1.333 333 333 333 333 259 318 465 024 989 563 971 757 889 69 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 1.333 333 333 333 333 259 318 465 024 989 563 971 757 889 69(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
1.333 333 333 333 333 259 318 465 024 989 563 971 757 889 69(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: 0.333 333 333 333 333 259 318 465 024 989 563 971 757 889 69.

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) 0.333 333 333 333 333 259 318 465 024 989 563 971 757 889 69 × 2 = 0 + 0.666 666 666 666 666 518 636 930 049 979 127 943 515 779 38;
  • 2) 0.666 666 666 666 666 518 636 930 049 979 127 943 515 779 38 × 2 = 1 + 0.333 333 333 333 333 037 273 860 099 958 255 887 031 558 76;
  • 3) 0.333 333 333 333 333 037 273 860 099 958 255 887 031 558 76 × 2 = 0 + 0.666 666 666 666 666 074 547 720 199 916 511 774 063 117 52;
  • 4) 0.666 666 666 666 666 074 547 720 199 916 511 774 063 117 52 × 2 = 1 + 0.333 333 333 333 332 149 095 440 399 833 023 548 126 235 04;
  • 5) 0.333 333 333 333 332 149 095 440 399 833 023 548 126 235 04 × 2 = 0 + 0.666 666 666 666 664 298 190 880 799 666 047 096 252 470 08;
  • 6) 0.666 666 666 666 664 298 190 880 799 666 047 096 252 470 08 × 2 = 1 + 0.333 333 333 333 328 596 381 761 599 332 094 192 504 940 16;
  • 7) 0.333 333 333 333 328 596 381 761 599 332 094 192 504 940 16 × 2 = 0 + 0.666 666 666 666 657 192 763 523 198 664 188 385 009 880 32;
  • 8) 0.666 666 666 666 657 192 763 523 198 664 188 385 009 880 32 × 2 = 1 + 0.333 333 333 333 314 385 527 046 397 328 376 770 019 760 64;
  • 9) 0.333 333 333 333 314 385 527 046 397 328 376 770 019 760 64 × 2 = 0 + 0.666 666 666 666 628 771 054 092 794 656 753 540 039 521 28;
  • 10) 0.666 666 666 666 628 771 054 092 794 656 753 540 039 521 28 × 2 = 1 + 0.333 333 333 333 257 542 108 185 589 313 507 080 079 042 56;
  • 11) 0.333 333 333 333 257 542 108 185 589 313 507 080 079 042 56 × 2 = 0 + 0.666 666 666 666 515 084 216 371 178 627 014 160 158 085 12;
  • 12) 0.666 666 666 666 515 084 216 371 178 627 014 160 158 085 12 × 2 = 1 + 0.333 333 333 333 030 168 432 742 357 254 028 320 316 170 24;
  • 13) 0.333 333 333 333 030 168 432 742 357 254 028 320 316 170 24 × 2 = 0 + 0.666 666 666 666 060 336 865 484 714 508 056 640 632 340 48;
  • 14) 0.666 666 666 666 060 336 865 484 714 508 056 640 632 340 48 × 2 = 1 + 0.333 333 333 332 120 673 730 969 429 016 113 281 264 680 96;
  • 15) 0.333 333 333 332 120 673 730 969 429 016 113 281 264 680 96 × 2 = 0 + 0.666 666 666 664 241 347 461 938 858 032 226 562 529 361 92;
  • 16) 0.666 666 666 664 241 347 461 938 858 032 226 562 529 361 92 × 2 = 1 + 0.333 333 333 328 482 694 923 877 716 064 453 125 058 723 84;
  • 17) 0.333 333 333 328 482 694 923 877 716 064 453 125 058 723 84 × 2 = 0 + 0.666 666 666 656 965 389 847 755 432 128 906 250 117 447 68;
  • 18) 0.666 666 666 656 965 389 847 755 432 128 906 250 117 447 68 × 2 = 1 + 0.333 333 333 313 930 779 695 510 864 257 812 500 234 895 36;
  • 19) 0.333 333 333 313 930 779 695 510 864 257 812 500 234 895 36 × 2 = 0 + 0.666 666 666 627 861 559 391 021 728 515 625 000 469 790 72;
  • 20) 0.666 666 666 627 861 559 391 021 728 515 625 000 469 790 72 × 2 = 1 + 0.333 333 333 255 723 118 782 043 457 031 250 000 939 581 44;
  • 21) 0.333 333 333 255 723 118 782 043 457 031 250 000 939 581 44 × 2 = 0 + 0.666 666 666 511 446 237 564 086 914 062 500 001 879 162 88;
  • 22) 0.666 666 666 511 446 237 564 086 914 062 500 001 879 162 88 × 2 = 1 + 0.333 333 333 022 892 475 128 173 828 125 000 003 758 325 76;
  • 23) 0.333 333 333 022 892 475 128 173 828 125 000 003 758 325 76 × 2 = 0 + 0.666 666 666 045 784 950 256 347 656 250 000 007 516 651 52;
  • 24) 0.666 666 666 045 784 950 256 347 656 250 000 007 516 651 52 × 2 = 1 + 0.333 333 332 091 569 900 512 695 312 500 000 015 033 303 04;
  • 25) 0.333 333 332 091 569 900 512 695 312 500 000 015 033 303 04 × 2 = 0 + 0.666 666 664 183 139 801 025 390 625 000 000 030 066 606 08;
  • 26) 0.666 666 664 183 139 801 025 390 625 000 000 030 066 606 08 × 2 = 1 + 0.333 333 328 366 279 602 050 781 250 000 000 060 133 212 16;
  • 27) 0.333 333 328 366 279 602 050 781 250 000 000 060 133 212 16 × 2 = 0 + 0.666 666 656 732 559 204 101 562 500 000 000 120 266 424 32;
  • 28) 0.666 666 656 732 559 204 101 562 500 000 000 120 266 424 32 × 2 = 1 + 0.333 333 313 465 118 408 203 125 000 000 000 240 532 848 64;
  • 29) 0.333 333 313 465 118 408 203 125 000 000 000 240 532 848 64 × 2 = 0 + 0.666 666 626 930 236 816 406 250 000 000 000 481 065 697 28;
  • 30) 0.666 666 626 930 236 816 406 250 000 000 000 481 065 697 28 × 2 = 1 + 0.333 333 253 860 473 632 812 500 000 000 000 962 131 394 56;
  • 31) 0.333 333 253 860 473 632 812 500 000 000 000 962 131 394 56 × 2 = 0 + 0.666 666 507 720 947 265 625 000 000 000 001 924 262 789 12;
  • 32) 0.666 666 507 720 947 265 625 000 000 000 001 924 262 789 12 × 2 = 1 + 0.333 333 015 441 894 531 250 000 000 000 003 848 525 578 24;
  • 33) 0.333 333 015 441 894 531 250 000 000 000 003 848 525 578 24 × 2 = 0 + 0.666 666 030 883 789 062 500 000 000 000 007 697 051 156 48;
  • 34) 0.666 666 030 883 789 062 500 000 000 000 007 697 051 156 48 × 2 = 1 + 0.333 332 061 767 578 125 000 000 000 000 015 394 102 312 96;
  • 35) 0.333 332 061 767 578 125 000 000 000 000 015 394 102 312 96 × 2 = 0 + 0.666 664 123 535 156 250 000 000 000 000 030 788 204 625 92;
  • 36) 0.666 664 123 535 156 250 000 000 000 000 030 788 204 625 92 × 2 = 1 + 0.333 328 247 070 312 500 000 000 000 000 061 576 409 251 84;
  • 37) 0.333 328 247 070 312 500 000 000 000 000 061 576 409 251 84 × 2 = 0 + 0.666 656 494 140 625 000 000 000 000 000 123 152 818 503 68;
  • 38) 0.666 656 494 140 625 000 000 000 000 000 123 152 818 503 68 × 2 = 1 + 0.333 312 988 281 250 000 000 000 000 000 246 305 637 007 36;
  • 39) 0.333 312 988 281 250 000 000 000 000 000 246 305 637 007 36 × 2 = 0 + 0.666 625 976 562 500 000 000 000 000 000 492 611 274 014 72;
  • 40) 0.666 625 976 562 500 000 000 000 000 000 492 611 274 014 72 × 2 = 1 + 0.333 251 953 125 000 000 000 000 000 000 985 222 548 029 44;
  • 41) 0.333 251 953 125 000 000 000 000 000 000 985 222 548 029 44 × 2 = 0 + 0.666 503 906 250 000 000 000 000 000 001 970 445 096 058 88;
  • 42) 0.666 503 906 250 000 000 000 000 000 001 970 445 096 058 88 × 2 = 1 + 0.333 007 812 500 000 000 000 000 000 003 940 890 192 117 76;
  • 43) 0.333 007 812 500 000 000 000 000 000 003 940 890 192 117 76 × 2 = 0 + 0.666 015 625 000 000 000 000 000 000 007 881 780 384 235 52;
  • 44) 0.666 015 625 000 000 000 000 000 000 007 881 780 384 235 52 × 2 = 1 + 0.332 031 250 000 000 000 000 000 000 015 763 560 768 471 04;
  • 45) 0.332 031 250 000 000 000 000 000 000 015 763 560 768 471 04 × 2 = 0 + 0.664 062 500 000 000 000 000 000 000 031 527 121 536 942 08;
  • 46) 0.664 062 500 000 000 000 000 000 000 031 527 121 536 942 08 × 2 = 1 + 0.328 125 000 000 000 000 000 000 000 063 054 243 073 884 16;
  • 47) 0.328 125 000 000 000 000 000 000 000 063 054 243 073 884 16 × 2 = 0 + 0.656 250 000 000 000 000 000 000 000 126 108 486 147 768 32;
  • 48) 0.656 250 000 000 000 000 000 000 000 126 108 486 147 768 32 × 2 = 1 + 0.312 500 000 000 000 000 000 000 000 252 216 972 295 536 64;
  • 49) 0.312 500 000 000 000 000 000 000 000 252 216 972 295 536 64 × 2 = 0 + 0.625 000 000 000 000 000 000 000 000 504 433 944 591 073 28;
  • 50) 0.625 000 000 000 000 000 000 000 000 504 433 944 591 073 28 × 2 = 1 + 0.250 000 000 000 000 000 000 000 001 008 867 889 182 146 56;
  • 51) 0.250 000 000 000 000 000 000 000 001 008 867 889 182 146 56 × 2 = 0 + 0.500 000 000 000 000 000 000 000 002 017 735 778 364 293 12;
  • 52) 0.500 000 000 000 000 000 000 000 002 017 735 778 364 293 12 × 2 = 1 + 0.000 000 000 000 000 000 000 000 004 035 471 556 728 586 24;
  • 53) 0.000 000 000 000 000 000 000 000 004 035 471 556 728 586 24 × 2 = 0 + 0.000 000 000 000 000 000 000 000 008 070 943 113 457 172 48;

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:


0.333 333 333 333 333 259 318 465 024 989 563 971 757 889 69(10) =

0.0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0(2)

5. Positive number before normalization:

1.333 333 333 333 333 259 318 465 024 989 563 971 757 889 69(10) =

1.0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 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:


1.333 333 333 333 333 259 318 465 024 989 563 971 757 889 69(10) =

1.0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0(2) =

1.0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 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):
1.0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 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. 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0 =

0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101


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) =
0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101


Decimal number 1.333 333 333 333 333 259 318 465 024 989 563 971 757 889 69 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1111 1111 - 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101

Share

» Convert decimal 1.333 333 333 333 333 259 318 465 024 989 563 971 757 888 73 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: Show, Don’t Tell, the Strategic Power of Action. Gemini Notebook YouTube Video of 9.21 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