Characteristics of scientific notation

A number is written in scientific notation when it looks like this:

$a \times 10^n$ , where:

  • $a$ is a single digit number or decimal between $1$ and $10$ (not including $10$).
  • $n$ is an integer (positive, negative, or zero).

Examples:

  • $4.2 \times 10^5 = 420{,}000$
  • $6.7 \times 10^{-3} = 0.0067$
  • $9.0 \times 10^0 = 9$

Key points:

  • The first factor is always at least $1$ but less than $10$.
  • The second factor is a power of $10$.
  • Positive exponents make large numbers.
  • Negative exponents make small numbers.

When scientific notation is useful

Scientific notation is helpful when numbers are too large or too small to work with easily.

It is useful:

  • When writing numbers with many zeros.
  • When comparing quantities across very different scales.
  • When estimating size, distance, or measurement in science and technology.

It allows us to:

  • Quickly see the size of a number.
  • Perform calculations more efficiently.
  • Communicate values clearly without writing out every zero.

Real-world examples

Here are some places where scientific notation is used:

  1. Astronomy
    The mass of Earth is about $5.97 \times 10^{24}$ kilograms. Writing out all those zeros would be very inconvenient.

  2. Medicine
    A single bacterium might be about $2 \times 10^{-6}$ meters long. Scientific notation makes this tiny length easy to express.

  3. Technology
    A computer might perform $3.2 \times 10^9$ operations per second. Writing the full number, $3{,}200{,}000{,}000$, takes more space and is harder to read.