1.999 999 999 999 999 888 977 697 537 484 345 957 636 833 190 917 968 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 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 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 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 7 × 2 = 1 + 0.999 999 999 999 999 777 955 395 074 968 691 915 273 666 381 835 937 4;
  • 2) 0.999 999 999 999 999 777 955 395 074 968 691 915 273 666 381 835 937 4 × 2 = 1 + 0.999 999 999 999 999 555 910 790 149 937 383 830 547 332 763 671 874 8;
  • 3) 0.999 999 999 999 999 555 910 790 149 937 383 830 547 332 763 671 874 8 × 2 = 1 + 0.999 999 999 999 999 111 821 580 299 874 767 661 094 665 527 343 749 6;
  • 4) 0.999 999 999 999 999 111 821 580 299 874 767 661 094 665 527 343 749 6 × 2 = 1 + 0.999 999 999 999 998 223 643 160 599 749 535 322 189 331 054 687 499 2;
  • 5) 0.999 999 999 999 998 223 643 160 599 749 535 322 189 331 054 687 499 2 × 2 = 1 + 0.999 999 999 999 996 447 286 321 199 499 070 644 378 662 109 374 998 4;
  • 6) 0.999 999 999 999 996 447 286 321 199 499 070 644 378 662 109 374 998 4 × 2 = 1 + 0.999 999 999 999 992 894 572 642 398 998 141 288 757 324 218 749 996 8;
  • 7) 0.999 999 999 999 992 894 572 642 398 998 141 288 757 324 218 749 996 8 × 2 = 1 + 0.999 999 999 999 985 789 145 284 797 996 282 577 514 648 437 499 993 6;
  • 8) 0.999 999 999 999 985 789 145 284 797 996 282 577 514 648 437 499 993 6 × 2 = 1 + 0.999 999 999 999 971 578 290 569 595 992 565 155 029 296 874 999 987 2;
  • 9) 0.999 999 999 999 971 578 290 569 595 992 565 155 029 296 874 999 987 2 × 2 = 1 + 0.999 999 999 999 943 156 581 139 191 985 130 310 058 593 749 999 974 4;
  • 10) 0.999 999 999 999 943 156 581 139 191 985 130 310 058 593 749 999 974 4 × 2 = 1 + 0.999 999 999 999 886 313 162 278 383 970 260 620 117 187 499 999 948 8;
  • 11) 0.999 999 999 999 886 313 162 278 383 970 260 620 117 187 499 999 948 8 × 2 = 1 + 0.999 999 999 999 772 626 324 556 767 940 521 240 234 374 999 999 897 6;
  • 12) 0.999 999 999 999 772 626 324 556 767 940 521 240 234 374 999 999 897 6 × 2 = 1 + 0.999 999 999 999 545 252 649 113 535 881 042 480 468 749 999 999 795 2;
  • 13) 0.999 999 999 999 545 252 649 113 535 881 042 480 468 749 999 999 795 2 × 2 = 1 + 0.999 999 999 999 090 505 298 227 071 762 084 960 937 499 999 999 590 4;
  • 14) 0.999 999 999 999 090 505 298 227 071 762 084 960 937 499 999 999 590 4 × 2 = 1 + 0.999 999 999 998 181 010 596 454 143 524 169 921 874 999 999 999 180 8;
  • 15) 0.999 999 999 998 181 010 596 454 143 524 169 921 874 999 999 999 180 8 × 2 = 1 + 0.999 999 999 996 362 021 192 908 287 048 339 843 749 999 999 998 361 6;
  • 16) 0.999 999 999 996 362 021 192 908 287 048 339 843 749 999 999 998 361 6 × 2 = 1 + 0.999 999 999 992 724 042 385 816 574 096 679 687 499 999 999 996 723 2;
  • 17) 0.999 999 999 992 724 042 385 816 574 096 679 687 499 999 999 996 723 2 × 2 = 1 + 0.999 999 999 985 448 084 771 633 148 193 359 374 999 999 999 993 446 4;
  • 18) 0.999 999 999 985 448 084 771 633 148 193 359 374 999 999 999 993 446 4 × 2 = 1 + 0.999 999 999 970 896 169 543 266 296 386 718 749 999 999 999 986 892 8;
  • 19) 0.999 999 999 970 896 169 543 266 296 386 718 749 999 999 999 986 892 8 × 2 = 1 + 0.999 999 999 941 792 339 086 532 592 773 437 499 999 999 999 973 785 6;
  • 20) 0.999 999 999 941 792 339 086 532 592 773 437 499 999 999 999 973 785 6 × 2 = 1 + 0.999 999 999 883 584 678 173 065 185 546 874 999 999 999 999 947 571 2;
  • 21) 0.999 999 999 883 584 678 173 065 185 546 874 999 999 999 999 947 571 2 × 2 = 1 + 0.999 999 999 767 169 356 346 130 371 093 749 999 999 999 999 895 142 4;
  • 22) 0.999 999 999 767 169 356 346 130 371 093 749 999 999 999 999 895 142 4 × 2 = 1 + 0.999 999 999 534 338 712 692 260 742 187 499 999 999 999 999 790 284 8;
  • 23) 0.999 999 999 534 338 712 692 260 742 187 499 999 999 999 999 790 284 8 × 2 = 1 + 0.999 999 999 068 677 425 384 521 484 374 999 999 999 999 999 580 569 6;
  • 24) 0.999 999 999 068 677 425 384 521 484 374 999 999 999 999 999 580 569 6 × 2 = 1 + 0.999 999 998 137 354 850 769 042 968 749 999 999 999 999 999 161 139 2;
  • 25) 0.999 999 998 137 354 850 769 042 968 749 999 999 999 999 999 161 139 2 × 2 = 1 + 0.999 999 996 274 709 701 538 085 937 499 999 999 999 999 998 322 278 4;
  • 26) 0.999 999 996 274 709 701 538 085 937 499 999 999 999 999 998 322 278 4 × 2 = 1 + 0.999 999 992 549 419 403 076 171 874 999 999 999 999 999 996 644 556 8;
  • 27) 0.999 999 992 549 419 403 076 171 874 999 999 999 999 999 996 644 556 8 × 2 = 1 + 0.999 999 985 098 838 806 152 343 749 999 999 999 999 999 993 289 113 6;
  • 28) 0.999 999 985 098 838 806 152 343 749 999 999 999 999 999 993 289 113 6 × 2 = 1 + 0.999 999 970 197 677 612 304 687 499 999 999 999 999 999 986 578 227 2;
  • 29) 0.999 999 970 197 677 612 304 687 499 999 999 999 999 999 986 578 227 2 × 2 = 1 + 0.999 999 940 395 355 224 609 374 999 999 999 999 999 999 973 156 454 4;
  • 30) 0.999 999 940 395 355 224 609 374 999 999 999 999 999 999 973 156 454 4 × 2 = 1 + 0.999 999 880 790 710 449 218 749 999 999 999 999 999 999 946 312 908 8;
  • 31) 0.999 999 880 790 710 449 218 749 999 999 999 999 999 999 946 312 908 8 × 2 = 1 + 0.999 999 761 581 420 898 437 499 999 999 999 999 999 999 892 625 817 6;
  • 32) 0.999 999 761 581 420 898 437 499 999 999 999 999 999 999 892 625 817 6 × 2 = 1 + 0.999 999 523 162 841 796 874 999 999 999 999 999 999 999 785 251 635 2;
  • 33) 0.999 999 523 162 841 796 874 999 999 999 999 999 999 999 785 251 635 2 × 2 = 1 + 0.999 999 046 325 683 593 749 999 999 999 999 999 999 999 570 503 270 4;
  • 34) 0.999 999 046 325 683 593 749 999 999 999 999 999 999 999 570 503 270 4 × 2 = 1 + 0.999 998 092 651 367 187 499 999 999 999 999 999 999 999 141 006 540 8;
  • 35) 0.999 998 092 651 367 187 499 999 999 999 999 999 999 999 141 006 540 8 × 2 = 1 + 0.999 996 185 302 734 374 999 999 999 999 999 999 999 998 282 013 081 6;
  • 36) 0.999 996 185 302 734 374 999 999 999 999 999 999 999 998 282 013 081 6 × 2 = 1 + 0.999 992 370 605 468 749 999 999 999 999 999 999 999 996 564 026 163 2;
  • 37) 0.999 992 370 605 468 749 999 999 999 999 999 999 999 996 564 026 163 2 × 2 = 1 + 0.999 984 741 210 937 499 999 999 999 999 999 999 999 993 128 052 326 4;
  • 38) 0.999 984 741 210 937 499 999 999 999 999 999 999 999 993 128 052 326 4 × 2 = 1 + 0.999 969 482 421 874 999 999 999 999 999 999 999 999 986 256 104 652 8;
  • 39) 0.999 969 482 421 874 999 999 999 999 999 999 999 999 986 256 104 652 8 × 2 = 1 + 0.999 938 964 843 749 999 999 999 999 999 999 999 999 972 512 209 305 6;
  • 40) 0.999 938 964 843 749 999 999 999 999 999 999 999 999 972 512 209 305 6 × 2 = 1 + 0.999 877 929 687 499 999 999 999 999 999 999 999 999 945 024 418 611 2;
  • 41) 0.999 877 929 687 499 999 999 999 999 999 999 999 999 945 024 418 611 2 × 2 = 1 + 0.999 755 859 374 999 999 999 999 999 999 999 999 999 890 048 837 222 4;
  • 42) 0.999 755 859 374 999 999 999 999 999 999 999 999 999 890 048 837 222 4 × 2 = 1 + 0.999 511 718 749 999 999 999 999 999 999 999 999 999 780 097 674 444 8;
  • 43) 0.999 511 718 749 999 999 999 999 999 999 999 999 999 780 097 674 444 8 × 2 = 1 + 0.999 023 437 499 999 999 999 999 999 999 999 999 999 560 195 348 889 6;
  • 44) 0.999 023 437 499 999 999 999 999 999 999 999 999 999 560 195 348 889 6 × 2 = 1 + 0.998 046 874 999 999 999 999 999 999 999 999 999 999 120 390 697 779 2;
  • 45) 0.998 046 874 999 999 999 999 999 999 999 999 999 999 120 390 697 779 2 × 2 = 1 + 0.996 093 749 999 999 999 999 999 999 999 999 999 998 240 781 395 558 4;
  • 46) 0.996 093 749 999 999 999 999 999 999 999 999 999 998 240 781 395 558 4 × 2 = 1 + 0.992 187 499 999 999 999 999 999 999 999 999 999 996 481 562 791 116 8;
  • 47) 0.992 187 499 999 999 999 999 999 999 999 999 999 996 481 562 791 116 8 × 2 = 1 + 0.984 374 999 999 999 999 999 999 999 999 999 999 992 963 125 582 233 6;
  • 48) 0.984 374 999 999 999 999 999 999 999 999 999 999 992 963 125 582 233 6 × 2 = 1 + 0.968 749 999 999 999 999 999 999 999 999 999 999 985 926 251 164 467 2;
  • 49) 0.968 749 999 999 999 999 999 999 999 999 999 999 985 926 251 164 467 2 × 2 = 1 + 0.937 499 999 999 999 999 999 999 999 999 999 999 971 852 502 328 934 4;
  • 50) 0.937 499 999 999 999 999 999 999 999 999 999 999 971 852 502 328 934 4 × 2 = 1 + 0.874 999 999 999 999 999 999 999 999 999 999 999 943 705 004 657 868 8;
  • 51) 0.874 999 999 999 999 999 999 999 999 999 999 999 943 705 004 657 868 8 × 2 = 1 + 0.749 999 999 999 999 999 999 999 999 999 999 999 887 410 009 315 737 6;
  • 52) 0.749 999 999 999 999 999 999 999 999 999 999 999 887 410 009 315 737 6 × 2 = 1 + 0.499 999 999 999 999 999 999 999 999 999 999 999 774 820 018 631 475 2;
  • 53) 0.499 999 999 999 999 999 999 999 999 999 999 999 774 820 018 631 475 2 × 2 = 0 + 0.999 999 999 999 999 999 999 999 999 999 999 999 549 640 037 262 950 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 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 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 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 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