Department of Mathematics --- College of Science --- University of Utah

Mathematics 1010 online


Sadly, most people learn mathematics by thinking of it as a bunch of recipes to handle certain problems. Actually mathematics is a web of facts, concepts, and logical reasoning, and a vastly more efficient approach to understanding it is to focus on the principles and connections rather than on isolated facts.

Isaac Newton, considered by many the greatest scientist that ever lived, and certainly one of the greatest mathematicians that ever lived, put it this way in a letter to Nathaniel Hawes on 25 May 1694:

You want to be one who reasons nimbly and efficiently, rather than a vulgar mechanic!

Sprinkled through these pages are explicit descriptions of some of the principles that can be used in building and understanding not just the curriculum of Intermediate Algebra, but all of mathematics. They are typeset in green for better visibility, and they are all listed on this particular page for your reference. Some are much more profound than others, but all of them will empower you tremendously if you consciously apply them and if you actively look for them and observe them in action as you go through this course.

Building Mathematics

Reduce your problem to one you have solved before.

This principle is used ubiquitously in mathematics. It has a major consequence: in order to understand a piece of mathematics you have to understand what preceded it. Following is a more elaborate version of the same principle:

Introduce concepts in a simple context and then generalize them in such a way that rules and facts that are true in the simple context remain true in the more general context.

You can see this principle in action for example in the way we build the number system or define powers with exponents other than natural numbers.


Always Have Expectations

Before you attempt the solution of a problem or start studying a subject, it pays to think explicitly about what you expect your solutions to look like, or what you expect to learn. There are two possibilities: your expectations are met, or they are not. In the first case you feel reassured and on top of things which is nice. In the second case there are two possibilities. Either you made a mistake and now that you are alert to the fact you can recover from it. Or, and this is the most exciting possibility, your expectations were based on some misunderstanding and now you have a chance to improve your understanding and learn something new.

Solving Equations

Applying the same operation on both sides of a valid equation gives another valid equation. Think of an equation as one of those old fashioned scales where you match an unknown weight on one side with a collection known weights on the other. The weights on the two sides are in balance. If you do the same thing to the weight on each side they will still be in balance.

To solve an equation figure out what bothers you and then apply a suitable operation on both sides of the equation to get rid of it.

This is the only way to solve an equation. You don't need to memorize techniques for example for each item in the following list of problems taken from our textbook: linear equations in standard form, linear equations in non-standard form, linear equations involving decimals, linear equations containing fractions, linear equations: special cases, linear equations using ratios, linear equations involving proportions, linear equations in percent problems. There is a similar litany for quadratic equations, and one for radical equations. Don't believe any of this, just understand how to manipulate algebraic expressions and use common sense and the above principle.

When you have solved a problem observe this principle:

Check your answers

Always Check Your Answers

There are many reasons to check your answers:

Solving Problems

To solve a difficult problem solve a simpler problem first and apply what you learn to the more difficult problem.

Of course you may have to iterate this technique and build a whole hierarchy of problems to solve a truly difficult problem. Sometimes mathematicians spend a life time just doing that.

Students often look at a problem and decide instantly that they don't know how to solve it. Don't do that. If everything you can do you can do in 5 minutes you cannot do very much. Some of the problems in this course do require significant effort on your part which may include going back over your notes. looking for simpler yet related problems, or simplifying the problem at hand to make it more tractable.

Do the easy stuff first

When you need to solve several problems work on the easy ones first. They may teach you something that will help you solve the more difficult problems.

Obviously this principle applies to doing homework and taking exams. There is rarely a reason to do a set of problems in the precise sequence in which they are listed.

Complex Numbers

This is a minor principle that nonetheless makes work with complex numbers straightforward:

You work with complex numbers as you would with ordinary algebraic expressions containing the variable i except that you replace i2 with -1 wherever it occurs.