Number Flavors Part II: The Reals


Image credit: “an apple a day…” by Michael Verhoef on flickr.

 

In the last post we talked about rational numbers. Rational numbers are numbers that are integers divided by integers. So, 3, 4/5, -10, and 0.99 are all rational numbers. It’s not hard to think of real-world uses for these numbers: 3 pennies, 4/5 of a cake, a -$10 balance, 0.99 yards of string. And we saw that if I write down any integer, fraction, or decimal, it will be a rational number.

So are there numbers other than rational numbers? Weird as it may seem, the answer is yes. I’m going to introduce two more categories of numbers: real numbers and complex numbers. You probably learned about these in school at some point. But if you’re not sure what they mean, this post may help.

integer_to_complex

The integers are a sub-category of the rational numbers, which are a sub-category of the real numbers, which are a sub-category of the complex numbers. (Colors chosen to match my synesthesia.)

Wikipedia says a real number is “a value that represents a quantity along a continuous line.” Let’s break that sentence down into pieces. A “value” is a number. We won’t be more specific than that here. “Line” means number line, like the one people sometimes draw for the integers.

"NumberLineIntegers" by Ezra Katz - Own work. Licensed under CC BY-SA 3.0 via Wikimedia Commons.

The idea of the integer number line is that all of the integers can be put in order: the next integer after 0 is 1, and the integer after that is 2, and so on. In the other direction, the integer one less than 0 is -1, the integer one less than -1 is -2, and so on. The fact that the number line goes on forever is okay, because we don’t have to draw the whole thing to know what the pattern is. Each number is 1 more than the number to its left, and 1 less than the number to its right.

The set of integers is discrete. In short that means that there’s a gap between 2 and 3. If you drove your car down the integer number line, there would be a pause between driving past -5 and driving past -4.

Continuous is the opposite of discrete. A number line is continuous if there are no gaps between numbers.

We’ll discuss how that works in the next post.

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