On this and the following pages, you will find useful resources to help your learning process - no matter which teacher you have or which textbook you use.

On this page, we begin by reviewing some basic facts about different types of numbers and arithmetic operations with real numbers.


The basic and advanced learning objectives listed below are meant to give you an idea of the material you should learn about this section. These are mainly intended to be used in a course which uses an Active Learning approach, where students are required to “read ahead” before each class - but can equally be used in a more traditional course setting, for example as “after-class” review pages.

Unless your teacher gives you specific instructions, it is up to you to decide how much of the listed resources you need to read or watch - you probably do not need to go through all of it. You might also want to look at the General Study Tips & Tricks page for some recommendations on how to effectively study with a math textbook and videos.

Basic Learning Objectives

These are the tasks you should be able to perform with reasonable fluency when you arrive at your next class meeting. Important new vocabulary words are indicated in italics.

  • Understand the classification of numbers into natural, whole, rational, irrational, and real numbers.
  • Understand the order of arithmetic precedence, using the GEMS (replacing the older PEMDAS) mnemonic rule.
  • Be able to correctly evaluate expressions involving many arithmetic operations.

Advanced Learning Objectives

In addition to mastering the basic objectives, here are the tasks you should be able to perform after class, with practice:

  • Recognize algebraic expressions as opposed to numerical expressions.
  • Recognize constants and variables in a given algebraic expression.
  • Evaluate or simplify a given algebraic expression.

To prepare for class

  • Watch the following video (by Sjoberg Math, 4m59) which explains the classification of numbers into natural, whole, rational, irrational, and real numbers, including a bit of their history:

  • Watch the following video (by MooMooMath and Science, 3m51s) which gives examples of how to determine the type of a given number:

  • Watch the following videos (by Mrs. Smith MJHS Math, each 4m) which explain the mnemonic rule GEMS (which replaces the older rule PEMDAS) to determine the correct order of arithmetic operations:

  • Review, as needed, the content on these pages covering Sec IV material for more information on arithmetic operations of real numbers:

After class

  • Watch the following video (by Anywhere Math, 9m56s) which explains algebraic expressions and how to distinguish constants and variables:

  • Watch the following video (by Anywhere Math, 10m16s) which shows examples on how to evaluate or simplify a given algebraic expression:

Beyond this course

  • (Optional) Watch the following video (by Matt Parker & Numberphile, 14m26s) which explains that the classification of numbers actually is more involved than it may seem, by talking about constructable, algebraic, transcendental, computable, and normal numbers:

  • (Optional) Watch the following video (by Up and Atom, 9m02s) which gives a brief introduction to the the concepts of imaginary and complex numbers:


Gabriel Indurskis Avatar Gabriel Indurskis






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