-0.000 000 000 742 147 676 646 708 07 Converted to 32 Bit Single Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.000 000 000 742 147 676 646 708 07(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
-0.000 000 000 742 147 676 646 708 07(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:

|-0.000 000 000 742 147 676 646 708 07| = 0.000 000 000 742 147 676 646 708 07


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: 0.000 000 000 742 147 676 646 708 07.

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) 0.000 000 000 742 147 676 646 708 07 × 2 = 0 + 0.000 000 001 484 295 353 293 416 14;
  • 2) 0.000 000 001 484 295 353 293 416 14 × 2 = 0 + 0.000 000 002 968 590 706 586 832 28;
  • 3) 0.000 000 002 968 590 706 586 832 28 × 2 = 0 + 0.000 000 005 937 181 413 173 664 56;
  • 4) 0.000 000 005 937 181 413 173 664 56 × 2 = 0 + 0.000 000 011 874 362 826 347 329 12;
  • 5) 0.000 000 011 874 362 826 347 329 12 × 2 = 0 + 0.000 000 023 748 725 652 694 658 24;
  • 6) 0.000 000 023 748 725 652 694 658 24 × 2 = 0 + 0.000 000 047 497 451 305 389 316 48;
  • 7) 0.000 000 047 497 451 305 389 316 48 × 2 = 0 + 0.000 000 094 994 902 610 778 632 96;
  • 8) 0.000 000 094 994 902 610 778 632 96 × 2 = 0 + 0.000 000 189 989 805 221 557 265 92;
  • 9) 0.000 000 189 989 805 221 557 265 92 × 2 = 0 + 0.000 000 379 979 610 443 114 531 84;
  • 10) 0.000 000 379 979 610 443 114 531 84 × 2 = 0 + 0.000 000 759 959 220 886 229 063 68;
  • 11) 0.000 000 759 959 220 886 229 063 68 × 2 = 0 + 0.000 001 519 918 441 772 458 127 36;
  • 12) 0.000 001 519 918 441 772 458 127 36 × 2 = 0 + 0.000 003 039 836 883 544 916 254 72;
  • 13) 0.000 003 039 836 883 544 916 254 72 × 2 = 0 + 0.000 006 079 673 767 089 832 509 44;
  • 14) 0.000 006 079 673 767 089 832 509 44 × 2 = 0 + 0.000 012 159 347 534 179 665 018 88;
  • 15) 0.000 012 159 347 534 179 665 018 88 × 2 = 0 + 0.000 024 318 695 068 359 330 037 76;
  • 16) 0.000 024 318 695 068 359 330 037 76 × 2 = 0 + 0.000 048 637 390 136 718 660 075 52;
  • 17) 0.000 048 637 390 136 718 660 075 52 × 2 = 0 + 0.000 097 274 780 273 437 320 151 04;
  • 18) 0.000 097 274 780 273 437 320 151 04 × 2 = 0 + 0.000 194 549 560 546 874 640 302 08;
  • 19) 0.000 194 549 560 546 874 640 302 08 × 2 = 0 + 0.000 389 099 121 093 749 280 604 16;
  • 20) 0.000 389 099 121 093 749 280 604 16 × 2 = 0 + 0.000 778 198 242 187 498 561 208 32;
  • 21) 0.000 778 198 242 187 498 561 208 32 × 2 = 0 + 0.001 556 396 484 374 997 122 416 64;
  • 22) 0.001 556 396 484 374 997 122 416 64 × 2 = 0 + 0.003 112 792 968 749 994 244 833 28;
  • 23) 0.003 112 792 968 749 994 244 833 28 × 2 = 0 + 0.006 225 585 937 499 988 489 666 56;
  • 24) 0.006 225 585 937 499 988 489 666 56 × 2 = 0 + 0.012 451 171 874 999 976 979 333 12;
  • 25) 0.012 451 171 874 999 976 979 333 12 × 2 = 0 + 0.024 902 343 749 999 953 958 666 24;
  • 26) 0.024 902 343 749 999 953 958 666 24 × 2 = 0 + 0.049 804 687 499 999 907 917 332 48;
  • 27) 0.049 804 687 499 999 907 917 332 48 × 2 = 0 + 0.099 609 374 999 999 815 834 664 96;
  • 28) 0.099 609 374 999 999 815 834 664 96 × 2 = 0 + 0.199 218 749 999 999 631 669 329 92;
  • 29) 0.199 218 749 999 999 631 669 329 92 × 2 = 0 + 0.398 437 499 999 999 263 338 659 84;
  • 30) 0.398 437 499 999 999 263 338 659 84 × 2 = 0 + 0.796 874 999 999 998 526 677 319 68;
  • 31) 0.796 874 999 999 998 526 677 319 68 × 2 = 1 + 0.593 749 999 999 997 053 354 639 36;
  • 32) 0.593 749 999 999 997 053 354 639 36 × 2 = 1 + 0.187 499 999 999 994 106 709 278 72;
  • 33) 0.187 499 999 999 994 106 709 278 72 × 2 = 0 + 0.374 999 999 999 988 213 418 557 44;
  • 34) 0.374 999 999 999 988 213 418 557 44 × 2 = 0 + 0.749 999 999 999 976 426 837 114 88;
  • 35) 0.749 999 999 999 976 426 837 114 88 × 2 = 1 + 0.499 999 999 999 952 853 674 229 76;
  • 36) 0.499 999 999 999 952 853 674 229 76 × 2 = 0 + 0.999 999 999 999 905 707 348 459 52;
  • 37) 0.999 999 999 999 905 707 348 459 52 × 2 = 1 + 0.999 999 999 999 811 414 696 919 04;
  • 38) 0.999 999 999 999 811 414 696 919 04 × 2 = 1 + 0.999 999 999 999 622 829 393 838 08;
  • 39) 0.999 999 999 999 622 829 393 838 08 × 2 = 1 + 0.999 999 999 999 245 658 787 676 16;
  • 40) 0.999 999 999 999 245 658 787 676 16 × 2 = 1 + 0.999 999 999 998 491 317 575 352 32;
  • 41) 0.999 999 999 998 491 317 575 352 32 × 2 = 1 + 0.999 999 999 996 982 635 150 704 64;
  • 42) 0.999 999 999 996 982 635 150 704 64 × 2 = 1 + 0.999 999 999 993 965 270 301 409 28;
  • 43) 0.999 999 999 993 965 270 301 409 28 × 2 = 1 + 0.999 999 999 987 930 540 602 818 56;
  • 44) 0.999 999 999 987 930 540 602 818 56 × 2 = 1 + 0.999 999 999 975 861 081 205 637 12;
  • 45) 0.999 999 999 975 861 081 205 637 12 × 2 = 1 + 0.999 999 999 951 722 162 411 274 24;
  • 46) 0.999 999 999 951 722 162 411 274 24 × 2 = 1 + 0.999 999 999 903 444 324 822 548 48;
  • 47) 0.999 999 999 903 444 324 822 548 48 × 2 = 1 + 0.999 999 999 806 888 649 645 096 96;
  • 48) 0.999 999 999 806 888 649 645 096 96 × 2 = 1 + 0.999 999 999 613 777 299 290 193 92;
  • 49) 0.999 999 999 613 777 299 290 193 92 × 2 = 1 + 0.999 999 999 227 554 598 580 387 84;
  • 50) 0.999 999 999 227 554 598 580 387 84 × 2 = 1 + 0.999 999 998 455 109 197 160 775 68;
  • 51) 0.999 999 998 455 109 197 160 775 68 × 2 = 1 + 0.999 999 996 910 218 394 321 551 36;
  • 52) 0.999 999 996 910 218 394 321 551 36 × 2 = 1 + 0.999 999 993 820 436 788 643 102 72;
  • 53) 0.999 999 993 820 436 788 643 102 72 × 2 = 1 + 0.999 999 987 640 873 577 286 205 44;
  • 54) 0.999 999 987 640 873 577 286 205 44 × 2 = 1 + 0.999 999 975 281 747 154 572 410 88;

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:


0.000 000 000 742 147 676 646 708 07(10) =

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

6. Positive number before normalization:

0.000 000 000 742 147 676 646 708 07(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:


0.000 000 000 742 147 676 646 708 07(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 -0.000 000 000 742 147 676 646 708 07 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