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Discussing Numbers: The Number Line, Types of Numbers, Sets, Inequalities, and Intervals October 23, 2006

Posted by LearningNerd in Arithmetic, Math, Mondays, Number Theory, Pre-Algebra, Science.

I said I’d start with the basics, so here they are!

Sections in this post: The Number Line and Absolute Value, Types of Numbers, Sets, Inequalities, and Intervals and a Final Review.

By the way, this post includes three videos. They’re a bit long and not particularly entertaining, but they’re useful if you’d like to take a break from reading.

The Number Line and Absolute Value

It all starts with the number line, which, as the name implies, is simply a line marked with numbers. Math teachers love to use number lines to teach everything from arithmetic to calculus, and though I remember the number line as it appeared on many repetitive worksheets, I realize that it can actually be useful. Like an outline, the number line is a great way to visualize the abstract ideas you’re working with: smaller numbers on the left, bigger numbers on the right. For an example, check out this nifty tool that compares two numbers using a number line.

Next, absolute value: the distance of a number from zero (the origin). This one’s pretty straightforward. See Absolute Value for a complete overview.

Types of Numbers

A set is a group of things (in this case, numbers) considered as a whole. Now, a lot can be said about sets (see set theory), but for now, that simple definition will do.

number system is a set of numbers that can be used in arithmetic operations (like addition and subtraction). As we all learned in school a long, long time ago, there are different types of numbers like positive numbers, negative numbers, even numbers and odd numbers. (Those types of numbers can be listed in sets, but they aren’t considered number systems. I looked for a formal explanation of this, but I’ve yet to find one.)

There are several types of numbers or number systems commonly taught in math classes:

See Number Types for a basic overview, and/or watch this video:

I’ll be learning more about irrational and complex numbers later; they aren’t important until you get to more advanced mathematics.  


Like I already mentioned, a set is just a group of things considered as a whole. When discussing numbers, mathematicians save time by using set notation. The whole thing’s pretty simple, but it uses some symbols that you don’t see too often. See Set Notation for a review, and be sure to bookmark the page for reference. The following video also explains set notation:

Just for fun, here’s an interesting math problem I came across: “Which is greater, the number of rational numbers between 0 and 1 or the number of rational numbers between 0 and 2?” See the answer and explanation here. (By the way, I got it wrong. :roll: )


The inequality signs (like < and >) are used to compare numbers. They’re pretty straightforward; 13 is less than 27 (13 < 27), two is greater than negative three (2 > -3), and so on. Rest assured, it does get more complicated.

For kids or anyone who wants some practice, here are a couple interactive games and quizzes:

Intervals and a Final Review

An interval is basically just a set of numbers containing every number between two numbers. As with sets, intervals have their own notation; see Interval Notation for a summary of the basics and remember to bookmark the page for reference.

The following video reviews inequalities, intervals, and set notation:

As a final review, check out this interactive example of Set and Interval Notation, where you can move an interval on a number line and see how the notation changes.


1. Carlene Riley - October 30, 2006

these are great lessons, i would love to have more of your maths lessons. i have learnt a lot of the things i could not understand. thank you , hope to haer from you real soon.


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