Scientific Notation
Writing very large and very small numbers without counting zeros.
The explanation
Scientific notation writes a number as (something between 1 and 10) × 10^power.
93,000,000 becomes 9.3 × 10⁷. Count how many places the decimal moved left: seven.
0.00042 becomes 4.2 × 10⁻⁴. Moving right gives a negative power.
To multiply, multiply the fronts and add the powers. To divide, divide the fronts and subtract the powers.
Afterwards, check the front number is still between 1 and 10. If you get 34 × 10⁵, fix it to 3.4 × 10⁶.
Standard form is a × 10ⁿ with 1 ≤ |a| < 10 and n an integer. The constraint on a makes the representation unique, which is what allows two numbers to be compared by exponent first and mantissa second.
Arithmetic follows the exponent laws:
(a×10ᵐ)(b×10ⁿ) = ab×10ᵐ⁺ⁿ (a×10ᵐ)/(b×10ⁿ) = (a/b)×10ᵐ⁻ⁿ
Addition and subtraction require a common exponent first, exactly as adding fractions requires a common denominator — the powers of ten are the units, and only like units combine.
Renormalising after arithmetic is a required step, since ab may land outside [1, 10). Beyond notation, the exponent is a statement about magnitude: comparing 10⁷ and 10⁴ tells you one quantity is a thousand times the other without any further computation, which is the basis of order-of-magnitude estimation in science.
Worked example
Compute (6.4 × 10⁵)(5 × 10⁻⁸) in scientific notation.
- Multiply the fronts: 6.4 × 5 = 32.
- Add the exponents: 5 + (−8) = −3.
- So far 32 × 10⁻³, which is not standard form.
- Renormalise: 32 = 3.2 × 10¹, so 3.2 × 10⁻².
Answer: 3.2 × 10⁻²
Common mistakes
- Leaving an answer like 32 × 10⁻³ in non-standard form.
- Adding exponents when adding the numbers rather than multiplying them.