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Mathematics › 📐 Grade 8 › Scientific Notation: Writing Huge and Tiny Numbers Compactly
📐 Grade 8 · Lesson 13 of 15

Scientific Notation: Writing Huge and Tiny Numbers Compactly

One digit, a decimal point, and a power of ten — that's all it takes to write a number as small as an atom or as large as a galaxy.

Grade 8Pre-Algebra / Algebra 1
Scientific Notation: Writing Huge and Tiny Numbers Compactly — illustration
💡
The big idea: Some numbers in science are so large or so small that writing every digit is unwieldy — the distance to a star, the width of a virus. Scientific notation solves this by writing any number as a single digit from 1 to 9, a decimal part, and a power of ten that keeps track of the place value. Sliding the decimal point one place is the same as multiplying or dividing by 10, which is exactly why powers of ten do all the work.
🎯 By the end, you'll be able to
  • Write a large or small number in scientific notation, a × 10ⁿ with 1 ≤ a < 10
  • Convert a number from scientific notation back to standard form
  • Explain what the sign and size of the exponent tell you about the number
  • Multiply numbers written in scientific notation
📎 You should already know
  • Powers and exponents
  • Place value and decimals

Numbers too big — or too small — to write out comfortably

The Sun is about 150,000,000,000 metres from Earth. A grain of sand's width is about 0.0005 metres. Both numbers are correct, but both are clumsy: it's easy to miscount a zero, and hard to compare sizes at a glance. Scientific notation was invented exactly for numbers like these.

🔑 The form of scientific notation
A number in scientific notation is written as a × 10n, where a is a number with exactly one nonzero digit before the decimal point (so 1 ≤ a < 10), and n is an integer — the power of ten that rescales it back to the original size.
\[ a \times 10^{n}, \quad 1 \le a < 10 \]
a holds the digits; 10ⁿ holds the size. A positive n means a large number; a negative n means a small number less than 1.

Converting a large number

To write a large number in scientific notation, move the decimal point left until only one nonzero digit remains before it. The exponent n is positive and equals how many places you moved the decimal point.

📝 Worked example: Write 4,500,000 in scientific notation.
  1. Place the decimal point after the first nonzero digit: 4.5.
  2. Count how many places the decimal point moved from the end of 4500000. to right after the 4 — that's 6 places.
  3. Since we moved the decimal point left across a large number, the exponent is positive 6.
✓ 4,500,000 = <strong>4.5 &times; 10<sup>6</sup></strong>.
🎮 Scientific-Notation Zoom LIVE
Zoom across powers of ten to write huge and tiny numbers compactly.

Converting a small number

Numbers smaller than 1 work the same way, but the decimal point moves the other direction — to the right, past the leading zeros — and the exponent comes out negative. A negative exponent doesn't mean a negative number; it means “divide by that power of ten,” which makes the value smaller than 1.

📝 Worked example: Write 0.00032 in scientific notation.
  1. Move the decimal point right until only one nonzero digit sits before it: 0.00032 becomes 3.2.
  2. Count the places moved: the decimal point crossed 4 places to get from 0.00032 to 3.2.
  3. Since the original number was smaller than 1, the exponent is negative 4.
✓ 0.00032 = <strong>3.2 &times; 10<sup>&minus;4</sup></strong>.
⚠️ Keep a between 1 and 10
A common mistake is writing something like 32 × 103 or 0.32 × 106. Neither is proper scientific notation, because a must satisfy 1 ≤ a < 10. Always adjust until exactly one nonzero digit sits before the decimal point, updating the exponent to compensate.

Multiplying in scientific notation

Scientific notation makes multiplication easier, not harder: multiply the a-parts together, and add the exponents on the powers of ten — the same rule used for any product of powers with the same base.

\[ (a \times 10^{m})(b \times 10^{n}) = (a \times b) \times 10^{m+n} \]
Multiply the front numbers; add the exponents.
📝 Worked example: Multiply (2 &times; 10<sup>3</sup>) &times; (3 &times; 10<sup>5</sup>).
  1. Multiply the front numbers: 2 × 3 = 6.
  2. Add the exponents: 3 + 5 = 8.
  3. Since 6 is already between 1 and 10, no further adjustment is needed.
✓ (2 &times; 10<sup>3</sup>) &times; (3 &times; 10<sup>5</sup>) = <strong>6 &times; 10<sup>8</sup></strong>.

Check your understanding

1. Write 67,000 in scientific notation.
Moving the decimal point from 67000. to 6.7 crosses 4 places, so 67,000 = 6.7 × 10⁴.
2. Write 0.0089 in scientific notation.
Moving the decimal point from 0.0089 to 8.9 crosses 3 places to the right, and the number is less than 1, so the exponent is −3.
3. Which of these is correctly written in scientific notation?
Only 3.2 × 10⁵ has a leading digit a with 1 ≤ a < 10; the others break that rule or use an invalid exponent.
4. What is (2 × 10⁴) × (4 × 10³)?
Multiply the front numbers: 2 × 4 = 8. Add the exponents: 4 + 3 = 7. Result: 8 × 10⁷.
5. Which number is larger: 5 × 10³ or 5 × 10⁻³?
A positive exponent (10³ = 1000) makes the number much larger than 1, while a negative exponent (10⁻³ = 0.001) makes it much smaller than 1.
✅ Key takeaways
  • Scientific notation writes any number as a × 10ⁿ, with a single nonzero leading digit (1 ≤ a < 10).
  • For large numbers, move the decimal point left and use a positive exponent.
  • For numbers smaller than 1, move the decimal point right and use a negative exponent.
  • The exponent counts exactly how many places the decimal point moved.
  • To multiply numbers in scientific notation, multiply the leading numbers and add the exponents.