-0.000 000 000 742 147 676 646 690 9 Converted to 32 Bit Single Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.000 000 000 742 147 676 646 690 9(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
-0.000 000 000 742 147 676 646 690 9(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. Start with the positive version of the number:

|-0.000 000 000 742 147 676 646 690 9| = 0.000 000 000 742 147 676 646 690 9


2. 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;

3. 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)


4. Convert to binary (base 2) the fractional part: 0.000 000 000 742 147 676 646 690 9.

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) 0.000 000 000 742 147 676 646 690 9 × 2 = 0 + 0.000 000 001 484 295 353 293 381 8;
  • 2) 0.000 000 001 484 295 353 293 381 8 × 2 = 0 + 0.000 000 002 968 590 706 586 763 6;
  • 3) 0.000 000 002 968 590 706 586 763 6 × 2 = 0 + 0.000 000 005 937 181 413 173 527 2;
  • 4) 0.000 000 005 937 181 413 173 527 2 × 2 = 0 + 0.000 000 011 874 362 826 347 054 4;
  • 5) 0.000 000 011 874 362 826 347 054 4 × 2 = 0 + 0.000 000 023 748 725 652 694 108 8;
  • 6) 0.000 000 023 748 725 652 694 108 8 × 2 = 0 + 0.000 000 047 497 451 305 388 217 6;
  • 7) 0.000 000 047 497 451 305 388 217 6 × 2 = 0 + 0.000 000 094 994 902 610 776 435 2;
  • 8) 0.000 000 094 994 902 610 776 435 2 × 2 = 0 + 0.000 000 189 989 805 221 552 870 4;
  • 9) 0.000 000 189 989 805 221 552 870 4 × 2 = 0 + 0.000 000 379 979 610 443 105 740 8;
  • 10) 0.000 000 379 979 610 443 105 740 8 × 2 = 0 + 0.000 000 759 959 220 886 211 481 6;
  • 11) 0.000 000 759 959 220 886 211 481 6 × 2 = 0 + 0.000 001 519 918 441 772 422 963 2;
  • 12) 0.000 001 519 918 441 772 422 963 2 × 2 = 0 + 0.000 003 039 836 883 544 845 926 4;
  • 13) 0.000 003 039 836 883 544 845 926 4 × 2 = 0 + 0.000 006 079 673 767 089 691 852 8;
  • 14) 0.000 006 079 673 767 089 691 852 8 × 2 = 0 + 0.000 012 159 347 534 179 383 705 6;
  • 15) 0.000 012 159 347 534 179 383 705 6 × 2 = 0 + 0.000 024 318 695 068 358 767 411 2;
  • 16) 0.000 024 318 695 068 358 767 411 2 × 2 = 0 + 0.000 048 637 390 136 717 534 822 4;
  • 17) 0.000 048 637 390 136 717 534 822 4 × 2 = 0 + 0.000 097 274 780 273 435 069 644 8;
  • 18) 0.000 097 274 780 273 435 069 644 8 × 2 = 0 + 0.000 194 549 560 546 870 139 289 6;
  • 19) 0.000 194 549 560 546 870 139 289 6 × 2 = 0 + 0.000 389 099 121 093 740 278 579 2;
  • 20) 0.000 389 099 121 093 740 278 579 2 × 2 = 0 + 0.000 778 198 242 187 480 557 158 4;
  • 21) 0.000 778 198 242 187 480 557 158 4 × 2 = 0 + 0.001 556 396 484 374 961 114 316 8;
  • 22) 0.001 556 396 484 374 961 114 316 8 × 2 = 0 + 0.003 112 792 968 749 922 228 633 6;
  • 23) 0.003 112 792 968 749 922 228 633 6 × 2 = 0 + 0.006 225 585 937 499 844 457 267 2;
  • 24) 0.006 225 585 937 499 844 457 267 2 × 2 = 0 + 0.012 451 171 874 999 688 914 534 4;
  • 25) 0.012 451 171 874 999 688 914 534 4 × 2 = 0 + 0.024 902 343 749 999 377 829 068 8;
  • 26) 0.024 902 343 749 999 377 829 068 8 × 2 = 0 + 0.049 804 687 499 998 755 658 137 6;
  • 27) 0.049 804 687 499 998 755 658 137 6 × 2 = 0 + 0.099 609 374 999 997 511 316 275 2;
  • 28) 0.099 609 374 999 997 511 316 275 2 × 2 = 0 + 0.199 218 749 999 995 022 632 550 4;
  • 29) 0.199 218 749 999 995 022 632 550 4 × 2 = 0 + 0.398 437 499 999 990 045 265 100 8;
  • 30) 0.398 437 499 999 990 045 265 100 8 × 2 = 0 + 0.796 874 999 999 980 090 530 201 6;
  • 31) 0.796 874 999 999 980 090 530 201 6 × 2 = 1 + 0.593 749 999 999 960 181 060 403 2;
  • 32) 0.593 749 999 999 960 181 060 403 2 × 2 = 1 + 0.187 499 999 999 920 362 120 806 4;
  • 33) 0.187 499 999 999 920 362 120 806 4 × 2 = 0 + 0.374 999 999 999 840 724 241 612 8;
  • 34) 0.374 999 999 999 840 724 241 612 8 × 2 = 0 + 0.749 999 999 999 681 448 483 225 6;
  • 35) 0.749 999 999 999 681 448 483 225 6 × 2 = 1 + 0.499 999 999 999 362 896 966 451 2;
  • 36) 0.499 999 999 999 362 896 966 451 2 × 2 = 0 + 0.999 999 999 998 725 793 932 902 4;
  • 37) 0.999 999 999 998 725 793 932 902 4 × 2 = 1 + 0.999 999 999 997 451 587 865 804 8;
  • 38) 0.999 999 999 997 451 587 865 804 8 × 2 = 1 + 0.999 999 999 994 903 175 731 609 6;
  • 39) 0.999 999 999 994 903 175 731 609 6 × 2 = 1 + 0.999 999 999 989 806 351 463 219 2;
  • 40) 0.999 999 999 989 806 351 463 219 2 × 2 = 1 + 0.999 999 999 979 612 702 926 438 4;
  • 41) 0.999 999 999 979 612 702 926 438 4 × 2 = 1 + 0.999 999 999 959 225 405 852 876 8;
  • 42) 0.999 999 999 959 225 405 852 876 8 × 2 = 1 + 0.999 999 999 918 450 811 705 753 6;
  • 43) 0.999 999 999 918 450 811 705 753 6 × 2 = 1 + 0.999 999 999 836 901 623 411 507 2;
  • 44) 0.999 999 999 836 901 623 411 507 2 × 2 = 1 + 0.999 999 999 673 803 246 823 014 4;
  • 45) 0.999 999 999 673 803 246 823 014 4 × 2 = 1 + 0.999 999 999 347 606 493 646 028 8;
  • 46) 0.999 999 999 347 606 493 646 028 8 × 2 = 1 + 0.999 999 998 695 212 987 292 057 6;
  • 47) 0.999 999 998 695 212 987 292 057 6 × 2 = 1 + 0.999 999 997 390 425 974 584 115 2;
  • 48) 0.999 999 997 390 425 974 584 115 2 × 2 = 1 + 0.999 999 994 780 851 949 168 230 4;
  • 49) 0.999 999 994 780 851 949 168 230 4 × 2 = 1 + 0.999 999 989 561 703 898 336 460 8;
  • 50) 0.999 999 989 561 703 898 336 460 8 × 2 = 1 + 0.999 999 979 123 407 796 672 921 6;
  • 51) 0.999 999 979 123 407 796 672 921 6 × 2 = 1 + 0.999 999 958 246 815 593 345 843 2;
  • 52) 0.999 999 958 246 815 593 345 843 2 × 2 = 1 + 0.999 999 916 493 631 186 691 686 4;
  • 53) 0.999 999 916 493 631 186 691 686 4 × 2 = 1 + 0.999 999 832 987 262 373 383 372 8;
  • 54) 0.999 999 832 987 262 373 383 372 8 × 2 = 1 + 0.999 999 665 974 524 746 766 745 6;

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).


5. 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:


0.000 000 000 742 147 676 646 690 9(10) =

0.0000 0000 0000 0000 0000 0000 0000 0011 0010 1111 1111 1111 1111 11(2)

6. Positive number before normalization:

0.000 000 000 742 147 676 646 690 9(10) =

0.0000 0000 0000 0000 0000 0000 0000 0011 0010 1111 1111 1111 1111 11(2)

7. Normalize the binary representation of the number.

Shift the decimal mark 31 positions to the right, so that only one non zero digit remains to the left of it:


0.000 000 000 742 147 676 646 690 9(10) =

0.0000 0000 0000 0000 0000 0000 0000 0011 0010 1111 1111 1111 1111 11(2) =

0.0000 0000 0000 0000 0000 0000 0000 0011 0010 1111 1111 1111 1111 11(2) × 20 =

1.1001 0111 1111 1111 1111 111(2) × 2-31


8. 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 1 (a negative number)

Exponent (unadjusted): -31

Mantissa (not normalized):
1.1001 0111 1111 1111 1111 111


9. Adjust the exponent.

Use the 8 bit excess/bias notation:


Exponent (adjusted) =

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

-31 + 2(8-1) - 1 =

(-31 + 127)(10) =

96(10)


10. 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;
  • 96 ÷ 2 = 48 + 0;
  • 48 ÷ 2 = 24 + 0;
  • 24 ÷ 2 = 12 + 0;
  • 12 ÷ 2 = 6 + 0;
  • 6 ÷ 2 = 3 + 0;
  • 3 ÷ 2 = 1 + 1;
  • 1 ÷ 2 = 0 + 1;

11. Construct the base 2 representation of the adjusted exponent.

Take all the remainders starting from the bottom of the list constructed above.


Exponent (adjusted) =

96(10) =

0110 0000(2)


12. 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. 100 1011 1111 1111 1111 1111 =

100 1011 1111 1111 1111 1111


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

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

Exponent (8 bits) =
0110 0000

Mantissa (23 bits) =
100 1011 1111 1111 1111 1111


Decimal number -0.000 000 000 742 147 676 646 690 9 converted to 32 bit single precision IEEE 754 binary floating point representation:

1 - 0110 0000 - 100 1011 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