0.000 000 001 862 645 149 230 956 84 Converted to 32 Bit Single Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 001 862 645 149 230 956 84(10) to 32 bit single precision IEEE 754 binary floating point representation standard (1 bit for sign, 8 bits for exponent, 23 bits for mantissa)

What are the steps to convert decimal number
0.000 000 001 862 645 149 230 956 84(10) to 32 bit single precision IEEE 754 binary floating point representation (1 bit for sign, 8 bits for exponent, 23 bits for mantissa)

1. First, convert to binary (in base 2) the integer part: 0.
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;
  • 0 ÷ 2 = 0 + 0;

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.

0(10) =

0(2)


3. Convert to binary (base 2) the fractional part: 0.000 000 001 862 645 149 230 956 84.

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) 0.000 000 001 862 645 149 230 956 84 × 2 = 0 + 0.000 000 003 725 290 298 461 913 68;
  • 2) 0.000 000 003 725 290 298 461 913 68 × 2 = 0 + 0.000 000 007 450 580 596 923 827 36;
  • 3) 0.000 000 007 450 580 596 923 827 36 × 2 = 0 + 0.000 000 014 901 161 193 847 654 72;
  • 4) 0.000 000 014 901 161 193 847 654 72 × 2 = 0 + 0.000 000 029 802 322 387 695 309 44;
  • 5) 0.000 000 029 802 322 387 695 309 44 × 2 = 0 + 0.000 000 059 604 644 775 390 618 88;
  • 6) 0.000 000 059 604 644 775 390 618 88 × 2 = 0 + 0.000 000 119 209 289 550 781 237 76;
  • 7) 0.000 000 119 209 289 550 781 237 76 × 2 = 0 + 0.000 000 238 418 579 101 562 475 52;
  • 8) 0.000 000 238 418 579 101 562 475 52 × 2 = 0 + 0.000 000 476 837 158 203 124 951 04;
  • 9) 0.000 000 476 837 158 203 124 951 04 × 2 = 0 + 0.000 000 953 674 316 406 249 902 08;
  • 10) 0.000 000 953 674 316 406 249 902 08 × 2 = 0 + 0.000 001 907 348 632 812 499 804 16;
  • 11) 0.000 001 907 348 632 812 499 804 16 × 2 = 0 + 0.000 003 814 697 265 624 999 608 32;
  • 12) 0.000 003 814 697 265 624 999 608 32 × 2 = 0 + 0.000 007 629 394 531 249 999 216 64;
  • 13) 0.000 007 629 394 531 249 999 216 64 × 2 = 0 + 0.000 015 258 789 062 499 998 433 28;
  • 14) 0.000 015 258 789 062 499 998 433 28 × 2 = 0 + 0.000 030 517 578 124 999 996 866 56;
  • 15) 0.000 030 517 578 124 999 996 866 56 × 2 = 0 + 0.000 061 035 156 249 999 993 733 12;
  • 16) 0.000 061 035 156 249 999 993 733 12 × 2 = 0 + 0.000 122 070 312 499 999 987 466 24;
  • 17) 0.000 122 070 312 499 999 987 466 24 × 2 = 0 + 0.000 244 140 624 999 999 974 932 48;
  • 18) 0.000 244 140 624 999 999 974 932 48 × 2 = 0 + 0.000 488 281 249 999 999 949 864 96;
  • 19) 0.000 488 281 249 999 999 949 864 96 × 2 = 0 + 0.000 976 562 499 999 999 899 729 92;
  • 20) 0.000 976 562 499 999 999 899 729 92 × 2 = 0 + 0.001 953 124 999 999 999 799 459 84;
  • 21) 0.001 953 124 999 999 999 799 459 84 × 2 = 0 + 0.003 906 249 999 999 999 598 919 68;
  • 22) 0.003 906 249 999 999 999 598 919 68 × 2 = 0 + 0.007 812 499 999 999 999 197 839 36;
  • 23) 0.007 812 499 999 999 999 197 839 36 × 2 = 0 + 0.015 624 999 999 999 998 395 678 72;
  • 24) 0.015 624 999 999 999 998 395 678 72 × 2 = 0 + 0.031 249 999 999 999 996 791 357 44;
  • 25) 0.031 249 999 999 999 996 791 357 44 × 2 = 0 + 0.062 499 999 999 999 993 582 714 88;
  • 26) 0.062 499 999 999 999 993 582 714 88 × 2 = 0 + 0.124 999 999 999 999 987 165 429 76;
  • 27) 0.124 999 999 999 999 987 165 429 76 × 2 = 0 + 0.249 999 999 999 999 974 330 859 52;
  • 28) 0.249 999 999 999 999 974 330 859 52 × 2 = 0 + 0.499 999 999 999 999 948 661 719 04;
  • 29) 0.499 999 999 999 999 948 661 719 04 × 2 = 0 + 0.999 999 999 999 999 897 323 438 08;
  • 30) 0.999 999 999 999 999 897 323 438 08 × 2 = 1 + 0.999 999 999 999 999 794 646 876 16;
  • 31) 0.999 999 999 999 999 794 646 876 16 × 2 = 1 + 0.999 999 999 999 999 589 293 752 32;
  • 32) 0.999 999 999 999 999 589 293 752 32 × 2 = 1 + 0.999 999 999 999 999 178 587 504 64;
  • 33) 0.999 999 999 999 999 178 587 504 64 × 2 = 1 + 0.999 999 999 999 998 357 175 009 28;
  • 34) 0.999 999 999 999 998 357 175 009 28 × 2 = 1 + 0.999 999 999 999 996 714 350 018 56;
  • 35) 0.999 999 999 999 996 714 350 018 56 × 2 = 1 + 0.999 999 999 999 993 428 700 037 12;
  • 36) 0.999 999 999 999 993 428 700 037 12 × 2 = 1 + 0.999 999 999 999 986 857 400 074 24;
  • 37) 0.999 999 999 999 986 857 400 074 24 × 2 = 1 + 0.999 999 999 999 973 714 800 148 48;
  • 38) 0.999 999 999 999 973 714 800 148 48 × 2 = 1 + 0.999 999 999 999 947 429 600 296 96;
  • 39) 0.999 999 999 999 947 429 600 296 96 × 2 = 1 + 0.999 999 999 999 894 859 200 593 92;
  • 40) 0.999 999 999 999 894 859 200 593 92 × 2 = 1 + 0.999 999 999 999 789 718 401 187 84;
  • 41) 0.999 999 999 999 789 718 401 187 84 × 2 = 1 + 0.999 999 999 999 579 436 802 375 68;
  • 42) 0.999 999 999 999 579 436 802 375 68 × 2 = 1 + 0.999 999 999 999 158 873 604 751 36;
  • 43) 0.999 999 999 999 158 873 604 751 36 × 2 = 1 + 0.999 999 999 998 317 747 209 502 72;
  • 44) 0.999 999 999 998 317 747 209 502 72 × 2 = 1 + 0.999 999 999 996 635 494 419 005 44;
  • 45) 0.999 999 999 996 635 494 419 005 44 × 2 = 1 + 0.999 999 999 993 270 988 838 010 88;
  • 46) 0.999 999 999 993 270 988 838 010 88 × 2 = 1 + 0.999 999 999 986 541 977 676 021 76;
  • 47) 0.999 999 999 986 541 977 676 021 76 × 2 = 1 + 0.999 999 999 973 083 955 352 043 52;
  • 48) 0.999 999 999 973 083 955 352 043 52 × 2 = 1 + 0.999 999 999 946 167 910 704 087 04;
  • 49) 0.999 999 999 946 167 910 704 087 04 × 2 = 1 + 0.999 999 999 892 335 821 408 174 08;
  • 50) 0.999 999 999 892 335 821 408 174 08 × 2 = 1 + 0.999 999 999 784 671 642 816 348 16;
  • 51) 0.999 999 999 784 671 642 816 348 16 × 2 = 1 + 0.999 999 999 569 343 285 632 696 32;
  • 52) 0.999 999 999 569 343 285 632 696 32 × 2 = 1 + 0.999 999 999 138 686 571 265 392 64;
  • 53) 0.999 999 999 138 686 571 265 392 64 × 2 = 1 + 0.999 999 998 277 373 142 530 785 28;

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:


0.000 000 001 862 645 149 230 956 84(10) =

0.0000 0000 0000 0000 0000 0000 0000 0111 1111 1111 1111 1111 1111 1(2)

5. Positive number before normalization:

0.000 000 001 862 645 149 230 956 84(10) =

0.0000 0000 0000 0000 0000 0000 0000 0111 1111 1111 1111 1111 1111 1(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 30 positions to the right, so that only one non zero digit remains to the left of it:


0.000 000 001 862 645 149 230 956 84(10) =

0.0000 0000 0000 0000 0000 0000 0000 0111 1111 1111 1111 1111 1111 1(2) =

0.0000 0000 0000 0000 0000 0000 0000 0111 1111 1111 1111 1111 1111 1(2) × 20 =

1.1111 1111 1111 1111 1111 111(2) × 2-30


7. 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): -30

Mantissa (not normalized):
1.1111 1111 1111 1111 1111 111


8. Adjust the exponent.

Use the 8 bit excess/bias notation:


Exponent (adjusted) =

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

-30 + 2(8-1) - 1 =

(-30 + 127)(10) =

97(10)


9. Convert the adjusted exponent from the decimal (base 10) to 8 bit binary.

Use the same technique of repeatedly dividing by 2:


  • division = quotient + remainder;
  • 97 ÷ 2 = 48 + 1;
  • 48 ÷ 2 = 24 + 0;
  • 24 ÷ 2 = 12 + 0;
  • 12 ÷ 2 = 6 + 0;
  • 6 ÷ 2 = 3 + 0;
  • 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) =

97(10) =

0110 0001(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 23 bits, only if necessary (not the case here).


Mantissa (normalized) =

1. 111 1111 1111 1111 1111 1111 =

111 1111 1111 1111 1111 1111


12. 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) =
0110 0001

Mantissa (23 bits) =
111 1111 1111 1111 1111 1111


Decimal number 0.000 000 001 862 645 149 230 956 84 converted to 32 bit single precision IEEE 754 binary floating point representation:

0 - 0110 0001 - 111 1111 1111 1111 1111 1111

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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