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Scientific Notation

Scientific Notation

A standardized way of writing real numbers. In scientific notation, all real numbers are written in the form a·10b, where 1 ≤ a < 10 and b is an integer. For example, 351 is written 3.51·102 in scientific notation.

Key Formula

a×10ba \times 10^b
Where:
  • aa = A decimal number where $1 \leq a < 10$
  • bb = An integer exponent (positive, negative, or zero)

Worked Example

Problem: Write 0.00032 in scientific notation.
Step 1: Move the decimal point to the right until you have a number between 1 and 10. Moving the decimal 4 places right gives 3.2.
0.00032→3.20.00032 \rightarrow 3.2
Step 2: Because you moved the decimal 4 places to the right, the exponent is −4-4. A rightward shift means a negative exponent.
b=−4b = -4
Step 3: Combine the coefficient and the power of 10.
0.00032=3.2×10−40.00032 = 3.2 \times 10^{-4}
Answer: 0.00032=3.2×10−40.00032 = 3.2 \times 10^{-4}

Why It Matters

Scientific notation makes it practical to work with extremely large or small quantities, such as the distance to a star (4.24×10134.24 \times 10^{13} km) or the mass of a proton (1.67×10−271.67 \times 10^{-27} kg). It also reduces errors by eliminating long strings of zeros and makes multiplying or dividing such numbers straightforward—you simply add or subtract the exponents.

Common Mistakes

Mistake: Using a coefficient outside the range 1≤a<101 \leq a < 10, such as writing 32×10332 \times 10^{3} instead of 3.2×1043.2 \times 10^{4}.
Correction: Always adjust the decimal point so the coefficient is at least 1 but less than 10, then update the exponent accordingly.

Related Terms

  • Real Numbers — The set of numbers scientific notation represents
  • Integers — The exponent b must be an integer
  • Exponent — The power of 10 used in the notation
  • Powers of 10 — The base-10 scaling factor in the form