Skill: Write large and small numbers in scientific notation

Scientific notation writes a nonzero number as

$$a \times 10^n$$

where $1 \le a < 10$ and $n$ is an integer.

Steps

1. Move the decimal point until the number begins with exactly one nonzero digit.
2. Count how many places the decimal point moved.
3. Use:
   • a positive exponent when the original number is large, usually greater than $10$;
   • a negative exponent when the original number is between $0$ and $1$.
4. Remember: the exponent’s negative sign does not make the number negative. It tells you to move the decimal point left when changing back to standard form.

Worked example

Write $0.00056$ in scientific notation.

Step 1: Move the decimal point so one nonzero digit is to its left:

$$0.00056 \longrightarrow 5.6$$

Step 2: Count the places moved.

The decimal point moved $4$ places to the right.

Step 3: Choose the exponent.

Because the original number is between $0$ and $1$, use a negative exponent:

$$\boxed{0.00056 = 5.6 \times 10^{-4}}$$

To check, $10^{-4}$ means move the decimal point in $5.6$ four places to the left, giving $0.00056$.

For a large number, the decimal point moves left to create the number $a$, so the exponent is positive. For example, moving the decimal point $6$ places left would give an exponent of $6$.