Very large or very small numbers can be expressed in the simplest form by using scientific notation.

In this notation any number can be written as the product of a number $x$  such that $1\le x<10$  and a power of $10$ .

Concept: Scientific notation is a method to express very large or very small numbers in a compact and easily manageable form. Any number can be written as the product of two parts:

1. A decimal number $x$ such that $1 \leq x < 10$.
2. A power of 10.

The general form of scientific notation is:

x×10n

Where:

• $x$ is the coefficient ($1 \leq x < 10$).
• $n$ is an integer representing the power of 10.

Steps to Express a Number in Scientific Notation:

1. Locate the Decimal Point: Identify the position of the decimal point in the original number.
2. Adjust $x$$x$: Move the decimal point to create a new number between 1 and 10.
3. Count the Moves: Count the number of places the decimal point was moved. This determines the power of 10 ($n$):
   • If the decimal point is moved to the left, $n$ is positive.
   • If the decimal point is moved to the right, $n$ is negative.
4. Combine: Write the number as $x \times 10^n$.

Examples:

Example 1: Large Number

Express 5,600,000 in scientific notation.

• Original number: $5,600,000$
• Move the decimal point 6 places to the left to get $5.6$.
• Power of 10: $6$ (because the decimal moved left).
• Scientific notation: $5.6 \times 10^6$.

Example 2: Small Number

Express 0.00042 in scientific notation.

• Original number: $0.00042$
• Move the decimal point 4 places to the right to get $4.2$.
• Power of 10: $-4$ (because the decimal moved right).
• Scientific notation: $4.2 \times 10^{-4}$.

Example 3: Negative Large Number

Express -123,000 in scientific notation.

• Original number: $-123,000$
• Move the decimal point 5 places to the left to get $-1.23$.
• Power of 10: $5$.
• Scientific notation: $-1.23 \times 10^5$.

Example 4: Negative Small Number

Express -0.00789 in scientific notation.

• Original number: $-0.00789$
• Move the decimal point 3 places to the right to get $-7.89$.
• Power of 10: $-3$.
• Scientific notation: $-7.89 \times 10^{-3}$.