Standard form is a method of writing very large or very small numbers succinctly, commonly used in scientific, mathematical, and engineering contexts. It simplifies numbers to a format of $A \times 10^n$, where $A$ is a coefficient between 1 and 10 (including 1 but not 10), and $n$ is an integer indicating the power of 10.

What is Standard Form?

The standard form notation $A \times 10^n$ allows us to express numbers in a more manageable way, especially when dealing with extremely large or small values.

Key Concepts

• Coefficient $(A)$: A number between 1 and 10.
• Power of 10 $(10^n)$: Indicates how many places the decimal point has been moved. $n$ can be positive, negative, or zero.

Converting Numbers into Standard Form

Converting a number into standard form involves two steps:

1. Find the Coefficient: Adjust the decimal point so there is only one non-zero digit to the left.

2. Determine the Power of 10: Count the decimal moves. This count, $n$, is positive if moved left, negative if right.

Example 1: Large Number

• Original: 123,456
• Standard Form: $1.23456 \times 10^5$

Example 2: Small Number

• Original: 0.00789
• Standard Form: $7.89 \times 10^{-3}$

Converting Numbers Out of Standard Form

To convert from standard form to a normal number:

1. Move the Decimal Point: Based on $n$, move the decimal point right (if $n$ is positive) or left (if $n$ is negative).

2. Adjust with Zeros: Add zeros as needed to fill gaps.

Example 1: From Standard Form

• Standard Form: $3.65 \times 10^4$
• Normal Number: 36,500

Example 2: From Standard Form

• Standard Form: $4.2 \times 10^{-3}$
• Normal Number: 0.0042

Practice Questions

1. Convert 5,000,000 into standard form.

Solution:

2. Convert $8.2 \times 10^6$ into a normal number.

Solution:

3. Calculate $2.5 \times 10^3 \times 4 \times 10^2$ in standard form.

Solution: