Base ten decimal number 1 000 000 011 000 000 000 000 000 000 000 converted to 32 bit single precision IEEE 754 binary floating point standard

How to convert the decimal number 1 000 000 011 000 000 000 000 000 000 000(10)
to
32 bit single precision IEEE 754 binary floating point
(1 bit for sign, 8 bits for exponent, 23 bits for mantissa)

1. Divide the number repeatedly by 2, keeping track of each remainder, until we get a quotient that is equal to zero:

  • division = quotient + remainder;
  • 1 000 000 011 000 000 000 000 000 000 000 ÷ 2 = 500 000 005 500 000 000 000 000 000 000 + 0;
  • 500 000 005 500 000 000 000 000 000 000 ÷ 2 = 250 000 002 750 000 000 000 000 000 000 + 0;
  • 250 000 002 750 000 000 000 000 000 000 ÷ 2 = 125 000 001 375 000 000 000 000 000 000 + 0;
  • 125 000 001 375 000 000 000 000 000 000 ÷ 2 = 62 500 000 687 500 000 000 000 000 000 + 0;
  • 62 500 000 687 500 000 000 000 000 000 ÷ 2 = 31 250 000 343 750 000 000 000 000 000 + 0;
  • 31 250 000 343 750 000 000 000 000 000 ÷ 2 = 15 625 000 171 875 000 000 000 000 000 + 0;
  • 15 625 000 171 875 000 000 000 000 000 ÷ 2 = 7 812 500 085 937 500 000 000 000 000 + 0;
  • 7 812 500 085 937 500 000 000 000 000 ÷ 2 = 3 906 250 042 968 750 000 000 000 000 + 0;
  • 3 906 250 042 968 750 000 000 000 000 ÷ 2 = 1 953 125 021 484 375 000 000 000 000 + 0;
  • 1 953 125 021 484 375 000 000 000 000 ÷ 2 = 976 562 510 742 187 500 000 000 000 + 0;
  • 976 562 510 742 187 500 000 000 000 ÷ 2 = 488 281 255 371 093 750 000 000 000 + 0;
  • 488 281 255 371 093 750 000 000 000 ÷ 2 = 244 140 627 685 546 875 000 000 000 + 0;
  • 244 140 627 685 546 875 000 000 000 ÷ 2 = 122 070 313 842 773 437 500 000 000 + 0;
  • 122 070 313 842 773 437 500 000 000 ÷ 2 = 61 035 156 921 386 718 750 000 000 + 0;
  • 61 035 156 921 386 718 750 000 000 ÷ 2 = 30 517 578 460 693 359 375 000 000 + 0;
  • 30 517 578 460 693 359 375 000 000 ÷ 2 = 15 258 789 230 346 679 687 500 000 + 0;
  • 15 258 789 230 346 679 687 500 000 ÷ 2 = 7 629 394 615 173 339 843 750 000 + 0;
  • 7 629 394 615 173 339 843 750 000 ÷ 2 = 3 814 697 307 586 669 921 875 000 + 0;
  • 3 814 697 307 586 669 921 875 000 ÷ 2 = 1 907 348 653 793 334 960 937 500 + 0;
  • 1 907 348 653 793 334 960 937 500 ÷ 2 = 953 674 326 896 667 480 468 750 + 0;
  • 953 674 326 896 667 480 468 750 ÷ 2 = 476 837 163 448 333 740 234 375 + 0;
  • 476 837 163 448 333 740 234 375 ÷ 2 = 238 418 581 724 166 870 117 187 + 1;
  • 238 418 581 724 166 870 117 187 ÷ 2 = 119 209 290 862 083 435 058 593 + 1;
  • 119 209 290 862 083 435 058 593 ÷ 2 = 59 604 645 431 041 717 529 296 + 1;
  • 59 604 645 431 041 717 529 296 ÷ 2 = 29 802 322 715 520 858 764 648 + 0;
  • 29 802 322 715 520 858 764 648 ÷ 2 = 14 901 161 357 760 429 382 324 + 0;
  • 14 901 161 357 760 429 382 324 ÷ 2 = 7 450 580 678 880 214 691 162 + 0;
  • 7 450 580 678 880 214 691 162 ÷ 2 = 3 725 290 339 440 107 345 581 + 0;
  • 3 725 290 339 440 107 345 581 ÷ 2 = 1 862 645 169 720 053 672 790 + 1;
  • 1 862 645 169 720 053 672 790 ÷ 2 = 931 322 584 860 026 836 395 + 0;
  • 931 322 584 860 026 836 395 ÷ 2 = 465 661 292 430 013 418 197 + 1;
  • 465 661 292 430 013 418 197 ÷ 2 = 232 830 646 215 006 709 098 + 1;
  • 232 830 646 215 006 709 098 ÷ 2 = 116 415 323 107 503 354 549 + 0;
  • 116 415 323 107 503 354 549 ÷ 2 = 58 207 661 553 751 677 274 + 1;
  • 58 207 661 553 751 677 274 ÷ 2 = 29 103 830 776 875 838 637 + 0;
  • 29 103 830 776 875 838 637 ÷ 2 = 14 551 915 388 437 919 318 + 1;
  • 14 551 915 388 437 919 318 ÷ 2 = 7 275 957 694 218 959 659 + 0;
  • 7 275 957 694 218 959 659 ÷ 2 = 3 637 978 847 109 479 829 + 1;
  • 3 637 978 847 109 479 829 ÷ 2 = 1 818 989 423 554 739 914 + 1;
  • 1 818 989 423 554 739 914 ÷ 2 = 909 494 711 777 369 957 + 0;
  • 909 494 711 777 369 957 ÷ 2 = 454 747 355 888 684 978 + 1;
  • 454 747 355 888 684 978 ÷ 2 = 227 373 677 944 342 489 + 0;
  • 227 373 677 944 342 489 ÷ 2 = 113 686 838 972 171 244 + 1;
  • 113 686 838 972 171 244 ÷ 2 = 56 843 419 486 085 622 + 0;
  • 56 843 419 486 085 622 ÷ 2 = 28 421 709 743 042 811 + 0;
  • 28 421 709 743 042 811 ÷ 2 = 14 210 854 871 521 405 + 1;
  • 14 210 854 871 521 405 ÷ 2 = 7 105 427 435 760 702 + 1;
  • 7 105 427 435 760 702 ÷ 2 = 3 552 713 717 880 351 + 0;
  • 3 552 713 717 880 351 ÷ 2 = 1 776 356 858 940 175 + 1;
  • 1 776 356 858 940 175 ÷ 2 = 888 178 429 470 087 + 1;
  • 888 178 429 470 087 ÷ 2 = 444 089 214 735 043 + 1;
  • 444 089 214 735 043 ÷ 2 = 222 044 607 367 521 + 1;
  • 222 044 607 367 521 ÷ 2 = 111 022 303 683 760 + 1;
  • 111 022 303 683 760 ÷ 2 = 55 511 151 841 880 + 0;
  • 55 511 151 841 880 ÷ 2 = 27 755 575 920 940 + 0;
  • 27 755 575 920 940 ÷ 2 = 13 877 787 960 470 + 0;
  • 13 877 787 960 470 ÷ 2 = 6 938 893 980 235 + 0;
  • 6 938 893 980 235 ÷ 2 = 3 469 446 990 117 + 1;
  • 3 469 446 990 117 ÷ 2 = 1 734 723 495 058 + 1;
  • 1 734 723 495 058 ÷ 2 = 867 361 747 529 + 0;
  • 867 361 747 529 ÷ 2 = 433 680 873 764 + 1;
  • 433 680 873 764 ÷ 2 = 216 840 436 882 + 0;
  • 216 840 436 882 ÷ 2 = 108 420 218 441 + 0;
  • 108 420 218 441 ÷ 2 = 54 210 109 220 + 1;
  • 54 210 109 220 ÷ 2 = 27 105 054 610 + 0;
  • 27 105 054 610 ÷ 2 = 13 552 527 305 + 0;
  • 13 552 527 305 ÷ 2 = 6 776 263 652 + 1;
  • 6 776 263 652 ÷ 2 = 3 388 131 826 + 0;
  • 3 388 131 826 ÷ 2 = 1 694 065 913 + 0;
  • 1 694 065 913 ÷ 2 = 847 032 956 + 1;
  • 847 032 956 ÷ 2 = 423 516 478 + 0;
  • 423 516 478 ÷ 2 = 211 758 239 + 0;
  • 211 758 239 ÷ 2 = 105 879 119 + 1;
  • 105 879 119 ÷ 2 = 52 939 559 + 1;
  • 52 939 559 ÷ 2 = 26 469 779 + 1;
  • 26 469 779 ÷ 2 = 13 234 889 + 1;
  • 13 234 889 ÷ 2 = 6 617 444 + 1;
  • 6 617 444 ÷ 2 = 3 308 722 + 0;
  • 3 308 722 ÷ 2 = 1 654 361 + 0;
  • 1 654 361 ÷ 2 = 827 180 + 1;
  • 827 180 ÷ 2 = 413 590 + 0;
  • 413 590 ÷ 2 = 206 795 + 0;
  • 206 795 ÷ 2 = 103 397 + 1;
  • 103 397 ÷ 2 = 51 698 + 1;
  • 51 698 ÷ 2 = 25 849 + 0;
  • 25 849 ÷ 2 = 12 924 + 1;
  • 12 924 ÷ 2 = 6 462 + 0;
  • 6 462 ÷ 2 = 3 231 + 0;
  • 3 231 ÷ 2 = 1 615 + 1;
  • 1 615 ÷ 2 = 807 + 1;
  • 807 ÷ 2 = 403 + 1;
  • 403 ÷ 2 = 201 + 1;
  • 201 ÷ 2 = 100 + 1;
  • 100 ÷ 2 = 50 + 0;
  • 50 ÷ 2 = 25 + 0;
  • 25 ÷ 2 = 12 + 1;
  • 12 ÷ 2 = 6 + 0;
  • 6 ÷ 2 = 3 + 0;
  • 3 ÷ 2 = 1 + 1;
  • 1 ÷ 2 = 0 + 1;

2. Construct the base 2 representation of the integer part of the number, by taking all the remainders starting from the bottom of the list constructed above:

1 000 000 011 000 000 000 000 000 000 000(10) =


1100 1001 1111 0010 1100 1001 1111 0010 0100 1001 0110 0001 1111 0110 0101 0110 1010 1101 0000 1110 0000 0000 0000 0000 0000(2)

3. Normalize the binary representation of the number, shifting the decimal mark 99 positions to the left so that only one non zero digit remains to the left of it:

1 000 000 011 000 000 000 000 000 000 000(10) =


1100 1001 1111 0010 1100 1001 1111 0010 0100 1001 0110 0001 1111 0110 0101 0110 1010 1101 0000 1110 0000 0000 0000 0000 0000(2) =


1100 1001 1111 0010 1100 1001 1111 0010 0100 1001 0110 0001 1111 0110 0101 0110 1010 1101 0000 1110 0000 0000 0000 0000 0000(2) × 20 =


1.1001 0011 1110 0101 1001 0011 1110 0100 1001 0010 1100 0011 1110 1100 1010 1101 0101 1010 0001 1100 0000 0000 0000 0000 000(2) × 299

Up to this moment, there are the following elements that would feed into the 32 bit single precision IEEE 754 binary floating point representation:

Sign: 0 (a positive number)


Exponent (unadjusted): 99


Mantissa (not normalized): 1.1001 0011 1110 0101 1001 0011 1110 0100 1001 0010 1100 0011 1110 1100 1010 1101 0101 1010 0001 1100 0000 0000 0000 0000 000

4. Adjust the exponent in 8 bit excess/bias notation and then convert it from decimal (base 10) to 8 bit binary, by using the same technique of repeatedly dividing by 2:

Exponent (adjusted) =


Exponent (unadjusted) + 2(8-1) - 1 =


99 + 2(8-1) - 1 =


(99 + 127)(10) =


226(10)


  • division = quotient + remainder;
  • 226 ÷ 2 = 113 + 0;
  • 113 ÷ 2 = 56 + 1;
  • 56 ÷ 2 = 28 + 0;
  • 28 ÷ 2 = 14 + 0;
  • 14 ÷ 2 = 7 + 0;
  • 7 ÷ 2 = 3 + 1;
  • 3 ÷ 2 = 1 + 1;
  • 1 ÷ 2 = 0 + 1;

Exponent (adjusted) =


226(10) =


1110 0010(2)

5. Normalize mantissa, remove the leading (the leftmost) bit, since it's allways 1 (and the decimal point, if the case) then adjust its length to 23 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. 100 1001 1111 0010 1100 1001 1111 0010 0100 1001 0110 0001 1111 0110 0101 0110 1010 1101 0000 1110 0000 0000 0000 0000 0000 =


100 1001 1111 0010 1100 1001

Conclusion:

The three elements that make up the number's 32 bit single precision IEEE 754 binary floating point representation:

Sign (1 bit) =
0 (a positive number)


Exponent (8 bits) =
1110 0010


Mantissa (23 bits) =
100 1001 1111 0010 1100 1001

Number 1 000 000 011 000 000 000 000 000 000 000, a decimal, converted from decimal system (base 10)
to
32 bit single precision IEEE 754 binary floating point:


0 - 1110 0010 - 100 1001 1111 0010 1100 1001

(32 bits IEEE 754)
  • Sign (1 bit):

    • 0

      31
  • Exponent (8 bits):

    • 1

      30
    • 1

      29
    • 1

      28
    • 0

      27
    • 0

      26
    • 0

      25
    • 1

      24
    • 0

      23
  • Mantissa (23 bits):

    • 1

      22
    • 0

      21
    • 0

      20
    • 1

      19
    • 0

      18
    • 0

      17
    • 1

      16
    • 1

      15
    • 1

      14
    • 1

      13
    • 1

      12
    • 0

      11
    • 0

      10
    • 1

      9
    • 0

      8
    • 1

      7
    • 1

      6
    • 0

      5
    • 0

      4
    • 1

      3
    • 0

      2
    • 0

      1
    • 1

      0

Convert decimal numbers from base ten to 32 bit single precision IEEE 754 binary floating point standard

A number in 32 bit single precision IEEE 754 binary floating point standard representation requires three building elements: sign (it takes 1 bit and it's either 0 for positive or 1 for negative numbers), exponent (8 bits), mantissa (23 bits)

Latest decimal numbers converted from base ten to 32 bit single precision IEEE 754 floating point binary standard representation

How to convert decimal numbers from base ten to 32 bit single precision IEEE 754 binary floating point standard

Follow the steps below to convert a base 10 decimal number to 32 bit single 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 base ten 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 of the previous dividing 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 previous multiplying operations, starting from the top of the constructed list 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, by shifting the decimal point (or if you prefer, the decimal mark) "n" positions either to the left or to the right, so that only one non zero digit remains to the left of the decimal point.
  • 7. Adjust the exponent in 8 bit excess/bias notation and then convert it from decimal (base 10) to 8 bit binary, by using the same technique of repeatedly dividing by 2, as shown above:
    Exponent (adjusted) = Exponent (unadjusted) + 2(8-1) - 1
  • 8. Normalize mantissa, remove the leading (leftmost) bit, since it's allways '1' (and the decimal sign if the case) and adjust its length to 23 bits, either by removing the excess bits from the right (losing precision...) or by adding extra '0' bits 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 -25.347 from decimal system (base ten) to 32 bit single precision IEEE 754 binary floating point:

  • 1. Start with the positive version of the number:

    |-25.347| = 25.347

  • 2. First convert the integer part, 25. Divide it repeatedly by 2, keeping track of each remainder, until we get a quotient that is equal to zero:
    • division = quotient + remainder;
    • 25 ÷ 2 = 12 + 1;
    • 12 ÷ 2 = 6 + 0;
    • 6 ÷ 2 = 3 + 0;
    • 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:

    25(10) = 1 1001(2)

  • 4. Then convert the fractional part, 0.347. 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) 0.347 × 2 = 0 + 0.694;
    • 2) 0.694 × 2 = 1 + 0.388;
    • 3) 0.388 × 2 = 0 + 0.776;
    • 4) 0.776 × 2 = 1 + 0.552;
    • 5) 0.552 × 2 = 1 + 0.104;
    • 6) 0.104 × 2 = 0 + 0.208;
    • 7) 0.208 × 2 = 0 + 0.416;
    • 8) 0.416 × 2 = 0 + 0.832;
    • 9) 0.832 × 2 = 1 + 0.664;
    • 10) 0.664 × 2 = 1 + 0.328;
    • 11) 0.328 × 2 = 0 + 0.656;
    • 12) 0.656 × 2 = 1 + 0.312;
    • 13) 0.312 × 2 = 0 + 0.624;
    • 14) 0.624 × 2 = 1 + 0.248;
    • 15) 0.248 × 2 = 0 + 0.496;
    • 16) 0.496 × 2 = 0 + 0.992;
    • 17) 0.992 × 2 = 1 + 0.984;
    • 18) 0.984 × 2 = 1 + 0.968;
    • 19) 0.968 × 2 = 1 + 0.936;
    • 20) 0.936 × 2 = 1 + 0.872;
    • 21) 0.872 × 2 = 1 + 0.744;
    • 22) 0.744 × 2 = 1 + 0.488;
    • 23) 0.488 × 2 = 0 + 0.976;
    • 24) 0.976 × 2 = 1 + 0.952;
    • We didn't get any fractional part that was equal to zero. But we had enough iterations (over Mantissa limit = 23) 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:

    0.347(10) = 0.0101 1000 1101 0100 1111 1101(2)

  • 6. Summarizing - the positive number before normalization:

    25.347(10) = 1 1001.0101 1000 1101 0100 1111 1101(2)

  • 7. Normalize the binary representation of the number, shifting the decimal point 4 positions to the left so that only one non-zero digit stays to the left of the decimal point:

    25.347(10) =
    1 1001.0101 1000 1101 0100 1111 1101(2) =
    1 1001.0101 1000 1101 0100 1111 1101(2) × 20 =
    1.1001 0101 1000 1101 0100 1111 1101(2) × 24

  • 8. Up to this moment, there are the following elements that would feed into the 32 bit single precision IEEE 754 binary floating point:

    Sign: 1 (a negative number)

    Exponent (unadjusted): 4

    Mantissa (not-normalized): 1.1001 0101 1000 1101 0100 1111 1101

  • 9. Adjust the exponent in 8 bit excess/bias notation and then convert it from decimal (base 10) to 8 bit binary (base 2), by using the same technique of repeatedly dividing it by 2, as already demonstrated above:

    Exponent (adjusted) = Exponent (unadjusted) + 2(8-1) - 1 = (4 + 127)(10) = 131(10) =
    1000 0011(2)

  • 10. Normalize the mantissa, remove the leading (leftmost) bit, since it's allways '1' (and the decimal point) and adjust its length to 23 bits, by removing the excess bits from the right (losing precision...):

    Mantissa (not-normalized): 1.1001 0101 1000 1101 0100 1111 1101

    Mantissa (normalized): 100 1010 1100 0110 1010 0111

  • Conclusion:

    Sign (1 bit) = 1 (a negative number)

    Exponent (8 bits) = 1000 0011

    Mantissa (23 bits) = 100 1010 1100 0110 1010 0111

  • Number -25.347, converted from the decimal system (base 10) to 32 bit single precision IEEE 754 binary floating point =


    1 - 1000 0011 - 100 1010 1100 0110 1010 0111