Writing Numbers in Scientific Notation

**Scientific notation** is a way to write very large or very small numbers in a compact form: $$a \times 10^n$$ Where: - $a$ is a number between $1$ and $10$ (including $1$, but not $10$): $1 \le a < 10$ - $n$ is an integer (positive, negative, or zero) **Examples:** - $6,020,000,000,000,000,000,000,000 = 6.02 \times 10^{23}$ (Avogadro's number) - $0.000000001 = 1 \times 10^{-9}$ (one billionth) The exponent $n$ tells you how many places to move the decimal point.

Converting a Large Number

Write $45,600,000$ in scientific notation.

Place the decimal after the first non-zero digit: Move the decimal to get a number between 1 and 10: $4.56$ = $4.56$

Count decimal places moved: Original: $45600000.0$ → $4.56$ means we moved 7 places left = 7 places

Write the power of 10: Moving left = positive exponent = $10^7$

Combine into scientific notation: $4.56 \times 10^7$ = $4.56 \times 10^7$

Answer: $45,600,000 = 4.56 \times 10^7$

Converting a Small Number

Write $0.00082$ in scientific notation.

Find the first non-zero digit: Skip the zeros: $0.000\underline{8}2$, first non-zero digit is 8 = 8 is the first significant digit

Place decimal after the first non-zero digit: We need $8.2$ (a number between 1 and 10) = $8.2$

Count decimal places moved: From $0.00082$ to $8.2$: moved 4 places right = 4 places right

Write the power of 10: Moving right = negative exponent = $10^{-4}$

Combine into scientific notation: $8.2 \times 10^{-4}$ = $8.2 \times 10^{-4}$

Answer: $0.00082 = 8.2 \times 10^{-4}$

Correcting Improper Scientific Notation

Rewrite $32.5 \times 10^4$ in proper scientific notation.

Identify the problem: $32.5$ is not between 1 and 10 (it's greater than 10) = Need to adjust the coefficient

Adjust the coefficient: Move decimal one place left: $32.5 \rightarrow 3.25$ = $3.25$

Compensate the exponent: We divided by 10, so multiply the power by 10: $10^4 \times 10 = 10^5$ = $10^5$

Write correct form: $3.25 \times 10^5$ = $3.25 \times 10^5$

Answer: $32.5 \times 10^4 = 3.25 \times 10^5$

Converting a Number Between 1 and 10

Write $7.5$ in scientific notation.

Check if the number is already between 1 and 10: $1 \le 7.5 < 10$ ✓ = Already in correct form

Determine the exponent: No decimal movement needed, so exponent is 0 = $10^0 = 1$

Write in scientific notation: $7.5 \times 10^0$ = $7.5 \times 10^0$

Answer: $7.5 = 7.5 \times 10^0$

Scientists, engineers, and mathematicians work with numbers that would be impossible to write out fully: - **Astronomy**: The distance to the nearest star is about $40,000,000,000,000$ km. In scientific notation: $4 \times 10^{13}$ km - **Biology**: A human cell is about $0.00001$ meters wide. In scientific notation: $1 \times 10^{-5}$ m - **Computing**: A modern computer can perform $1,000,000,000,000$ calculations per second: $1 \times 10^{12}$ operations/sec - **Chemistry**: The mass of a hydrogen atom is $0.00000000000000000000000000167$ kg: $1.67 \times 10^{-27}$ kg Without scientific notation, calculations with these numbers would be extremely error-prone!

Mistake: Writing $12.5 \times 10^3$ instead of $1.25 \times 10^4$

Why: The coefficient must be between 1 and 10. The number 12.5 is greater than 10.

Correct: Always adjust so the coefficient $a$ satisfies $1 \le a < 10$. If $a \ge 10$, move the decimal left and increase the exponent.

Mistake: Using the wrong sign for the exponent

Why: Students confuse which direction gives positive vs negative exponents.

Correct: Large numbers (move decimal LEFT) → positive exponent. Small numbers less than 1 (move decimal RIGHT) → negative exponent.

Mistake: Forgetting to count all the zeros

Why: When counting places, especially with many zeros, it's easy to miscount.

Correct: Write out each digit position or use place value: $0.000045$ has 5 places to move (count: $0.00004.5$).

Mistake: Writing $0.82 \times 10^{-3}$ instead of $8.2 \times 10^{-4}$

Why: The coefficient 0.82 is less than 1, which violates the rule $1 \le a < 10$.

Correct: The coefficient must be at least 1. Move the decimal right to get $8.2$, then adjust the exponent to $-4$.

Astronomy: Distances in Space

Astronomers use scientific notation to express vast cosmic distances that would be impractical to write in standard form.

The distance to the Andromeda Galaxy is approximately $2,400,000,000,000,000,000,000$ meters, written as $2.4 \times 10^{21}$ m.

Microbiology: Sizes of Organisms

Biologists use scientific notation to describe organisms too small to see with the naked eye.

A typical bacterium is about $0.000002$ meters long, written as $2 \times 10^{-6}$ m (2 micrometers).

Computer Science: Data Storage

Tech companies measure data in bytes, with prefixes representing powers of 10.

A terabyte is $1,000,000,000,000$ bytes, or $1 \times 10^{12}$ bytes.

Scientific notation expresses numbers as $a \times 10^n$ where $1 \le a < 10$

For large numbers, move decimal LEFT and use POSITIVE exponent

For small numbers (less than 1), move decimal RIGHT and use NEGATIVE exponent

Count the decimal places moved to determine the exponent value

Always check that your coefficient is between 1 and 10

Q: Why can't the coefficient be 10 or greater?

A: If $a \ge 10$, you can always rewrite it by moving the decimal and adjusting the exponent. For example, $10 \times 10^5 = 1 \times 10^6$. Having one standard form prevents confusion.

Q: What if my number is exactly 1, like 1,000,000?

A: Write it as $1 \times 10^6$. The coefficient 1 satisfies $1 \le a < 10$.

Q: Is $3 \times 10^0$ a valid scientific notation?

A: Yes! Since $10^0 = 1$, this equals 3. Any number between 1 and 10 can be written with exponent 0.

Exit Presentation

Writing Numbers in Scientific Notation

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Arithmetic / Scientific Notation

Writing Numbers in Scientific Notation

Learn how to convert large and small numbers into scientific notation and understand when to use this format.

Intermediate25 min

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