Lesson 6. Floating Point Arithmetic
Lesson Objective
- Be able to carry out floating point arithmetic.
- Be able to carry out floating point subtraction.
Lesson Notes
Floating Point Arithmetic
- Work out where the binary point goes in each number.
- Line both numbers up on the normal binary number line. The binary points need to be in the same place.
- Now add them as normal.
1. These are the floating point numbers we are adding.
| Mantissa |
Exponent |
| -1 |
|
1/2 |
1/4 |
1/8 |
1/16 |
-4 |
2 |
1 |
| 0 |
• |
1 |
0 |
1 |
0 |
0 |
1 |
1 |
| 0 |
• |
1 |
0 |
1 |
1 |
0 |
0 |
1 |
2. Put the point in the correct place. Using the Exponent.
3. Convert to fixed point. Line up the decimal points. Add 0's in the extra places.
| -8 |
4 |
2 |
1 |
|
1/2 |
1/4 |
1/8 |
1/16 |
| 0 |
1 |
0 |
1 |
• |
0 |
0 |
0 |
0 |
| 0 |
0 |
0 |
1 |
• |
0 |
1 |
1 |
0 |
4. Normal addition on both numbers.
|
|
1 |
|
|
|
|
|
|
|
|
|
0 |
1 |
0 |
1 |
• |
0 |
0 |
0 |
0 |
? |
| + |
0 |
0 |
0 |
1 |
• |
0 |
1 |
1 |
0 |
? |
|
0 |
1 |
1 |
0 |
• |
0 |
1 |
1 |
0 |
? |
5. Normalise the binary number. Move the decimal point 3 spaces to the left. Positive number, msb must be 0, 2nd msb must be 1. Exponent is +3 because the point would have to move 3 spaces to the right to return the original value.
| Mantissa |
Exponent |
| -1 |
|
1/2 |
1/4 |
1/8 |
1/16 |
1/32 |
1/64 |
1/128 |
-4 |
2 |
1 |
| 0 |
• |
1 |
1 |
0 |
0 |
1 |
1 |
0 |
0 |
1 |
1 |
Floating Point Subtraction
- To subtract a floating point number from another, first convert them both to fixed point.
- Find the two’s complement of the number to be subtracted.
- Add the two numbers.
- Convert result to normalised floating point.
1. These are the floating point numbers we are subtracting.
| Mantissa |
Exponent |
| -1 |
|
1/2 |
1/4 |
1/8 |
1/16 |
-4 |
2 |
1 |
| 0 |
• |
1 |
1 |
0 |
0 |
0 |
1 |
0 |
| 0 |
• |
1 |
0 |
1 |
0 |
0 |
1 |
1 |
2. Put the point in the correct place. Using the Exponent.
3. Convert to fixed point. Line up the decimal points. Add 0's in the extra places.
| -8 |
4 |
2 |
1 |
|
1/2 |
1/4 |
1/8 |
1/16 |
| 0 |
0 |
1 |
1 |
• |
0 |
0 |
0 |
0 |
| 0 |
1 |
0 |
1 |
• |
0 |
0 |
0 |
0 |
4. Normal subtraction of the numbers. Flip and add one. Digits to the right of the decimal point stay the same.
|
0 |
1 |
0 |
1 |
• |
0 |
0 |
0 |
0 |
5 |
|
1 |
0 |
1 |
0 |
• |
0 |
0 |
0 |
0 |
|
| + |
0 |
0 |
0 |
1 |
• |
|
|
|
|
|
|
1 |
0 |
1 |
1 |
• |
0 |
0 |
0 |
0 |
-5 |
5. Add the positive and negative number.
|
|
1 |
|
|
|
|
|
|
|
|
|
0 |
0 |
1 |
1 |
• |
0 |
0 |
0 |
0 |
3 |
| + |
1 |
0 |
1 |
1 |
• |
0 |
0 |
0 |
0 |
-5 |
|
1 |
1 |
1 |
0 |
• |
0 |
0 |
0 |
0 |
-2 |
6. Normalise the numbers.
| Mantissa |
Exponent |
| -1 |
|
1/2 |
1/4 |
1/8 |
1/16 |
1/32 |
1/64 |
1/128 |
-4 |
2 |
1 |
| 1 |
• |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
1 |
0 |
