[Math Lair] Scientific Notation

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When working with real-world numbers in fields such as physics or astronomy, very small or very large quantities frequently appear. For example, the total number of particles in your body is roughly 10,000,000,000,000,000,000,000,000,000, and, in quantum physics, the Planck length is 0.00000000000000000000000000000000001616199 metres,

Rather than express numbers with large numbers of leading or trailing zeroes such as illustrated above, scientific notation is often used to represent such quantities. It is a compact and easy-to-use method of expressing very large and very small numbers. In scientific notation, a number is written as a number between 1 and 9.999... multiplied by a power of ten (expressed in exponential notation). To convert a number to scientific notation, find the largest power of 10 no greater than that number, and factor that power from that number. Express the power of 10 in exponential notation.

Some examples:

  1. The Sun is 93,000,000 miles from the Earth. Let's write 93,000,000 in scientific notation. The largest power of 10 no greater than 93,000,000 is 10,000,000 or 107. 93,000,000 divided by 107 is 9.3. So, 93,000,000 in scientific notation would be 9.3 × 107.
  2. The size of an adenovirus, which causes colds in humans, is 0.00000008 metres. The next-largest power of 10 is 10−8, so this can be written 8 × 10-8.

Another advantage of scientific notation is that it can make it easier to determine how many significant figures are in a measurement (see accuracy and precision for more information about significant figures).