Magnitude

Orders of Magnitude

In everyday language, “orders of magnitude” connotes the idea of the expansion of size in some significant way. In science, there is a more precise definition.

Scientifically speaking, an order of magnitude of a number usually refers to the approximate value of that number, measured in powers of ten—in other words, where that number sits on a scale that measures ones, tens, hundreds, etc. So 10 and 20 are the same order of magnitude, but 100 is one order of magnitude larger. Between 10 and 1,000, there are two orders of magnitude, and so on.

Let’s think of this another way. If you recall a number line from school, it looks something like this:

Each step you take on this number line involves the change of one (move to the right to add one to your value, or subtract one if you move to the left).

With orders of magnitude, the math is a bit different. Instead of adding (or subtracting) a set amount for every step you take on the number line, you multiply (or divide) by a prescribed amount. It looks something like this:

This is powerful. On the first number line, we went up by a value of 5 in 5 steps. Simple enough. But in the second number line, we moved up by a value of 100,000 in the same number of steps. We need this turbocharged way of changing values in order to depict the awesome range of magnitudes, both large and small, that we’ll be exploring in our known Universe.

Linear versus Logarithmic Scales

Because scientists, engineers, and other technicians so often need to present values that stretch over many orders of magnitude, they have developed a visual way to show this. It’s called the logarithmic scale.

Let’s look at a run-of-the-mill ruler like the one you used in elementary school. For every big dark line on the ruler, you are taught to add one inch or centimeter. If you move the same distance again, you add another inch or centimeter. Three steps equal three inches or centimeters, and so on. This is called a linear scale, meaning one move is the equivalent of adding—or subtracting—one unit of distance (in the case of a ruler, that is one inch or centimeter).

In a logarithmic scale, you don’t add or subtract for every step, you multiply or divide depending whether you are moving to the right or to the left. The most common form of a logarithmic scale uses the number ten as the base, multiplied by itself one, two, three, etc., times.

So the first step on our logarithmic ruler represents 10 × 1, which is 10. The second step on this logarithmic ruler is 10 × 10, or 100. The third step is 10 × 10 × 10, which equals 1,000, which means we’ve moved up three orders of magnitude. In other words, after only taking three steps on a logarithmic scale, we’ve quickly reached 1,000, as opposed to the thousand steps that would have been required on a ruler using a linear scale.

This is what makes logarithmic scales so helpful when depicting the magnitude of things or behaviors that have a huge range. In this book, we will make it clear whether we’re talking about a linear or a logarithmic scale with each different phenomenon.

Scientific Notation

You can see that as the logarithmic scale continues to grow, we will quickly be awash in a sea of zeroes representing huge numerical values. Fortunately, science has a simpler system for expressing these values.

Let’s say you have a million dollars. You could choose to express your vast fortune in writing as $1,000,000. Likewise, a billion dollars could be written as $1,000,000,000, and a trillion would be $1,000,000,000,000, and so on.

What happens if you want to multiply your trillion dollars by another million? That amount turns out to be $1,000,000,000,000,000,000. Writing out all those zeros starts to become a real pain. Although with a million trillion dollars on hand, you could probably hire someone to write it for you, there is another way.

Enter scientific notation. In our example, when a trillion dollars (a one followed by twelve zeroes) was multiplied by one million (with six zeroes), the amount came out to be the number 1 followed by 18 zeros. Another way to write that is this:

10¹⁸

The 18 in this example is called the exponent, and it means that the number 1 is followed by 18 zeroes. If the exponent were 999, it would mean that the number you were expressing is a one followed by 999 zeroes.

The key to understanding scientific notation in this book is that for every number the exponent goes up, you are multiplying by a factor of ten. Conversely, if the exponent goes down by one, you are dividing by ten.

Multiplying with Exponents

You might have noticed that when we multiplied a million by a trillion, we ended up with the exponent of 18. This is also the same as adding the exponents (10⁶ and 10¹²) of the million and trillion. That’s the way to quickly and accurately multiply two numbers in scientific notation—simply add the two exponents together to get the new value.

There will be times in this book that you will see numbers in scientific notation in which the exponent has a negative sign in front of it like this:

10⁻¹⁸

When that negative sign appears, it means we have to move the other direction on our order-of-magnitude number line, which is to say we divide by ten. Once again in this example, there are 18 zeroes, but the decimal point is to the left of them and the one is on the right. This value, therefore, translates into

0.0000000000000000001

There are many examples in this book where things with very tiny values may surprise and interest you. Note that, no matter how large the negative exponent gets, the number never becomes negative. It just gets closer to zero.

Units, Equations, and Other Fun Stuff

Another very important tool for understanding and expressing magnitude is units.

We use units all of the time in our daily lives, even if we aren’t fully aware of it. How fast is your car going? (Typically we think in miles or kilometers per hour.) How long has it been since something happened? (Given in the number of seconds, minutes, hours, etc.) What is the temperature outside? (Degrees in Fahrenheit or Celsius, most often.) A unit is a way to quantify something in a reliable and consistent form. To put it another way: it’s how we measure things.

Systems of units developed in various pockets of discipline and geography over a wide span of time, which led to the cornucopia of units for practically every physical measurement. Recognizing what a mess this can make, some experts got together in an international consortium to sort it all out.

The current system is called the International System of Units, but is better known by the initials SI (for the French version of the name, Le System International d’Unites). At the heart of the SI is the metric system. The building block for measuring distance or length is the meter, and for mass it’s the kilogram, and so on. There are seven “base” units, each of which covers a singular physical property: length, mass, time, electric current, temperature, quantity of substance, and intensity of light.

SI Units

Throughout this book, we will use SI units. This will allow you to know that we are comparing the proverbial apples to apples instead of apples to oranges and let you appreciate the differences in magnitude at a relatively quick glance.