Scientific notation
The idea
Numbers in science get awkward fast — a red blood cell is about 0.000008 m wide, and the Sun is about 150000000 km away. Scientific notation rewrites any such number as a value between 1 and 10 times a power of ten: 8 × 10⁻⁶ and 1.5 × 10⁸. You already know place value and how multiplying by 10 slides the decimal point; the exponent simply records how many slides, with negative exponents meaning slides toward smaller.
Treat the two parts separately and the arithmetic becomes light: to divide, divide the front numbers and subtract the exponents. One care point: the front must stay between 1 and 10, so a result like 0.25 × 10⁴ needs one more slide to become 2.5 × 10³. The trap to dodge is judging size by the front number alone — 9 × 10³ is far smaller than 2 × 10⁶, because the exponent, not the front, sets the scale.
Worked example
A red blood cell is about 8 × 10⁻⁶ m wide. About how many cells would fit side by side across a 2 cm scratch, which is 2 × 10⁻² m long?
- The number of cells is the scratch length divided by one cell width: (2 × 10⁻²) ÷ (8 × 10⁻⁶).
- Handle the parts separately: the fronts give 2 ÷ 8 = 0.25, and the powers of ten give 10⁻² ÷ 10⁻⁶ = 10 to the power −2 − (−6) = 10⁴.
- The raw result 0.25 × 10⁴ is not in proper form, since 0.25 is below 1; slide once to get 2.5 × 10³, which is 2500.
- Check by multiplying back: 2500 cells × 8 × 10⁻⁶ m = 20000 × 10⁻⁶ m = 2 × 10⁻² m, exactly the scratch length.
Answer. About 2.5 × 10³, or 2500, red blood cells would fit across the 2 cm scratch.
Check your understanding
- Why must the front number stay between 1 and 10, and what goes wrong with comparisons if it does not?
- How does a negative exponent on the 10 differ in meaning from a negative sign on the whole number?
- Why do the exponents subtract when you divide two powers of ten — what is happening to the factors?
- How would you quickly decide which of 7 × 10⁵ and 3 × 10⁶ is larger, and what is the general rule?
Build the foundations first
Scientific notation builds on these concepts. If any feel shaky, start there.