Prerequisites: M23, M16

Key Concepts

scientific notation
powers of 10
standard form
order of magnitude

Scientific Notation

The distance from Earth to the Sun is about 150,000,000 kilometers. The diameter of a hydrogen atom is about 0.00000000012 meters. Writing out all those zeros is tedious and error-prone. Scientific notation gives us a compact, precise way to express very large and very small numbers.

The Format

A number in scientific notation has the form:

a \times 10 n

where a (the coefficient) satisfies 1 ≤ a < 10, and n (the exponent) is an integer.

Component	Rule	Example
Coefficient (a)	Must be at least 1 and less than 10	3.2, 7.08, 1.0
Exponent (n)	Positive for large numbers, negative for small	10⁶, 10^-4

Converting Standard Form to Scientific Notation

Move the decimal point until you have a number between 1 and 10. Count how many places you moved it:

Moved left → exponent is positive (number was large)
Moved right → exponent is negative (number was small)

Worked Example 1: Large Number

Write 47,300,000 in scientific notation.

Place the decimal after the first non-zero digit: 4.7300000
Count places moved: the decimal moved 7 places to the left.
Write: 4.73 × 10⁷

Worked Example 2: Small Number

Write 0.000062 in scientific notation.

Place the decimal after the first non-zero digit: 6.2
Count places moved: the decimal moved 5 places to the right.
Since the original number was small, the exponent is negative: 6.2 × 10^-5

Converting Scientific Notation to Standard Form

Move the decimal point in the direction indicated by the exponent:

Positive exponent → move decimal right (number gets larger)
Negative exponent → move decimal left (number gets smaller)

Worked Example 3: Back to Standard Form

Convert 8.04 × 10^-3 to standard form.

The exponent is -3, so move the decimal 3 places to the left.
8.04 → 0.00804

Multiplying and Dividing in Scientific Notation

These operations are especially clean in scientific notation:

Multiplication:

(a \times 10 m) \times (b \times 10 n) = (a \times b) \times 10 m+n

Multiply the coefficients, add the exponents.

Division:

(a \times 10 m) \div (b \times 10 n) = (a \div b) \times 10 m-n

Divide the coefficients, subtract the exponents.

Worked Example 4: Multiplication

(3.0 × 10⁴) × (2.5 × 10³)

Multiply coefficients: 3.0 × 2.5 = 7.5
Add exponents: 4 + 3 = 7
Result: 7.5 × 10⁷

Order of Magnitude

The order of magnitude of a number is the power of 10 closest to it. For example, 7,500 is on the order of 10⁴ (about 10,000). This is useful for quick comparisons: a million (10⁶) is three orders of magnitude larger than a thousand (10³).

Common Mistake: Coefficient Out of Range

If your multiplication gives a coefficient of 10 or more, adjust it. For example, (5 × 10³) × (4 × 10²) = 20 × 10⁵. Since 20 is not between 1 and 10, rewrite as 2.0 × 10⁶.

Practice Problems

1. Write 93,000,000 in scientific notation.

Show Solution

Move the decimal 7 places left: 9.3 × 10⁷

2. Write 0.0000407 in scientific notation.

Show Solution

Move the decimal 5 places right: 4.07 × 10^-5

3. Convert 6.1 × 10⁴ to standard form.

Show Solution

Move the decimal 4 places right: 61,000

4. Calculate (8 × 10⁵) × (3 × 10^-2). Express in scientific notation.

Show Solution

Coefficients: 8 × 3 = 24. Exponents: 5 + (-2) = 3. So 24 × 10³. Adjust coefficient: 2.4 × 10⁴.

5. Which is larger: 9.8 × 10⁷ or 3.1 × 10⁸?

Show Solution

Compare exponents first. 10⁸ > 10⁷, so 3.1 × 10⁸ (310,000,000) is larger than 9.8 × 10⁷ (98,000,000).

Lesson Summary

Scientific notation: a × 10ⁿ, where 1 ≤ a < 10.
Positive exponent = large number; negative exponent = small number.
Multiply: multiply coefficients, add exponents. Divide: divide coefficients, subtract exponents.
Always adjust the coefficient back into the range [1, 10) if needed.