significant figures rules for addition and subtraction
user
When adding or subtracting numbers, the result should have the same number of decimal places as the number with the fewest decimal places. Here are the steps to follow:

1. Identify the number with the fewest decimal places.
2. Perform the addition or subtraction.
3. Round the result to the same number of decimal places as the number with the fewest decimal places.

For example:
- 12.34 + 5.6 = 17.94, but since 5.6 has only one decimal place, the result should be rounded to 17.9.
- 10.005 - 2.1 = 7.905, but since 2.1 has only one decimal place, the result should be rounded to 7.9.

This ensures that the precision of the result does not exceed the least precise measurement.
assistant
I'm working with numbers in scientific notation. How do I apply the significant figures rules for addition and subtraction in this case?
user
When working with numbers in scientific notation for addition and subtraction, the key is to align the decimal points before performing the operations. Here are the steps to follow:

1. **Convert the numbers to the same power of 10**: This ensures that the decimal points are aligned.
2. **Add or subtract the coefficients**: Perform the addition or subtraction on the coefficients (the numbers before the multiplication by the power of 10).
3. **Round the result**: Round the result to the same number of decimal places as the number with the fewest decimal places.

### Example

Let's say you have the following numbers in scientific notation:
- \( 3.45 \times 10^2 \)
- \( 2.1 \times 10^2 \)

**Step 1: Convert to the same power of 10**
Both numbers are already in the same power of 10 (\(10^2\)).

**Step 2: Add or subtract the coefficients**
\[ 3.45 + 2.1 = 5.55 \]

**Step 3: Round the result**
The number \( 2.1 \) has only one decimal place, so the result should be rounded to one decimal place.
\[ 5.55 \approx 5.6 \]

So, the final result is:
\[ 5.6 \times 10^2 \]

### Another Example

Let's say you have:
- \( 4.56 \times 10^3 \)
- \( 7.8 \times 10^2 \)

**Step 1: Convert to the same power of 10**
Convert \( 7.8 \times 10^2 \) to \( 0.78 \times 10^3 \).

**Step 2: Add or subtract the coefficients**
\[ 4.56 + 0.78 = 5.34 \]

**Step 3: Round the result**
The number \( 0.78 \) has two decimal places, so the result should be rounded to two decimal places.
\[ 5.34 \]

So, the final result is:
\[ 5.34 \times 10^3 \]

By following these steps, you can ensure that your calculations maintain the appropriate precision.
assistant
Can you give an example of subtracting numbers in scientific notation? Specifically, \( 5.67 \times 10^4 \) - \( 3.4 \times 10^3 \)?
user
Certainly! Let's go through the steps to subtract the numbers \( 5.67 \times 10^4 \) and \( 3.4 \times 10^3 \).

### Step-by-Step Solution

1. **Convert the numbers to the same power of 10**:
   - \( 5.67 \times 10^4 \) is already in \( 10^4 \).
   - Convert \( 3.4 \times 10^3 \) to \( 0.34 \times 10^4 \).

2. **Subtract the coefficients**:
   - Subtract \( 0.34 \) from \( 5.67 \):
     \[ 5.67 - 0.34 = 5.33 \]

3. **Round the result**:
   - The number \( 0.34 \) has two decimal places, so the result should be rounded to two decimal places.
     \[ 5.33 \]

4. **Combine the result with the power of 10**:
   \[ 5.33 \times 10^4 \]

So, the final result of \( 5.67 \times 10^4 - 3.4 \times 10^3 \) is:
\[ 5.33 \times 10^4 \]
assistant