Scientific Notation

Scientific notation writes a nonzero number as a number between $1$ and $10$ times a power of $10$. It is a compact way to write numbers like $4{,}500{,}000$ or $0.00045$ without changing their value.

$$a \times 10^n$$

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

The condition on $a$ matters. The coefficient must stay between $1$ and $10$ in absolute value, so $45 \times 10^3$ is not standard scientific notation even though it equals $4.5 \times 10^4$.

What Scientific Notation Tells You

Every time you move the decimal point one place, you are multiplying or dividing by $10$. Scientific notation packages that place-value idea into a short form.

If you move the decimal to the left, the original number was at least $10$, so the exponent is positive. If you move the decimal to the right, the original number was between $0$ and $1$ in absolute value, so the exponent is negative.

That gives you a quick reading rule:

• Large numbers use positive powers of $10$.
• Small nonzero numbers use negative powers of $10$.

Worked Example: Write $0.00045$ In Scientific Notation

Move the decimal point until the leading number is between $1$ and $10$:

$$0.00045 \rightarrow 4.5$$

The decimal moved $4$ places to the right. Moving right means the exponent is negative, so

$$0.00045 = 4.5 \times 10^{-4}$$

You can check the value:

$$10^{-4} = \frac{1}{10^4} = \frac{1}{10000}$$

so

$$4.5 \times 10^{-4} = \frac{4.5}{10000} = 0.00045$$

This example shows the two decisions that matter most: first make the coefficient usable, then choose the sign of the exponent from the direction you moved the decimal.

Common Mistakes With Scientific Notation

1. Using a coefficient outside the standard range. For example, $45 \times 10^4$ is equivalent to a scientific-notation value, but it is not in standard form because $45$ is not between $1$ and $10$.
2. Reversing the exponent sign. A very small positive number needs a negative exponent, not a positive one.
3. Counting decimal moves incorrectly when zeros are involved.
4. Forgetting the nonzero condition. The usual form $a \times 10^n$ with $1 \le |a| < 10$ describes nonzero numbers; zero is usually written simply as $0$.

When Scientific Notation Is Used

Scientific notation is useful when place value gets hard to read. That happens often in science, engineering, measurement, and data work.

You will see it in values such as microscopic lengths, astronomical distances, and quantities that vary by many powers of $10$. It also makes calculations with very large or very small numbers easier to organize.

How To Read Scientific Notation Quickly

Read the coefficient first, then read the power of $10$ as a place-value instruction.

For example, in $6.2 \times 10^5$, the $6.2$ gives the leading size and $10^5$ shows the number is in the hundred-thousands range. In $6.2 \times 10^{-5}$, the same leading size is scaled down to a very small number.

Try Your Own Version

Try writing $7{,}200{,}000$ and $0.0000081$ in scientific notation. Then check whether your coefficient is between $1$ and $10$ and whether the exponent sign matches the direction you moved the decimal.