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Learn how to write very large or very small values with scientific notation, preserve significant figures, and convert units without losing meaning.

---

## When Zeros Get in the Way

Physics often uses values that are too large or too small to write comfortably. An area of $$0.000764 \text{ m}^2$$ is still readable, but an electron mass of about $$0.000000000000000000000000000000911 \text{ kg}$$ makes it easy to lose count of the zeros.

Visible text: Physics often uses values that are too large or too small to write comfortably. An area of is still readable, but an electron mass of about makes it easy to lose count of the zeros.

**Scientific notation** writes a value as a coefficient multiplied by a power of $$10$$.

Visible text: **Scientific notation** writes a value as a coefficient multiplied by a power of .

```math
a \times 10^n
```

For positive values, the standard form is:

```math
1 \le a < 10 \quad \text{and} \quad n \in \mathbb{Z}
```

The coefficient $$a$$ carries the significant figures. The exponent $$n$$ carries the scale.

Visible text: The coefficient carries the significant figures. The exponent carries the scale.

## The Decimal Point Moves, the Value Stays

The safest way to read scientific notation is to imagine moving the decimal point until the coefficient is between $$1$$ and $$10$$.

Visible text: The safest way to read scientific notation is to imagine moving the decimal point until the coefficient is between and .

| Ordinary value | Decimal movement | Scientific notation |
| :------------- | :--------------- | :------------------ |
| $$7\,640\,000 \text{ m}$$ | $$6$$ places left | $$7.64 \times 10^6 \text{ m}$$ |
| $$0.000764 \text{ m}^2$$ | $$4$$ places right | $$7.64 \times 10^{-4} \text{ m}^2$$ |
| $$0.000000911 \text{ kg}$$ | $$7$$ places right | $$9.11 \times 10^{-7} \text{ kg}$$ |

Visible text: | Ordinary value | Decimal movement | Scientific notation |
| :------------- | :--------------- | :------------------ |
| | places left | |
| | places right | |
| | places right | |

If the decimal point moves left, the exponent is positive. If the decimal point moves right, the exponent is negative.

```math
\begin{aligned}
7\,640\,000 &= 7.64 \times 10^6 \\
0.000764 &= 7.64 \times 10^{-4}
\end{aligned}
```

## The Exponent Is Not a Significant Figure

In scientific notation, the number of significant figures is read from the coefficient, not from the power of $$10$$.

Visible text: In scientific notation, the number of significant figures is read from the coefficient, not from the power of .

| Scientific notation | Significant figures | Reason |
| :------------------ | :------------------ | :----- |
| $$7.64 \times 10^{-4} \text{ m}^2$$ | $$3$$ | The coefficient is $$7.64$$ |
| $$1.0 \times 10^{-7} \text{ m}$$ | $$2$$ | The zero after the decimal point in $$1.0$$ is intentional |
| $$1 \times 10^3 \text{ m}$$ | $$1$$ | The coefficient is only $$1$$ |
| $$1.000 \times 10^3 \text{ m}$$ | $$4$$ | The zeros in $$1.000$$ express precision |

Visible text: | Scientific notation | Significant figures | Reason |
| :------------------ | :------------------ | :----- |
| | | The coefficient is |
| | | The zero after the decimal point in is intentional |
| | | The coefficient is only |
| | | The zeros in express precision |

That is why scientific notation is useful for measurement. The value $$1000 \text{ m}$$ can be ambiguous, but $$1.000 \times 10^3 \text{ m}$$ clearly has $$4$$ significant figures.

Visible text: That is why scientific notation is useful for measurement. The value can be ambiguous, but clearly has significant figures.

## The Order for Reporting Measurements

When scientific notation is used for a measurement result, do not reverse the order.

```math
\text{round to significant figures} \to \text{convert units} \to \text{write scientific notation}
```

For example, the calculated area of a bottle cap is $$7.641504 \text{ cm}^2$$. If the original diameter supports only $$3$$ significant figures, the area is written as:

Visible text: For example, the calculated area of a bottle cap is . If the original diameter supports only significant figures, the area is written as:

```math
7.641504 \text{ cm}^2 \to 7.64 \text{ cm}^2
```

Only after that do we convert the area to the International System of Units (SI). SI is the international measurement unit standard used in science.

```math
\begin{aligned}
1 \text{ cm} &= 10^{-2} \text{ m} \\
1 \text{ cm}^2 &= (10^{-2} \text{ m})^2 = 10^{-4} \text{ m}^2 \\
7.64 \text{ cm}^2 &= 7.64 \times 10^{-4} \text{ m}^2
\end{aligned}
```

So the SI scientific-notation result is $$7.64 \times 10^{-4} \text{ m}^2$$.

Visible text: So the SI scientific-notation result is .

> Be careful with powered units. The conversion factor is powered too. From $$\text{cm}$$ to $$\text{m}$$, the factor is $$10^{-2}$$, but from $$\text{cm}^2$$ to $$\text{m}^2$$, the factor is $$10^{-4}$$.

Visible text: > Be careful with powered units. The conversion factor is powered too. From to , the factor is , but from to , the factor is .

## Calculating Without Long Strings of Zeros

Scientific notation also keeps operations tidy because powers of $$10$$ can be grouped.

Visible text: Scientific notation also keeps operations tidy because powers of can be grouped.

Suppose a very small object has length $$2.5 \times 10^{-6} \text{ m}$$ and width $$4.0 \times 10^{-7} \text{ m}$$.

Visible text: Suppose a very small object has length and width .

```math
\begin{aligned}
A &= (2.5 \times 10^{-6})(4.0 \times 10^{-7}) \text{ m}^2 \\
&= (2.5 \times 4.0) \times 10^{-6+(-7)} \text{ m}^2 \\
&= 10.0 \times 10^{-13} \text{ m}^2 \\
&= 1.00 \times 10^{-12} \text{ m}^2
\end{aligned}
```

The last step matters: the coefficient $$10.0$$ is not in standard form because it is not less than $$10$$. We rewrite it as $$1.00 \times 10^{-12}$$ while preserving $$3$$ significant figures.

Visible text: The last step matters: the coefficient is not in standard form because it is not less than . We rewrite it as while preserving significant figures.

Measurement-value writing references from NIST Guide to the SI and OpenStax Significant Figures can be opened through [the NIST link](https://www.nist.gov/pml/special-publication-811/nist-guide-si-chapter-7-rules-and-style-conventions-expressing-values) and [the OpenStax link](https://openstax.org/books/university-physics-volume-1/pages/1-6-significant-figures).