Part II. NUMERICAL REASONING
6. Working with Numbers
Among the most fundamental operations we do with quantities is arithmetic. We can encounter the need for arithmetic in any phase of problem solving, from making a ballpark estimate in the Understand phase to computing and double-checking a final result in the Execute and Check phases. Once we have a solid grasp of the operations that are allowable and those that aren’t – for example, is it OK to add or subtract quantities expressed in different units or on different scales? – we may get down to business with performing basic operations.
Most of us probably feel comfortable with most of these operations, at least when they concern simple numbers. However, it becomes easy to make errors or overlook important steps when we’re dealing with extremely large or small numbers, or when unit conversions become necessary. One setting in which we often encounter such difficulties is in working with proportions, including concentrations, ratios, and percentages. Though quantities like these are often conceptually simple, working with them and converting among ways of expressing them can be challenging. This chapter highlights some concepts and techniques for working with these sorts of unwieldy numbers so that we can work confidently, avoid simple mistakes, and even catch more complex ones.
We begin with a method for doing arithmetic that can be used to simplify computations, or to approximate solutions when a back-of-the-envelope computation is all you need. The method is particularly powerful when computations involve very large or very small numbers. As such, it can be useful for making ballpark estimates in the early stages of problem-solving. Our method makes strategic use of scientific notation, which you’ve probably encountered in secondary science classes. The philosophical basis of scientific notation also leads to the notion of order of magnitude, a concept that can be useful for comparing quantities as well as for judging the the appropriateness of estimates. Along the way, we’ll compare some ways of expressing normalized quantities like concentrations and proportions, and review the rules for arithmetic with exponents.
6.1 Scientific Notation
In high school chemistry, we learn that there are more than 602 sextillion molecules in a mole of a chemical substance.[1] But we don’t normally see Avogadro’s constant written as some number of sextillions, nor do we see it elaborated with all of the 24 digits necessary to write it in integer form: it is difficult to keep track of all those digits when writing them, and even more difficult to keep track when you’re reading or comparing different numbers. Instead of writing the entire number out, we use the shorthand of scientific notation, where Avogadro’s constant looks more like 6.022×1023. In general, scientific notation has the form:
[latex]a = 10^{b}[/latex]
where a and b are sometimes called the mantissa and power, respectively. So Avogadro’s constan