2.01 Scientific notation

Consider the mass of the sun, which is approximately 1\,988\,000\,000\,000\,000\,000\,000\,000\,000\,000 \text{ kg}. That's a very large number. How do scientists deal with numbers so large?

One way to write this number is to use scientific notation, as shown below. Writing numbers in scientific notation will help shorten the amount of writing or typing when doing calculations.

A number is written in scientific notation if it has the form a \times 10^{n} where a is greater than or equal to 1 and less than 10, and n is an integer.

We can follow these steps in writing numbers in scientific notation.

1. Move the decimal point to the left so that it is right after the first non-zero digit from 1to 9.

   Where's the decimal point in 2\,680\,000? Because it's a whole number, the decimal point is understood to be at the end of the number: 2\,680\,000.

   The first non-zero number is 2. If we slide the decimal point 6 places from the end of the number to the right of the 2, we will get 2.68. We don’t need the extra zeroes. The number 2.68 is between 1 and 10 as we wanted.

2. Multiply by 10 to the power of the number of places the decimal moved.

   We moved 6 places to the left so we have 10^6.

We may also see standard form referred to as basic numeral form or be asked to write it as a decimal.

Using the definition above, we can rewrite the mass of the sun as 1.988 \times 10^{30}\text{ kg}. That takes much less space to write.

Note that for very large numbers, we move the decimal point to the left and have positive power. For very small numbers less than 1, we move the decimal point to the right and have a negative power.

Examples