  Float to Decimal Conversion ## The Conversion Procedure

The rules for converting a floating point number into decimal are simply to reverse of the decimal to floating point conversion:
1. If the original number is in hex, convert it to binary.
2. Separate into the sign, exponent, and mantissa fields.
3. Extract the mantissa from the mantissa field, and restore the leading one. You may also omit the trailing zeros.
4. Extract the exponent from the exponent field, and subtract the bias to recover the actual exponent of two. As before, the bias is 2k−1 − 1, where k is the number of bits in the exponent field, giving 3 for the 8-bit format and 127 for the 32-bit.
5. De-normalize the number: move the binary point so the exponent is 0, and the value of the number remains unchanged.
6. Convert the binary value to decimal. This is done just as with binary integers, but the place values right of the binary point are fractions.
7. Set the sign of the decimal number according to the sign bit of the original floating point number: make it negative for 1; leave positive for 0.

If the binary exponent is very large or small, you can convert the mantissa directly to decimal without de-normalizing. Then use a calculator to raise two to the exponent, and perform the multiplication. This will give an approximate answer, but is sufficient in most cases.

## Examples Using The Conversion Procedure

• Convert the 8-bit floating point number e7 (in hex) to decimal.
1. Convert: e716 = 111001112.
2. Seprate: 11100111
3. Mantissa: 1.0111
4. Exponent: 1102 = 610; 6 − 3 = 3.
5. De-normalize: 1.01112 × 23 = 1011.1
6. Convert: Exponents 23 22 21 20 2-1 Place Values 8 4 2 1 0.5 Bits 1 0 1 1 . 1 Value 8 + 2 + 1 + 0.5 = 11.5
7. Sign: negative.
Result: e7 is -11.5
• Convert the 8-bit floating point number 26 (in hex) to decimal.
1. Convert and separate: 2616 = 00100110 2
2. Exponent: 0102 = 210; 2 − 3 = -1.
3. Denormalize: 1.0112 × 2-1 = 0.1011.
4. Convert: Exponents 20 2-1 2-2 2-3 2-4 Place Values 1 0.5 0.25 0.125 0.0625 Bits 0 . 1 0 1 1 Value 0.5 + 0.125 + 0.0625 = 0.6875
5. Sign: positive
Result: 26 is 0.6875.
• Convert the 8-bit floating point number d3 (in hex) to decimal.
1. Convert and separate: d316 = 11010011 2
2. Exponent: 1012 = 510; 5 − 3 = 2.
3. Denormalize: 1.00112 × 22 = 100.11.
4. Convert: Exponents 22 21 20 2-1 2-2 Place Values 4 2 1 0.5 0.25 Bits 1 0 0 . 1 1 Value 4 + 0.5 + 0.25 = 4.75
5. Sign: negative
Result: d3 is -4.75.
• Convert the 32-bit floating point number 44361000 (in hex) to decimal.
1. Convert and separate: 4436100016 = 01000100001101100001000000000000 2
2. Exponent: 100010002 = 13610; 136 − 127 = 9.
3. Denormalize: 1.011011000012 × 29 = 1011011000.01.
4. Convert: Exponents 29 28 27 26 25 24 23 22 21 20 2-1 2-2 Place Values 512 256 128 64 32 16 8 4 2 1 0.5 0.25 Bits 1 0 1 1 0 1 1 0 0 0 . 0 1 Value 512 + 128 + 64 + 16 + 8 + 0.25 = 728.25
5. Sign: positive
Result: 44361000 is 728.25.
• Convert the 32-bit floating point number be580000 (in hex) to decimal.
1. Convert and separate: be58000016 = 10111110010110000000000000000000 2
2. Exponent: 011111002 = 12410; 124 − 127 = -3.
3. Denormalize: 1.10112 × 2-3 = 0.0011011.
4. Convert: Exponents 20 2-1 2-2 2-3 2-4 2-5 2-6 2-7 Place Values 1 0.5 0.25 0.125 0.0625 0.03125 0.015625 0.0078125 Bits 0 . 0 0 1 1 0 1 1 Value 0.125 + 0.0625 + 0.015625 + 0.0078125 = 0.2109375
5. Sign: negative
Result: be580000 is -0.2109375.
• Convert the 32-bit floating point number a3358000 (in hex) to decimal.
1. Convert and separate: a335800016 = 10100011001101011000000000000000 2
2. Exponent: 010001102 = 7010; 70 − 127 = -57.
3. Since the exponent is far from zero, convert the original (normalized) mantissa:  Exponents 20 2-1 2-2 2-3 2-4 2-5 2-6 2-7 2-8 Place Values 1 0.5 0.25 0.125 0.0625 0.03125 0.015625 0.0078125 0.00390625 Bits 1 . 0 1 1 0 1 0 1 1 Value 1 + 0.25 + 0.125 + 0.03125 + 0.0078125 + 0.00390625 = 1.41796875
4. Use calculator to find 1.41796875 × 2-57. You should get something like 9.83913471531 × 10-18 .
5. Sign: negative
Result: a3358000 is about -9.83913471531 × 10-18 .
• Convert the 32-bit floating point number 76650000 (in hex) to decimal.
1. Convert and separate: 7665000016 = 01110110011001010000000000000000 2
2. Exponent: 111011002 = 23610; 236 − 127 = 109.
3. Since the exponent is far from zero, convert the original (normalized) mantissa:  Exponents 20 2-1 2-2 2-3 2-4 2-5 2-6 2-7 Place Values 1 0.5 0.25 0.125 0.0625 0.03125 0.015625 0.0078125 Bits 1 . 1 1 0 0 1 0 1 Value 1 + 0.5 + 0.25 + 0.03125 + 0.0078125 = 1.7890625
4. Use calculator to find 1.7890625 × 2109. You should get something like 1.16116794981 × 1033 .
5. Sign: positive
Result: 76650000 is about 1.16116794981 × 1033 .