Question

Lesson 12, scientific notification

Answer 1

It seems you might be referring to a lesson on scientific notation. Scientific notation is a way to express very large or very small numbers in a compact form. Here’s an overview of the topic, including the basic principles, rules, and how to use scientific notation.

### What is Scientific Notation?

Scientific notation expresses numbers as a product of two factors:
1. A number between 1 and 10 (the significand or mantissa).
2. A power of 10.

The general form is:
\[ a \times 10^n \]
where:
- \( a \) is the significand (1 ≤ |a| < 10)
- \( n \) is an integer (the exponent)

### Converting to Scientific Notation

1. **Identify the significant figures:** Find the first non-zero digit in the number.
2. **Move the decimal point:** Shift the decimal point to the left or right until only one non-zero digit remains to its left.
3. **Count the decimal places moved:** This will determine the exponent.
- If you move the decimal to the left, the exponent is positive.
- If you move the decimal to the right, the exponent is negative.
4. **Write in the form:** Combine the significand with the power of ten.

**Example:**
Convert 45000 to scientific notation.
- Move the decimal point left 4 places: 4.5
- Since we moved left, the exponent is +4.
- So, 45000 = \( 4.5 \times 10^4 \).

### Converting from Scientific Notation to Standard Form

To convert from scientific notation back to standard form, you do the following:

1. Multiply the significand by \( 10^n \) (move the decimal point in the significand).
2. If \( n \) is positive, move the decimal point to the right.
3. If \( n \) is negative, move the decimal point to the left.

**Example:**
Convert \( 3.2 \times 10^3 \) to standard form.
- Move the decimal point 3 places to the right: 3200.

### Operations with Scientific Notation

**Multiplication:**
- Multiply the significands.
- Add the exponents of 10.

\[
(a \times 10^m) \times (b \times 10^n) = (a \times b) \times 10^{(m+n)}
\]

**Division:**
- Divide the significands.
- Subtract the exponents of 10.

\[
\frac{a \times 10^m}{b \times 10^n} = \left(\frac{a}{b}\right) \times 10^{(m-n)}
\]

**Example:**
Multiply \( 2.0 \times 10^3 \) and \( 3.0 \times 10^2 \).
- \( 2.0 \times 3.0 = 6.0 \)
- \( 3 + 2 = 5 \)
- So, \( (2.0 \times 10^3) \times (3.0 \times 10^2) = 6.0 \times 10^5 \).

### Conclusion

Understanding scientific notation is crucial for dealing with a variety of scientific and mathematical problems, especially those involving large or small numbers. Being able to convert to and from scientific notation, as well as perform operations with it, will enhance your ability to communicate and calculate effectively in scientific contexts.