In astronomy (and other sciences), it is often necessary to deal with very large or very small numbers. In fact, when numbers become truly large in everyday life, such as the national debt in the United States, we call them astronomical. Among the ideas astronomers must routinely deal with is that the Earth is 150,000,000,000 meters from the Sun, and the mass of the hydrogen atom is 0.00000000000000000000000000167 kilograms. No one in his or her right mind would want to continue writing so many zeros!

Instead, scientists have agreed on a kind of shorthand notation, which is not only easier to write, but (as we shall see) makes multiplication and division of large and small numbers much less difficult. If you have never used this powers-of-ten notation or scientific notation, it may take a bit of time to get used to it, but you will soon find it much easier than keeping track of all those zeros.

Writing Large Numbers

In scientific notation, we generally agree to have only one number to the left of the decimal point. If a number is not in this format, it must be changed. The number 6 is already in the right format, because for integers, we understand there to be a decimal point to the right of them. So 6 is really 6., and there is indeed only one number to the left of the decimal point. But the number 965 (which is 965.) has three numbers to the left of the decimal point, and is thus ripe for conversion.

To change 965 to proper form, we must make it 9.65 and then keep track of the change we have made. (Think of the number as a weekly salary and suddenly it makes a lot of difference whether we have $965 or $9.65.) We keep track of the number of places we moved the decimal point by expressing it as a power of ten. So 965 becomes 9.65 × 10² or 9.65 multiplied by ten to the second power. The small raised 2 is called an exponent, and it tells us how many times we moved the decimal point to the left.

Note that 10² also designates 10 squared, or 10 × 10, which equals 100. And 9.65 × 100 is just 965, the number we started with. Another way to look at scientific notation is that we separate out the messy numbers out front, and leave the smooth units of ten for the exponent to denote. So a number like 1,372,568 becomes 1.372568 times a million (10⁶) or 1.372568 times 10 multiplied by itself 6 times. We had to move the decimal point six places to the left (from its place after the 8) to get the number into the form where there is only one digit to the left of the decimal point.

The reason we call this powers-of-ten notation is that our counting system is based on increases of ten; each place in our numbering system is ten times greater than the place to the right of it. As you have probably learned, this got started because human beings have ten fingers and we started counting with them. (It is interesting to speculate that if we ever meet intelligent life-forms with only eight fingers, their counting system would probably be a powers-of-eight notation!)

So, in the example we started with, the number of meters from Earth to the Sun is 1.5 × 10¹¹. Elsewhere in the book, we mention that a string 1 light-year long would fit around Earth’s equator 236 million or 236,000,000 times. In scientific notation, this would become 2.36 × 10⁸. Now if you like expressing things in millions, as the annual reports of successful companies do, you might like to write this number as 236 × 10⁶. However, the usual convention is to have only one number to the left of the decimal point.

Writing Small Numbers

Now take a number like 0.00347, which is also not in the standard (agreed-to) form for scientific notation. To put it into that format, we must make the first part of it 3.47 by moving the decimal point three places to the right. Note that this motion to the right is the opposite of the motion to the left that we discussed above. To keep track, we call this change negative and put a minus sign in the exponent. Thus 0.00347 becomes 3.47 × 10⁻³.

In the example we gave at the beginning, the mass of the hydrogen atom would then be written as 1.67 × 10⁻²⁷ kg. In this system, one is written as 10⁰, a tenth as 10⁻¹, a hundredth as 10⁻², and so on. Note that any number, no matter how large or how small, can be expressed in scientific notation.

Multiplication and Division

Scientific notation is not only compact and convenient, it also simplifies arithmetic. To multiply two numbers expressed as powers of ten, you need only multiply the numbers out front and then add the exponents. If there are no numbers out front, as in 100 × 100,000, then you just add the exponents (in our notation, 10² × 10⁵ = 10⁷). When there are numbers out front, you have to multiply them, but they are much easier to deal with than numbers with many zeros in them.

Here’s an example:

[latex]\left(3\times {10}^{5}\right)\times \left(2\times {10}^{9}\right)=6\times {10}^{14}[/latex]

And here’s another example:

[latex]\begin{array}{cc}\hfill 0.04\times 6,000,000& =\left(4\times {10}^{-2}\right)\times \left(6\times {10}^{6}\right)\hfill \\ & =24\times {10}^{4}\hfill \\ & =2.4\times {10}^{5}\hfill \end{array}[/latex]

Note in the second example that when we added the exponents, we treated negative exponents as we do in regular arithmetic (−2 plus 6 equals 4). Also, notice that our first result had a 24 in it, which was not in the acceptable form, having two places to the left of the decimal point, and we therefore changed it to 2.4 and changed the exponent accordingly.

To divide, you divide the numbers out front and subtract the exponents. Here are several examples:

[latex]\begin{array}{ccc}\hfill \frac{1,000,000}{1000}& =\hfill & \frac{{10}^{6}}{{10}^{3}}={10}^{\left(6 - 3\right)}={10}^{3}\hfill \\ \hfill \frac{9\times {10}^{12}}{2\times {10}^{3}}& =\hfill & 4.5\times {10}^{9}\hfill \\ \hfill \frac{2.8\times {10}^{2}}{6.2\times {10}^{5}}& =\hfill & 0.452\times {10}^{\text{-}3}=4.52\times {10}^{-4}\hfill \end{array}[/latex]

In the last example, our first result was not in the standard form, so we had to change 0.452 into 4.52, and change the exponent accordingly.

If this is the first time that you have met scientific notation, we urge you to practice many examples using it. You might start by solving the exercises below. Like any new language, the notation looks complicated at first but gets easier as you practice it.