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Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/subject/high-school/10/physics/measurement/notation/en.mdx

Output docs content for large language models.

---

export const metadata = {
  title: "Scientific Notation",
  description:
    "Learn how to write very large or very small values with scientific notation, preserve significant figures, and convert units without losing meaning.",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "04/25/2026",
  subject: "Measurement in Scientific Work",
};

## When Zeros Get in the Way

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

**Scientific notation** writes a value as a coefficient multiplied by a power of <InlineMath math="10" />.

<BlockMath math="a \times 10^n" />

For positive values, the standard form is:

<BlockMath math="1 \le a < 10 \quad \text{and} \quad n \in \mathbb{Z}" />

The coefficient <InlineMath math="a" /> carries the significant figures. The exponent <InlineMath math="n" /> 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 <InlineMath math="1" /> and <InlineMath math="10" />.

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

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

<BlockMath 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 <InlineMath math="10" />.

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

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

## The Order for Reporting Measurements

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

<BlockMath 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 <InlineMath math="7.641504 \text{ cm}^2" />. If the original diameter supports only <InlineMath math="3" /> significant figures, the area is written as:

<BlockMath 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.

<BlockMath 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 <InlineMath math="7.64 \times 10^{-4} \text{ m}^2" />.

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

## Calculating Without Long Strings of Zeros

Scientific notation also keeps operations tidy because powers of <InlineMath math="10" /> can be grouped.

Suppose a very small object has length <InlineMath math="2.5 \times 10^{-6} \text{ m}" /> and width <InlineMath math="4.0 \times 10^{-7} \text{ m}" />.

<BlockMath 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 <InlineMath math="10.0" /> is not in standard form because it is not less than <InlineMath math="10" />. We rewrite it as <InlineMath math="1.00 \times 10^{-12}" /> while preserving <InlineMath math="3" /> 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).