1.1: Exercises
Use an index card or a piece of paper folded lengthwise, and cover up the right-hand column of the exercises below. Read each exercise in the left-hand column, answer it in your mind, then slide the index card down to reveal the answer and see if you’re right! For every exercise you missed, figure out why you missed it before moving on.
| Q | A |
|---|---|
| What’s the opposite of concrete? | Abstract. |
| What’s the opposite of discrete? | Continuous. |
| Consider a quantity of water in a glass. Would you call it abstract, or concrete? Discrete, or continuous? | Concrete, since it’s a real entity you can experience with the senses. Continuous, since it could be any number of ounces (or liters, or tablespoons, or whatever). The amount of water certainly doesn’t have to be an integer. (Food for thought: since all matter is ultimately comprised of atoms, are even substances like water discrete?) |
| Consider the number 27. Would you call it abstract, or concrete? Discrete, or continuous? | Abstract, since you can’t see or touch or smell “twenty-seven." Probably discrete, since it’s an integer, and when we think of whole numbers we think “discrete." (Food for thought: in real life, how would you know whether I meant the integer “27" or the decimal number “27.0?" And does it matter?) |
| Consider a bit in a computer’s memory. Would you call it abstract, or concrete? Discrete, or continuous? | Clearly it’s discrete. Abstract vs. concrete, though, is a little tricky. If we’re talking about the actual transistor and capacitor that’s physically present in the hardware, holding a tiny charge in some little chip, then it’s concrete. But if we’re talking about the value “1" that is conceptually part of the computer’s currently executing state, then it’s really abstract just like 27 was. In this book, we’ll always be talking about bits in this second, abstract sense. |
| If math isn’t just about numbers, what else is it about? | Any kind of abstract object that has properties we can reason about. |