Scientific Notation

What is Scientific Notation?

Scientific notation (also called standard form) is a way of writing very large or very small numbers using powers of 10. It makes these numbers easier to read, compare, and use in calculations.

Format: a × 10ⁿ, where 1 ≤ a < 10 and n is an integer (positive, negative, or zero).

Key terms:

• Coefficient (a): the number between 1 and 10
• Exponent (n): the power of 10 — tells you how far to move the decimal point
• Standard form: another name for scientific notation, used in the UK

Writing Large Numbers in Scientific Notation

Move the decimal point left until the coefficient is between 1 and 10. The number of places moved equals the positive exponent.

Steps:

1. Identify where the decimal point currently is (at the end of a whole number)
2. Count how many places you move it left to get a number between 1 and 10
3. Write the result as a × 10ⁿ

Example: 3,500,000

• Move decimal 6 places left: 3.5
• Result: 3.5 × 10⁶

Example: 92,000,000 (distance from Earth to Sun in km)

• Move decimal 7 places left: 9.2
• Result: 9.2 × 10⁷ km

Writing Small Numbers in Scientific Notation

Move the decimal point right until the coefficient is between 1 and 10. Use a negative exponent.

Steps:

1. Count how many places you move the decimal point right
2. Write the result as a × 10⁻ⁿ

Example: 0.000042

• Move decimal 5 places right: 4.2
• Result: 4.2 × 10⁻⁵

Example: 0.0000007

• Move decimal 7 places right: 7.0
• Result: 7 × 10⁻⁷

Converting Back to Standard Form

Positive exponent → move decimal right:

• 6.02 × 10³ → move 3 places right → 6,020
• 4.5 × 10⁵ → 450,000

Negative exponent → move decimal left:

• 5.1 × 10⁻³ → move 3 places left → 0.0051
• 3.0 × 10⁻⁶ → 0.000003

Multiplying in Scientific Notation

Multiply the coefficients together and add the exponents:

(3 × 10⁴) × (2 × 10³) = (3 × 2) × 10^(4+3) = 6 × 10⁷

If the result coefficient is not between 1 and 10, adjust: (5 × 10³) × (4 × 10²) = 20 × 10⁵ = 2 × 10⁶

Dividing in Scientific Notation

Divide the coefficients and subtract the exponents:

(8 × 10⁶) ÷ (4 × 10²) = (8 ÷ 4) × 10^(6-2) = 2 × 10⁴

Comparing Numbers in Scientific Notation

Compare the exponents first. The larger exponent means the larger number.

If exponents are equal, compare the coefficients:

• 4.2 × 10⁵ vs 8.1 × 10⁴ → 10⁵ > 10⁴, so 4.2 × 10⁵ is larger
• 7.3 × 10⁸ vs 2.1 × 10⁸ → same exponent, so compare 7.3 vs 2.1 → 7.3 × 10⁸ is larger

Why Use Scientific Notation?

Without it, calculations with extreme numbers become error-prone:

• The distance to the nearest star: 40,208,000,000,000 km vs 4.02 × 10¹³ km
• The mass of a proton: 0.000000000000000000000000001673 kg vs 1.673 × 10⁻²⁷ kg

Scientific fields — astronomy, chemistry, physics, biology — all rely on it daily.

Common Mistakes

• Writing the coefficient outside the range 1 to 10 (e.g. 35 × 10³ instead of 3.5 × 10⁴)
• Getting the sign of the exponent wrong for small numbers (forgetting the negative)
• Moving the decimal in the wrong direction when converting back
• Forgetting to adjust the exponent after multiplying or dividing coefficients

Tips and Tricks

• A positive exponent = large number (lots of zeros after the digits)
• A negative exponent = small number (zeros before the digits)
• The exponent tells you how many places the decimal moves — and in which direction
• When adding or subtracting in scientific notation, convert to the same power of 10 first
• Always check: is the coefficient between 1 and 10? If not, adjust it