Scientific Notation
Scientific notation expresses any number as a × 10ⁿ where 1 ≤ a < 10. Scientists use it to handle numbers like the speed of light or the mass of an atom.
The Format
Scientific notation: a × 10ⁿ
• a is a number from 1 to 9.999…
• n is an integer (positive for large numbers, negative for small)
Converting to Scientific Notation
Move the decimal until one non-zero digit is to the left. Count the moves — that is your exponent.
93,000,000 → 9.3 × 10⁷ (moved 7 places left)
0.000045 → 4.5 × 10⁻⁵ (moved 5 places right)
Multiplying in Scientific Notation
Multiply the a-values, add the exponents, then adjust if needed.
(3 × 10⁴) × (2 × 10³) = 6 × 10⁷
Real-World Scale
Speed of light ≈ 3 × 10⁸ m/s. Diameter of hydrogen atom ≈ 1.06 × 10⁻¹⁰ m. Scientific notation prevents mistakes when writing many zeros.
Converting Back to a Standard Number
Reading scientific notation is the reverse of writing it: the exponent tells you which way to move the decimal, and how far.
Positive exponent means a big number
4.7 × 10⁵ → move the decimal 5 places right → 470,000
Negative exponent means a small number
2.3 × 10⁻⁴ → move the decimal 4 places left → 0.00023
10⁻³ is 0.001, which is positive.
Dividing in Scientific Notation
Divide the front numbers, subtract the exponents, then fix the result so the front number sits between 1 and 10.
(8 × 10⁷) ÷ (2 × 10³)
Front numbers: 8 ÷ 2 = 4. Exponents: 7 − 3 = 4.
Answer: 4 × 10⁴
When the front number lands out of range
(3 × 10⁵) ÷ (6 × 10²) = 0.5 × 10³
0.5 is below 1, so shift once more: 5 × 10².
Adding and Subtracting
This one has an extra step that catches people out: the exponents must match before you can add.
(4 × 10³) + (5 × 10²)
- Rewrite the smaller one to match:
5 × 10² = 0.5 × 10³ - Now add the front numbers:
4 + 0.5 = 4.5 - Answer:
4.5 × 10³
Common Mistakes
- Leaving the front number outside the 1–9.99 range.
12 × 10³must become1.2 × 10⁴. - Moving the decimal the wrong way. Bigger number, positive exponent; smaller number, negative exponent.
- Adding exponents when adding numbers. Exponents only add during multiplication.
- Miscounting places. Count the jumps the decimal makes, not the number of zeros you see.
Where You Will Use This
- Chemistry, where Avogadro's number is
6.022 × 10²³. - Astronomy, for distances measured in trillions of kilometres.
- Biology and medicine, for the size of cells, viruses, and drug doses.
- Computing, where storage and processor speeds span many orders of magnitude.
Summary
- The format is
a × 10ⁿwith a between 1 and 10. - Positive exponents make large numbers; negative exponents make small ones.
- Multiply: multiply the fronts, add the exponents. Divide: divide the fronts, subtract the exponents.
- Add or subtract only after matching the exponents.
- Always tidy the answer back into proper form at the end.
FAQ
Is 12 × 10³ correct scientific notation? No — a must be less than 10. Rewrite as 1.2 × 10⁴.
What does 10⁰ equal? 1. So 7 × 10⁰ is just 7, which is why single-digit numbers rarely get written in scientific notation.
How is this different from engineering notation? Engineering notation restricts exponents to multiples of three, matching prefixes like kilo, mega, and milli. Scientific notation allows any integer exponent.
Why do calculators show E? 3.2E8 is the calculator's way of writing 3.2 × 10⁸. The E stands for exponent, not for the number e.
Do significant figures matter here? Yes. Scientific notation makes them obvious — 3.00 × 10⁸ claims three significant figures, while 3 × 10⁸ claims one.
Quick Quiz
Test what you just learned. Choose the best answer for each question.