1.999 999 999 999 999 888 977 697 537 484 345 957 636 833 190 917 968 741 7 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 1.999 999 999 999 999 888 977 697 537 484 345 957 636 833 190 917 968 741 7(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
1.999 999 999 999 999 888 977 697 537 484 345 957 636 833 190 917 968 741 7(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: 1.
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;
  • 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.

1(10) =

1(2)


3. Convert to binary (base 2) the fractional part: 0.999 999 999 999 999 888 977 697 537 484 345 957 636 833 190 917 968 741 7.

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) 0.999 999 999 999 999 888 977 697 537 484 345 957 636 833 190 917 968 741 7 × 2 = 1 + 0.999 999 999 999 999 777 955 395 074 968 691 915 273 666 381 835 937 483 4;
  • 2) 0.999 999 999 999 999 777 955 395 074 968 691 915 273 666 381 835 937 483 4 × 2 = 1 + 0.999 999 999 999 999 555 910 790 149 937 383 830 547 332 763 671 874 966 8;
  • 3) 0.999 999 999 999 999 555 910 790 149 937 383 830 547 332 763 671 874 966 8 × 2 = 1 + 0.999 999 999 999 999 111 821 580 299 874 767 661 094 665 527 343 749 933 6;
  • 4) 0.999 999 999 999 999 111 821 580 299 874 767 661 094 665 527 343 749 933 6 × 2 = 1 + 0.999 999 999 999 998 223 643 160 599 749 535 322 189 331 054 687 499 867 2;
  • 5) 0.999 999 999 999 998 223 643 160 599 749 535 322 189 331 054 687 499 867 2 × 2 = 1 + 0.999 999 999 999 996 447 286 321 199 499 070 644 378 662 109 374 999 734 4;
  • 6) 0.999 999 999 999 996 447 286 321 199 499 070 644 378 662 109 374 999 734 4 × 2 = 1 + 0.999 999 999 999 992 894 572 642 398 998 141 288 757 324 218 749 999 468 8;
  • 7) 0.999 999 999 999 992 894 572 642 398 998 141 288 757 324 218 749 999 468 8 × 2 = 1 + 0.999 999 999 999 985 789 145 284 797 996 282 577 514 648 437 499 998 937 6;
  • 8) 0.999 999 999 999 985 789 145 284 797 996 282 577 514 648 437 499 998 937 6 × 2 = 1 + 0.999 999 999 999 971 578 290 569 595 992 565 155 029 296 874 999 997 875 2;
  • 9) 0.999 999 999 999 971 578 290 569 595 992 565 155 029 296 874 999 997 875 2 × 2 = 1 + 0.999 999 999 999 943 156 581 139 191 985 130 310 058 593 749 999 995 750 4;
  • 10) 0.999 999 999 999 943 156 581 139 191 985 130 310 058 593 749 999 995 750 4 × 2 = 1 + 0.999 999 999 999 886 313 162 278 383 970 260 620 117 187 499 999 991 500 8;
  • 11) 0.999 999 999 999 886 313 162 278 383 970 260 620 117 187 499 999 991 500 8 × 2 = 1 + 0.999 999 999 999 772 626 324 556 767 940 521 240 234 374 999 999 983 001 6;
  • 12) 0.999 999 999 999 772 626 324 556 767 940 521 240 234 374 999 999 983 001 6 × 2 = 1 + 0.999 999 999 999 545 252 649 113 535 881 042 480 468 749 999 999 966 003 2;
  • 13) 0.999 999 999 999 545 252 649 113 535 881 042 480 468 749 999 999 966 003 2 × 2 = 1 + 0.999 999 999 999 090 505 298 227 071 762 084 960 937 499 999 999 932 006 4;
  • 14) 0.999 999 999 999 090 505 298 227 071 762 084 960 937 499 999 999 932 006 4 × 2 = 1 + 0.999 999 999 998 181 010 596 454 143 524 169 921 874 999 999 999 864 012 8;
  • 15) 0.999 999 999 998 181 010 596 454 143 524 169 921 874 999 999 999 864 012 8 × 2 = 1 + 0.999 999 999 996 362 021 192 908 287 048 339 843 749 999 999 999 728 025 6;
  • 16) 0.999 999 999 996 362 021 192 908 287 048 339 843 749 999 999 999 728 025 6 × 2 = 1 + 0.999 999 999 992 724 042 385 816 574 096 679 687 499 999 999 999 456 051 2;
  • 17) 0.999 999 999 992 724 042 385 816 574 096 679 687 499 999 999 999 456 051 2 × 2 = 1 + 0.999 999 999 985 448 084 771 633 148 193 359 374 999 999 999 998 912 102 4;
  • 18) 0.999 999 999 985 448 084 771 633 148 193 359 374 999 999 999 998 912 102 4 × 2 = 1 + 0.999 999 999 970 896 169 543 266 296 386 718 749 999 999 999 997 824 204 8;
  • 19) 0.999 999 999 970 896 169 543 266 296 386 718 749 999 999 999 997 824 204 8 × 2 = 1 + 0.999 999 999 941 792 339 086 532 592 773 437 499 999 999 999 995 648 409 6;
  • 20) 0.999 999 999 941 792 339 086 532 592 773 437 499 999 999 999 995 648 409 6 × 2 = 1 + 0.999 999 999 883 584 678 173 065 185 546 874 999 999 999 999 991 296 819 2;
  • 21) 0.999 999 999 883 584 678 173 065 185 546 874 999 999 999 999 991 296 819 2 × 2 = 1 + 0.999 999 999 767 169 356 346 130 371 093 749 999 999 999 999 982 593 638 4;
  • 22) 0.999 999 999 767 169 356 346 130 371 093 749 999 999 999 999 982 593 638 4 × 2 = 1 + 0.999 999 999 534 338 712 692 260 742 187 499 999 999 999 999 965 187 276 8;
  • 23) 0.999 999 999 534 338 712 692 260 742 187 499 999 999 999 999 965 187 276 8 × 2 = 1 + 0.999 999 999 068 677 425 384 521 484 374 999 999 999 999 999 930 374 553 6;
  • 24) 0.999 999 999 068 677 425 384 521 484 374 999 999 999 999 999 930 374 553 6 × 2 = 1 + 0.999 999 998 137 354 850 769 042 968 749 999 999 999 999 999 860 749 107 2;
  • 25) 0.999 999 998 137 354 850 769 042 968 749 999 999 999 999 999 860 749 107 2 × 2 = 1 + 0.999 999 996 274 709 701 538 085 937 499 999 999 999 999 999 721 498 214 4;
  • 26) 0.999 999 996 274 709 701 538 085 937 499 999 999 999 999 999 721 498 214 4 × 2 = 1 + 0.999 999 992 549 419 403 076 171 874 999 999 999 999 999 999 442 996 428 8;
  • 27) 0.999 999 992 549 419 403 076 171 874 999 999 999 999 999 999 442 996 428 8 × 2 = 1 + 0.999 999 985 098 838 806 152 343 749 999 999 999 999 999 998 885 992 857 6;
  • 28) 0.999 999 985 098 838 806 152 343 749 999 999 999 999 999 998 885 992 857 6 × 2 = 1 + 0.999 999 970 197 677 612 304 687 499 999 999 999 999 999 997 771 985 715 2;
  • 29) 0.999 999 970 197 677 612 304 687 499 999 999 999 999 999 997 771 985 715 2 × 2 = 1 + 0.999 999 940 395 355 224 609 374 999 999 999 999 999 999 995 543 971 430 4;
  • 30) 0.999 999 940 395 355 224 609 374 999 999 999 999 999 999 995 543 971 430 4 × 2 = 1 + 0.999 999 880 790 710 449 218 749 999 999 999 999 999 999 991 087 942 860 8;
  • 31) 0.999 999 880 790 710 449 218 749 999 999 999 999 999 999 991 087 942 860 8 × 2 = 1 + 0.999 999 761 581 420 898 437 499 999 999 999 999 999 999 982 175 885 721 6;
  • 32) 0.999 999 761 581 420 898 437 499 999 999 999 999 999 999 982 175 885 721 6 × 2 = 1 + 0.999 999 523 162 841 796 874 999 999 999 999 999 999 999 964 351 771 443 2;
  • 33) 0.999 999 523 162 841 796 874 999 999 999 999 999 999 999 964 351 771 443 2 × 2 = 1 + 0.999 999 046 325 683 593 749 999 999 999 999 999 999 999 928 703 542 886 4;
  • 34) 0.999 999 046 325 683 593 749 999 999 999 999 999 999 999 928 703 542 886 4 × 2 = 1 + 0.999 998 092 651 367 187 499 999 999 999 999 999 999 999 857 407 085 772 8;
  • 35) 0.999 998 092 651 367 187 499 999 999 999 999 999 999 999 857 407 085 772 8 × 2 = 1 + 0.999 996 185 302 734 374 999 999 999 999 999 999 999 999 714 814 171 545 6;
  • 36) 0.999 996 185 302 734 374 999 999 999 999 999 999 999 999 714 814 171 545 6 × 2 = 1 + 0.999 992 370 605 468 749 999 999 999 999 999 999 999 999 429 628 343 091 2;
  • 37) 0.999 992 370 605 468 749 999 999 999 999 999 999 999 999 429 628 343 091 2 × 2 = 1 + 0.999 984 741 210 937 499 999 999 999 999 999 999 999 998 859 256 686 182 4;
  • 38) 0.999 984 741 210 937 499 999 999 999 999 999 999 999 998 859 256 686 182 4 × 2 = 1 + 0.999 969 482 421 874 999 999 999 999 999 999 999 999 997 718 513 372 364 8;
  • 39) 0.999 969 482 421 874 999 999 999 999 999 999 999 999 997 718 513 372 364 8 × 2 = 1 + 0.999 938 964 843 749 999 999 999 999 999 999 999 999 995 437 026 744 729 6;
  • 40) 0.999 938 964 843 749 999 999 999 999 999 999 999 999 995 437 026 744 729 6 × 2 = 1 + 0.999 877 929 687 499 999 999 999 999 999 999 999 999 990 874 053 489 459 2;
  • 41) 0.999 877 929 687 499 999 999 999 999 999 999 999 999 990 874 053 489 459 2 × 2 = 1 + 0.999 755 859 374 999 999 999 999 999 999 999 999 999 981 748 106 978 918 4;
  • 42) 0.999 755 859 374 999 999 999 999 999 999 999 999 999 981 748 106 978 918 4 × 2 = 1 + 0.999 511 718 749 999 999 999 999 999 999 999 999 999 963 496 213 957 836 8;
  • 43) 0.999 511 718 749 999 999 999 999 999 999 999 999 999 963 496 213 957 836 8 × 2 = 1 + 0.999 023 437 499 999 999 999 999 999 999 999 999 999 926 992 427 915 673 6;
  • 44) 0.999 023 437 499 999 999 999 999 999 999 999 999 999 926 992 427 915 673 6 × 2 = 1 + 0.998 046 874 999 999 999 999 999 999 999 999 999 999 853 984 855 831 347 2;
  • 45) 0.998 046 874 999 999 999 999 999 999 999 999 999 999 853 984 855 831 347 2 × 2 = 1 + 0.996 093 749 999 999 999 999 999 999 999 999 999 999 707 969 711 662 694 4;
  • 46) 0.996 093 749 999 999 999 999 999 999 999 999 999 999 707 969 711 662 694 4 × 2 = 1 + 0.992 187 499 999 999 999 999 999 999 999 999 999 999 415 939 423 325 388 8;
  • 47) 0.992 187 499 999 999 999 999 999 999 999 999 999 999 415 939 423 325 388 8 × 2 = 1 + 0.984 374 999 999 999 999 999 999 999 999 999 999 998 831 878 846 650 777 6;
  • 48) 0.984 374 999 999 999 999 999 999 999 999 999 999 998 831 878 846 650 777 6 × 2 = 1 + 0.968 749 999 999 999 999 999 999 999 999 999 999 997 663 757 693 301 555 2;
  • 49) 0.968 749 999 999 999 999 999 999 999 999 999 999 997 663 757 693 301 555 2 × 2 = 1 + 0.937 499 999 999 999 999 999 999 999 999 999 999 995 327 515 386 603 110 4;
  • 50) 0.937 499 999 999 999 999 999 999 999 999 999 999 995 327 515 386 603 110 4 × 2 = 1 + 0.874 999 999 999 999 999 999 999 999 999 999 999 990 655 030 773 206 220 8;
  • 51) 0.874 999 999 999 999 999 999 999 999 999 999 999 990 655 030 773 206 220 8 × 2 = 1 + 0.749 999 999 999 999 999 999 999 999 999 999 999 981 310 061 546 412 441 6;
  • 52) 0.749 999 999 999 999 999 999 999 999 999 999 999 981 310 061 546 412 441 6 × 2 = 1 + 0.499 999 999 999 999 999 999 999 999 999 999 999 962 620 123 092 824 883 2;
  • 53) 0.499 999 999 999 999 999 999 999 999 999 999 999 962 620 123 092 824 883 2 × 2 = 0 + 0.999 999 999 999 999 999 999 999 999 999 999 999 925 240 246 185 649 766 4;

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:


0.999 999 999 999 999 888 977 697 537 484 345 957 636 833 190 917 968 741 7(10) =

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

5. Positive number before normalization:

1.999 999 999 999 999 888 977 697 537 484 345 957 636 833 190 917 968 741 7(10) =

1.1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 0(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 0 positions to the left, so that only one non zero digit remains to the left of it:


1.999 999 999 999 999 888 977 697 537 484 345 957 636 833 190 917 968 741 7(10) =

1.1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 0(2) =

1.1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 0(2) × 20


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): 0

Mantissa (not normalized):
1.1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 0


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

0 + 2(11-1) - 1 =

(0 + 1 023)(10) =

1 023(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;
  • 1 023 ÷ 2 = 511 + 1;
  • 511 ÷ 2 = 255 + 1;
  • 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;

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) =

1023(10) =

011 1111 1111(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. 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 0 =

1111 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) =
011 1111 1111

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


Decimal number 1.999 999 999 999 999 888 977 697 537 484 345 957 636 833 190 917 968 741 7 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1111 1111 - 1111 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