L4 · Scientific notation: expressing very large and very small numbers

Learning goal

“I can write very large and very small numbers in scientific notation, convert scientific notation back to standard form, and compare magnitudes - converting only, never doing arithmetic inside scientific notation.”

Scientific notation — a number written as a × 10^n, where a (the coefficient) obeys 1 ≤ a < 10 and n is an integer. Example: 4.5 × 10⁶.

Coefficient — the first factor in scientific notation. It must be at least 1 and less than 10: 4.5 × 10⁶ is in scientific notation; 45 × 10⁴ is NOT (45 is too big), even though it has the same value.

Standard form — the ordinary decimal writing of the number: 4,500,000.

Magnitude — the power-of-10 scale: positive exponents for large numbers (10⁶), negative exponents for numbers below 1 (10⁻⁴).

Worked example

Write 4,500,000 in scientific notation. Step 1: Place the decimal point so the coefficient is in [1, 10): 4.5. Step 2: Count the jumps from the original point (after the last 0) to the new position: 6 jumps left. Step 3: Large number -> positive exponent: 4,500,000 = 4.5 × 10⁶. Now a small one: write 0.00032 in scientific notation. Step 1: Coefficient in [1, 10): 3.2. Step 2: The point jumps 4 places RIGHT to get from 0.00032 to 3.2. Step 3: Small number (below 1) -> negative exponent: 0.00032 = 3.2 × 10⁻⁴. Say it plainly: the exponent's SIGN tells the size story (positive = large, negative = small); the coefficient's digits are the number's digits, in order.

Check yourself — explain it

Why is 45 × 10⁴ not scientific notation, even though it equals 450,000?

Paired problem — example still visible

Write 38,000 and 0.0061 in scientific notation -> 3.8 × 10⁴; 6.1 × 10⁻³.

Faded completion — last step is yours

"5,200,000 = ____ × 10^___" -> 5.2; 6

Ready to practise?

10 scored questions with a 3-rung hint ladder. Mastery needs 8/10 plus both gate questions.

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