0.176 147 460 937 499 999 999 999 999 999 998 107 637 409 382 4;
  • 36) 0.176 147 460 937 499 999 999 999 999 999 998 107 637 409 382 4 × 2 = 0 + 0.352 294 921 874 999 999 999 999 999 999 996 215 274 818 764 8;
  • 37) 0.352 294 921 874 999 999 999 999 999 999 996 215 274 818 764 8 × 2 = 0 + 0.704 589 843 749 999 999 999 999 999 999 992 430 549 637 529 6;
  • 38) 0.704 589 843 749 999 999 999 999 999 999 992 430 549 637 529 6 × 2 = 1 + 0.409 179 687 499 999 999 999 999 999 999 984 861 099 275 059 2;
  • 39) 0.409 179 687 499 999 999 999 999 999 999 984 861 099 275 059 2 × 2 = 0 + 0.818 359 374 999 999 999 999 999 999 999 969 722 198 550 118 4;
  • 40) 0.818 359 374 999 999 999 999 999 999 999 969 722 198 550 118 4 × 2 = 1 + 0.636 718 749 999 999 999 999 999 999 999 939 444 397 100 236 8;
  • 41) 0.636 718 749 999 999 999 999 999 999 999 939 444 397 100 236 8 × 2 = 1 + 0.273 437 499 999 999 999 999 999 999 999 878 888 794 200 473 6;
  • 42) 0.273 437 499 999 999 999 999 999 999 999 878 888 794 200 473 6 × 2 = 0 + 0.546 874 999 999 999 999 999 999 999 999 757 777 588 400 947 2;
  • 43) 0.546 874 999 999 999 999 999 999 999 999 757 777 588 400 947 2 × 2 = 1 + 0.093 749 999 999 999 999 999 999 999 999 515 555 176 801 894 4;
  • 44) 0.093 749 999 999 999 999 999 999 999 999 515 555 176 801 894 4 × 2 = 0 + 0.187 499 999 999 999 999 999 999 999 999 031 110 353 603 788 8;
  • 45) 0.187 499 999 999 999 999 999 999 999 999 031 110 353 603 788 8 × 2 = 0 + 0.374 999 999 999 999 999 999 999 999 998 062 220 707 207 577 6;
  • 46) 0.374 999 999 999 999 999 999 999 999 998 062 220 707 207 577 6 × 2 = 0 + 0.749 999 999 999 999 999 999 999 999 996 124 441 414 415 155 2;
  • 47) 0.749 999 999 999 999 999 999 999 999 996 124 441 414 415 155 2 × 2 = 1 + 0.499 999 999 999 999 999 999 999 999 992 248 882 828 830 310 4;
  • 48) 0.499 999 999 999 999 999 999 999 999 992 248 882 828 830 310 4 × 2 = 0 + 0.999 999 999 999 999 999 999 999 999 984 497 765 657 660 620 8;
  • 49) 0.999 999 999 999 999 999 999 999 999 984 497 765 657 660 620 8 × 2 = 1 + 0.999 999 999 999 999 999 999 999 999 968 995 531 315 321 241 6;
  • 50) 0.999 999 999 999 999 999 999 999 999 968 995 531 315 321 241 6 × 2 = 1 + 0.999 999 999 999 999 999 999 999 999 937 991 062 630 642 483 2;
  • 51) 0.999 999 999 999 999 999 999 999 999 937 991 062 630 642 483 2 × 2 = 1 + 0.999 999 999 999 999 999 999 999 999 875 982 125 261 284 966 4;
  • 52) 0.999 999 999 999 999 999 999 999 999 875 982 125 261 284 966 4 × 2 = 1 + 0.999 999 999 999 999 999 999 999 999 751 964 250 522 569 932 8;
  • 53) 0.999 999 999 999 999 999 999 999 999 751 964 250 522 569 932 8 × 2 = 1 + 0.999 999 999 999 999 999 999 999 999 503 928 501 045 139 865 6;

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:


0.141 592 653 589 793 115 997 963 468 544 185 161 590 576 116 8(10) =

0.0010 0100 0011 1111 0110 1010 1000 1000 1000 0101 1010 0010 1111 1(2)

5. Positive number before normalization:

3.141 592 653 589 793 115 997 963 468 544 185 161 590 576 116 8(10) =

11.0010 0100 0011 1111 0110 1010 1000 1000 1000 0101 1010 0010 1111 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:


3.141 592 653 589 793 115 997 963 468 544 185 161 590 576 116 8(10) =

11.0010 0100 0011 1111 0110 1010 1000 1000 1000 0101 1010 0010 1111 1(2) =

11.0010 0100 0011 1111 0110 1010 1000 1000 1000 0101 1010 0010 1111 1(2) × 20 =

1.1001 0010 0001 1111 1011 0101 0100 0100 0100 0010 1101 0001 0111 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):
1.1001 0010 0001 1111 1011 0101 0100 0100 0100 0010 1101 0001 0111 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. 1001 0010 0001 1111 1011 0101 0100 0100 0100 0010 1101 0001 0111 11 =

1001 0010 0001 1111 1011 0101 0100 0100 0100 0010 1101 0001 0111


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) =
1001 0010 0001 1111 1011 0101 0100 0100 0100 0010 1101 0001 0111


Decimal number 3.141 592 653 589 793 115 997 963 468 544 185 161 590 576 116 8 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 100 0000 0000 - 1001 0010 0001 1111 1011 0101 0100 0100 0100 0010 1101 0001 0111

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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