Scientific Notation Formula

The Formula

a \times 10^n where 1 \leq |a| < 10 and n \in \mathbb{Z}

When to use: Instead of writing out all the zeros in 93,000,000 or 0.000042, you slide the decimal point and count how many places it moved. The exponent on 10 keeps track of the shift.

Quick Example

93{,}000{,}000 = 9.3 \times 10^7 0.000042 = 4.2 \times 10^{-5}

Notation

a \times 10^n where a is the coefficient (between 1 and 10) and n is the exponent (positive for large numbers, negative for small numbers)

What This Formula Means

A way of writing very large or very small numbers as a \times 10^n, where 1 \leq |a| < 10 and n is an integer.

Instead of writing out all the zeros in 93,000,000 or 0.000042, you slide the decimal point and count how many places it moved. The exponent on 10 keeps track of the shift.

Worked Examples

Example 1

easy
Write 0.00047 in scientific notation.

Solution

  1. 1
    Move the decimal point right until we have a number between 1 and 10: 4.7.
  2. 2
    Count the places moved: 4 places to the right, so the exponent is -4.
  3. 3
    Result: 4.7 \times 10^{-4}.

Answer

4.7 \times 10^{-4}
Scientific notation expresses a number as a \times 10^n where 1 \leq a < 10. Moving the decimal right gives a negative exponent; moving it left gives a positive exponent.

Example 2

medium
Compute (3.0 \times 10^5) \times (2.0 \times 10^{-3}) and express the answer in scientific notation.

Common Mistakes

  • Writing the coefficient outside the range 1 \leq |a| < 10 (e.g., 25 \times 10^3 instead of 2.5 \times 10^4)
  • Using the wrong sign on the exponent (e.g., writing 0.003 as 3 \times 10^3 instead of 3 \times 10^{-3})
  • Forgetting to adjust the exponent when fixing the coefficient

Why This Formula Matters

Scientists and engineers work with numbers from the size of atoms (10^{-10} m) to galaxies (10^{21} m). Scientific notation makes these manageable.

Frequently Asked Questions

What is the Scientific Notation formula?

A way of writing very large or very small numbers as a \times 10^n, where 1 \leq |a| < 10 and n is an integer.

How do you use the Scientific Notation formula?

Instead of writing out all the zeros in 93,000,000 or 0.000042, you slide the decimal point and count how many places it moved. The exponent on 10 keeps track of the shift.

What do the symbols mean in the Scientific Notation formula?

a \times 10^n where a is the coefficient (between 1 and 10) and n is the exponent (positive for large numbers, negative for small numbers)

Why is the Scientific Notation formula important in Math?

Scientists and engineers work with numbers from the size of atoms (10^{-10} m) to galaxies (10^{21} m). Scientific notation makes these manageable.

What do students get wrong about Scientific Notation?

Determining the sign of the exponent: moving the decimal left gives a positive exponent (big numbers), right gives negative (small numbers).

What should I learn before the Scientific Notation formula?

Before studying the Scientific Notation formula, you should understand: exponent rules, place value, decimals.

← Back to Scientific Notation See All Examples Practice Problems