Convert the Number 406.054 687 7 to 32 Bit Single Precision IEEE 754 Binary Floating Point Representation Standard, From a Base 10 Decimal System Number. Detailed Explanations
Number 406.054 687 7(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: 406.
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;
- 406 ÷ 2 = 203 + 0;
- 203 ÷ 2 = 101 + 1;
- 101 ÷ 2 = 50 + 1;
- 50 ÷ 2 = 25 + 0;
- 25 ÷ 2 = 12 + 1;
- 12 ÷ 2 = 6 + 0;
- 6 ÷ 2 = 3 + 0;
- 3 ÷ 2 = 1 + 1;
- 1 ÷ 2 = 0 + 1;
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.
406(10) =
1 1001 0110(2)
3. Convert to binary (base 2) the fractional part: 0.054 687 7.
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) 0.054 687 7 × 2 = 0 + 0.109 375 4;
- 2) 0.109 375 4 × 2 = 0 + 0.218 750 8;
- 3) 0.218 750 8 × 2 = 0 + 0.437 501 6;
- 4) 0.437 501 6 × 2 = 0 + 0.875 003 2;
- 5) 0.875 003 2 × 2 = 1 + 0.750 006 4;
- 6) 0.750 006 4 × 2 = 1 + 0.500 012 8;
- 7) 0.500 012 8 × 2 = 1 + 0.000 025 6;
- 8) 0.000 025 6 × 2 = 0 + 0.000 051 2;
- 9) 0.000 051 2 × 2 = 0 + 0.000 102 4;
- 10) 0.000 102 4 × 2 = 0 + 0.000 204 8;
- 11) 0.000 204 8 × 2 = 0 + 0.000 409 6;
- 12) 0.000 409 6 × 2 = 0 + 0.000 819 2;
- 13) 0.000 819 2 × 2 = 0 + 0.001 638 4;
- 14) 0.001 638 4 × 2 = 0 + 0.003 276 8;
- 15) 0.003 276 8 × 2 = 0 + 0.006 553 6;
- 16) 0.006 553 6 × 2 = 0 + 0.013 107 2;
- 17) 0.013 107 2 × 2 = 0 + 0.026 214 4;
- 18) 0.026 214 4 × 2 = 0 + 0.052 428 8;
- 19) 0.052 428 8 × 2 = 0 + 0.104 857 6;
- 20) 0.104 857 6 × 2 = 0 + 0.209 715 2;
- 21) 0.209 715 2 × 2 = 0 + 0.419 430 4;
- 22) 0.419 430 4 × 2 = 0 + 0.838 860 8;
- 23) 0.838 860 8 × 2 = 1 + 0.677 721 6;
- 24) 0.677 721 6 × 2 = 1 + 0.355 443 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:
0.054 687 7(10) =
0.0000 1110 0000 0000 0000 0011(2)
5. Positive number before normalization:
406.054 687 7(10) =
1 1001 0110.0000 1110 0000 0000 0000 0011(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 8 positions to the left, so that only one non zero digit remains to the left of it:
406.054 687 7(10) =
1 1001 0110.0000 1110 0000 0000 0000 0011(2) =
1 1001 0110.0000 1110 0000 0000 0000 0011(2) × 20 =
1.1001 0110 0000 1110 0000 0000 0000 0011(2) × 28
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): 8
Mantissa (not normalized):
1.1001 0110 0000 1110 0000 0000 0000 0011
8. Adjust the exponent.
Use the 8 bit excess/bias notation:
Exponent (adjusted) =
Exponent (unadjusted) + 2(8-1) - 1 =
8 + 2(8-1) - 1 =
(8 + 127)(10) =
135(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;
- 135 ÷ 2 = 67 + 1;
- 67 ÷ 2 = 33 + 1;
- 33 ÷ 2 = 16 + 1;
- 16 ÷ 2 = 8 + 0;
- 8 ÷ 2 = 4 + 0;
- 4 ÷ 2 = 2 + 0;
- 2 ÷ 2 = 1 + 0;
- 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) =
135(10) =
1000 0111(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, by removing the excess bits, from the right (if any of the excess bits is set on 1, we are losing precision...).
Mantissa (normalized) =
1. 100 1011 0000 0111 0000 0000 0 0000 0011 =
100 1011 0000 0111 0000 0000
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) =
1000 0111
Mantissa (23 bits) =
100 1011 0000 0111 0000 0000
The base ten decimal number 406.054 687 7 converted and written in 32 bit single precision IEEE 754 binary floating point representation:
0 - 1000 0111 - 100 1011 0000 0111 0000 0000
Sign (1 bit):
Exponent (8 bits):
1
300
290
280
270
261
251
241
23
Mantissa (23 bits):
1
220
210
201
190
181
171
160
150
140
130
120
111
101
91
80
70
60
50
40
30
20
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