Scientific notation: A system, based on powers of 10, most useful for expressing very large and very small numbers.

Physicists like to measure the very big, the very small and everything in between. To express the results efficiently and clearly, they use scientific notation.

Scientific notation expresses a quantity as a number times a power of 10. Why is this useful? Here’s an example: the Earth is about 149,000,000,000 meters from the Sun. You could express that distance as we just did, with a long string of zeros, or you could use scientific notation to write it as 1.49×10¹¹ meters. The latter method has proven itself to be clearer and less prone to error.

The value on the left (1.49) is called the leading value. The power of 10 is typically chosen so the leading value is between one and 10. In the example immediately above, we multiplied by 10¹¹ so that we could use 1.49. We also could have written 14.9×10¹⁰ or 0.149×10¹² since all three values are equal, but a useful convention is to use a number between one and 10.

In case you have forgotten how to use exponents, here’s a quick review. Ten is the base number. Ten to the first power is 10; 10² is ten to the second (ten squared) or 100; ten to the third is 10 times 10 times 10, or 1000. A positive exponent tells you how many zeros to add after the one. When the exponent is zero, the value is one: 10⁰ equals one.

As mentioned, scientists also measure the very small. For example, a particle known as a muon has a mean lifespan of about 2.2 millionths of a second. Scientific notation provides a graceful way to express this number: 2.2×10⁻⁶ (2.2 times 10 to the negative sixth). To review the mathematics: ten to the minus one is 1/10; ten to the minus two is 1/100; ten to the minus three is 1/1000, and so forth.

You can also write 1.49×10¹¹ as 1.49e11. The two are equivalent. You may have seen this notation in computer spreadsheet programs such as Microsoft® Excel. We do not use this "e" notation in the text of the book, but if you submit answers to homework problems or interactive checkpoints, you will use it there.