-0.000 000 000 742 147 674 22 Converted to 32 Bit Single Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.000 000 000 742 147 674 22(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
-0.000 000 000 742 147 674 22(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:

|-0.000 000 000 742 147 674 22| = 0.000 000 000 742 147 674 22


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: 0.000 000 000 742 147 674 22.

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) 0.000 000 000 742 147 674 22 × 2 = 0 + 0.000 000 001 484 295 348 44;
  • 2) 0.000 000 001 484 295 348 44 × 2 = 0 + 0.000 000 002 968 590 696 88;
  • 3) 0.000 000 002 968 590 696 88 × 2 = 0 + 0.000 000 005 937 181 393 76;
  • 4) 0.000 000 005 937 181 393 76 × 2 = 0 + 0.000 000 011 874 362 787 52;
  • 5) 0.000 000 011 874 362 787 52 × 2 = 0 + 0.000 000 023 748 725 575 04;
  • 6) 0.000 000 023 748 725 575 04 × 2 = 0 + 0.000 000 047 497 451 150 08;
  • 7) 0.000 000 047 497 451 150 08 × 2 = 0 + 0.000 000 094 994 902 300 16;
  • 8) 0.000 000 094 994 902 300 16 × 2 = 0 + 0.000 000 189 989 804 600 32;
  • 9) 0.000 000 189 989 804 600 32 × 2 = 0 + 0.000 000 379 979 609 200 64;
  • 10) 0.000 000 379 979 609 200 64 × 2 = 0 + 0.000 000 759 959 218 401 28;
  • 11) 0.000 000 759 959 218 401 28 × 2 = 0 + 0.000 001 519 918 436 802 56;
  • 12) 0.000 001 519 918 436 802 56 × 2 = 0 + 0.000 003 039 836 873 605 12;
  • 13) 0.000 003 039 836 873 605 12 × 2 = 0 + 0.000 006 079 673 747 210 24;
  • 14) 0.000 006 079 673 747 210 24 × 2 = 0 + 0.000 012 159 347 494 420 48;
  • 15) 0.000 012 159 347 494 420 48 × 2 = 0 + 0.000 024 318 694 988 840 96;
  • 16) 0.000 024 318 694 988 840 96 × 2 = 0 + 0.000 048 637 389 977 681 92;
  • 17) 0.000 048 637 389 977 681 92 × 2 = 0 + 0.000 097 274 779 955 363 84;
  • 18) 0.000 097 274 779 955 363 84 × 2 = 0 + 0.000 194 549 559 910 727 68;
  • 19) 0.000 194 549 559 910 727 68 × 2 = 0 + 0.000 389 099 119 821 455 36;
  • 20) 0.000 389 099 119 821 455 36 × 2 = 0 + 0.000 778 198 239 642 910 72;
  • 21) 0.000 778 198 239 642 910 72 × 2 = 0 + 0.001 556 396 479 285 821 44;
  • 22) 0.001 556 396 479 285 821 44 × 2 = 0 + 0.003 112 792 958 571 642 88;
  • 23) 0.003 112 792 958 571 642 88 × 2 = 0 + 0.006 225 585 917 143 285 76;
  • 24) 0.006 225 585 917 143 285 76 × 2 = 0 + 0.012 451 171 834 286 571 52;
  • 25) 0.012 451 171 834 286 571 52 × 2 = 0 + 0.024 902 343 668 573 143 04;
  • 26) 0.024 902 343 668 573 143 04 × 2 = 0 + 0.049 804 687 337 146 286 08;
  • 27) 0.049 804 687 337 146 286 08 × 2 = 0 + 0.099 609 374 674 292 572 16;
  • 28) 0.099 609 374 674 292 572 16 × 2 = 0 + 0.199 218 749 348 585 144 32;
  • 29) 0.199 218 749 348 585 144 32 × 2 = 0 + 0.398 437 498 697 170 288 64;
  • 30) 0.398 437 498 697 170 288 64 × 2 = 0 + 0.796 874 997 394 340 577 28;
  • 31) 0.796 874 997 394 340 577 28 × 2 = 1 + 0.593 749 994 788 681 154 56;
  • 32) 0.593 749 994 788 681 154 56 × 2 = 1 + 0.187 499 989 577 362 309 12;
  • 33) 0.187 499 989 577 362 309 12 × 2 = 0 + 0.374 999 979 154 724 618 24;
  • 34) 0.374 999 979 154 724 618 24 × 2 = 0 + 0.749 999 958 309 449 236 48;
  • 35) 0.749 999 958 309 449 236 48 × 2 = 1 + 0.499 999 916 618 898 472 96;
  • 36) 0.499 999 916 618 898 472 96 × 2 = 0 + 0.999 999 833 237 796 945 92;
  • 37) 0.999 999 833 237 796 945 92 × 2 = 1 + 0.999 999 666 475 593 891 84;
  • 38) 0.999 999 666 475 593 891 84 × 2 = 1 + 0.999 999 332 951 187 783 68;
  • 39) 0.999 999 332 951 187 783 68 × 2 = 1 + 0.999 998 665 902 375 567 36;
  • 40) 0.999 998 665 902 375 567 36 × 2 = 1 + 0.999 997 331 804 751 134 72;
  • 41) 0.999 997 331 804 751 134 72 × 2 = 1 + 0.999 994 663 609 502 269 44;
  • 42) 0.999 994 663 609 502 269 44 × 2 = 1 + 0.999 989 327 219 004 538 88;
  • 43) 0.999 989 327 219 004 538 88 × 2 = 1 + 0.999 978 654 438 009 077 76;
  • 44) 0.999 978 654 438 009 077 76 × 2 = 1 + 0.999 957 308 876 018 155 52;
  • 45) 0.999 957 308 876 018 155 52 × 2 = 1 + 0.999 914 617 752 036 311 04;
  • 46) 0.999 914 617 752 036 311 04 × 2 = 1 + 0.999 829 235 504 072 622 08;
  • 47) 0.999 829 235 504 072 622 08 × 2 = 1 + 0.999 658 471 008 145 244 16;
  • 48) 0.999 658 471 008 145 244 16 × 2 = 1 + 0.999 316 942 016 290 488 32;
  • 49) 0.999 316 942 016 290 488 32 × 2 = 1 + 0.998 633 884 032 580 976 64;
  • 50) 0.998 633 884 032 580 976 64 × 2 = 1 + 0.997 267 768 065 161 953 28;
  • 51) 0.997 267 768 065 161 953 28 × 2 = 1 + 0.994 535 536 130 323 906 56;
  • 52) 0.994 535 536 130 323 906 56 × 2 = 1 + 0.989 071 072 260 647 813 12;
  • 53) 0.989 071 072 260 647 813 12 × 2 = 1 + 0.978 142 144 521 295 626 24;
  • 54) 0.978 142 144 521 295 626 24 × 2 = 1 + 0.956 284 289 042 591 252 48;

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:


0.000 000 000 742 147 674 22(10) =

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

6. Positive number before normalization:

0.000 000 000 742 147 674 22(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:


0.000 000 000 742 147 674 22(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 -0.000 000 000 742 147 674 22 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