1.6 Standard Form of a Number

Standard form (also known as scientific notation) is a way of expressing very large or very small numbers in the form: \[ a \times 10^n \quad \text{where } 1 \leq a < 10 \text{ and } n \text{ is an integer} \]

Key Concepts:

• For large numbers, \(n\) is positive and indicates how many places to move the decimal to the right.
• For small numbers, \(n\) is negative and indicates how many places to move the decimal to the left.

Examples:

Example 1: Converting a large number to standard form

Write \( 4500000 \) in standard form.

Solution:

Step 1: Move the decimal 6 places to the left: \[ 4500000 = 4.5 \times 10^6 \]

Example 2: Converting a small number to standard form

Write \( 0.00032 \) in standard form.

Solution:

Step 1: Move the decimal 4 places to the right: \[ 0.00032 = 3.2 \times 10^{-4} \]

Example 3: Multiplying numbers in standard form

Simplify: \[ (2.7 \times 10^{-4}) \times (6.3 \times 10^6) \]

Solution:

Step 1: Multiply the decimal parts: \[ 2.7 \times 6.3 = 17.01 \] Step 2: Multiply the powers of ten: \[ 10^{-4} \times 10^6 = 10^2 \] Step 3: Combine and write in standard form: \[ 17.01 \times 10^2 = 1.701 \times 10^3 \]