Convert the Number 0.132 812 3 to 32 Bit Single Precision IEEE 754 Binary Floating Point Representation Standard, From a Base 10 Decimal System Number. Detailed Explanations
Number 0.132 812 3(10) converted and written in 32 bit single precision IEEE 754 binary floating point representation (1 bit for sign, 8 bits for exponent, 23 bits for mantissa)
The first steps we'll go through to make the conversion:
Convert to binary (to base 2) the integer part of the number.
Convert to binary the fractional part of the number.
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: 0.132 812 3.
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) 0.132 812 3 × 2 = 0 + 0.265 624 6;
- 2) 0.265 624 6 × 2 = 0 + 0.531 249 2;
- 3) 0.531 249 2 × 2 = 1 + 0.062 498 4;
- 4) 0.062 498 4 × 2 = 0 + 0.124 996 8;
- 5) 0.124 996 8 × 2 = 0 + 0.249 993 6;
- 6) 0.249 993 6 × 2 = 0 + 0.499 987 2;
- 7) 0.499 987 2 × 2 = 0 + 0.999 974 4;
- 8) 0.999 974 4 × 2 = 1 + 0.999 948 8;
- 9) 0.999 948 8 × 2 = 1 + 0.999 897 6;
- 10) 0.999 897 6 × 2 = 1 + 0.999 795 2;
- 11) 0.999 795 2 × 2 = 1 + 0.999 590 4;
- 12) 0.999 590 4 × 2 = 1 + 0.999 180 8;
- 13) 0.999 180 8 × 2 = 1 + 0.998 361 6;
- 14) 0.998 361 6 × 2 = 1 + 0.996 723 2;
- 15) 0.996 723 2 × 2 = 1 + 0.993 446 4;
- 16) 0.993 446 4 × 2 = 1 + 0.986 892 8;
- 17) 0.986 892 8 × 2 = 1 + 0.973 785 6;
- 18) 0.973 785 6 × 2 = 1 + 0.947 571 2;
- 19) 0.947 571 2 × 2 = 1 + 0.895 142 4;
- 20) 0.895 142 4 × 2 = 1 + 0.790 284 8;
- 21) 0.790 284 8 × 2 = 1 + 0.580 569 6;
- 22) 0.580 569 6 × 2 = 1 + 0.161 139 2;
- 23) 0.161 139 2 × 2 = 0 + 0.322 278 4;
- 24) 0.322 278 4 × 2 = 0 + 0.644 556 8;
- 25) 0.644 556 8 × 2 = 1 + 0.289 113 6;
- 26) 0.289 113 6 × 2 = 0 + 0.578 227 2;
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...)
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:
0.132 812 3(10) =
0.0010 0001 1111 1111 1111 1100 10(2)
5. Positive number before normalization:
0.132 812 3(10) =
0.0010 0001 1111 1111 1111 1100 10(2)
The last steps we'll go through to make the conversion:
Normalize the binary representation of the number.
Adjust the exponent.
Convert the adjusted exponent from the decimal (base 10) to 8 bit binary.
Normalize the mantissa.
6. Normalize the binary representation of the number.
Shift the decimal mark 3 positions to the right, so that only one non zero digit remains to the left of it:
0.132 812 3(10) =
0.0010 0001 1111 1111 1111 1100 10(2) =
0.0010 0001 1111 1111 1111 1100 10(2) × 20 =
1.0000 1111 1111 1111 1110 010(2) × 2-3
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): -3
Mantissa (not normalized):
1.0000 1111 1111 1111 1110 010
8. Adjust the exponent.
Use the 8 bit excess/bias notation:
Exponent (adjusted) =
Exponent (unadjusted) + 2(8-1) - 1 =
-3 + 2(8-1) - 1 =
(-3 + 127)(10) =
124(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;
- 124 ÷ 2 = 62 + 0;
- 62 ÷ 2 = 31 + 0;
- 31 ÷ 2 = 15 + 1;
- 15 ÷ 2 = 7 + 1;
- 7 ÷ 2 = 3 + 1;
- 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) =
124(10) =
0111 1100(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. 000 0111 1111 1111 1111 0010 =
000 0111 1111 1111 1111 0010
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) =
0111 1100
Mantissa (23 bits) =
000 0111 1111 1111 1111 0010
The base ten decimal number 0.132 812 3 converted and written in 32 bit single precision IEEE 754 binary floating point representation:
0 - 0111 1100 - 000 0111 1111 1111 1111 0010
Sign (1 bit):
Exponent (8 bits):
0
301
291
281
271
261
250
240
23
Mantissa (23 bits):
0
220
210
200
191
181
171
161
151
141
131
121
111
101
91
81
71
61
51
40
30
21
10
0
Convert to 32 bit single precision IEEE 754 binary floating point representation standard
A number in 64 bit double 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 (11 bits), mantissa (52 bits)
The 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
Available Base Conversions Between Decimal and Binary Systems
Conversions Between Decimal System Numbers (Written in Base Ten) and Binary System Numbers (Base Two and Computer Representation):
1. Integer -> Binary
2. Decimal -> Binary
3. Binary -> Integer
4. Binary -> Decimal