Appendix B: Scientific Notation
In Appendix A, we described a new way to count. Now we would like to use this new way to count to learn how to write and discuss numbers.
To do this, we are going to try to answer the question, “What’s between the counting numbers?” When you first encountered arithmetic, this question was answered by learning about such topics as fractions, “mixed numbers”, and decimals. However, our base-ten logarithmic counting system has more than these between the different generations. Consider the jump we made between 100 and 101. The numbers 2, 3, 4, on up to 9 lie between these jumps. How do we access them in our new counting system?
The answer is scientific notation. This notation has two important parts: a coefficient and an exponent. We’ve already explored the properties of exponents, but now we get to explore all the “in between” possibilities by looking at a coefficient. At this point, I want to use stand-ins for numbers and so right here, just as in algebra, I will use letters to indicate that there are many possibilities for which precise numeral we might find. Here I use A to be the coefficient and B to be the exponent. The way we will write numbers now will be
A × 10B
So that we can make calculations easier, there are conventional restrictions placed on the precise values that A and B can have. You’ve already encountered the restrictions on B. The mathematical jargon is that B can be any integer, that is, any positive or negative whole number. A, on the other hand, can be any number between 1 and going up to 10, but not equal to ten. Why not equal to A = 10? Well, let’s see:
A × 10B = 10 × 10B = 1 × 10B+1
In other words, once A reaches the value of 10, you jump to the next order of magnitude and begin again with a new coefficient of 1.
This notational system for representing numbers is “Scientific Notation” and it is used regularly in scientific contexts as it handles large and small numbers easily by specifying an order of magnitude in B, but, with the coefficient A also allows for the representation of any possible number at that scale.
Scientific notation is sometimes taught to high school and college students along with a recipe for how to translate into decimal notation. There is usually some discussion of using the exponent B as a way to find out how many zeroes or decimal places after the decimal point there are in the number, depending on whether exponent B is negative or positive. The coefficient A is written down in the right "place" and then zeroes are used to fill in any empty spaces to the decimal point. This recipe works, but it might miss the way most people who use scientific notation think about such numbers. Scientific notation allows anyone who understands it to get important information at a glance about the size of a number. The first thing to keep in mind is that the most important number in scientific notation is the exponent B. This tells you what scale the number lives in. Take the number
3 × 108
Which is the approximate number of people living in the United States. The B = 8 exponent indicates that the number is in the realm of hundreds of millions. The A = 3 coefficient indicates that there are three of the hundreds of millions. Thus, the number is three hundred million. If necessary, we can translate the number
3 × 108 = 300 million
= 300,000,000
Translating the way we say a number in this fashion is an efficient way of converting between scientific notation and the way you first learned how to write numbers. Alternatively, using the technique of counting decimal places and filling up with zeroes, we could have built this same number, but following a recipe approach like that might miss the conceptual beauty of scientific notation that tells us that 108 is the scale of the number and 3 is how many of that size we have. This also makes it easy to identify a very small number like
7 × 10-5 = 7 hundred thousandths
= 7 / 100000 = 0.00007
A number that, in meters, is the average thickness of a human hair. It’s easier to read the number in scientific notation and translate it into English by identifying the scale and the coefficient than it is to count out the decimal places and zeroes.
Scientific notation helps make calculations efficient and less mentally taxing. And that is the whole point!
