Algebra

Evaluating Square Roots by Hand

Square roots commonly are irrational numbers, so that it is necessary to estimate them. Usually today, we use a calculator to find them. Many students, however, are curious about how they could do it without a calculator. Here I want to reverse the usual format of this blog, presenting a summary I have written that …

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Numerical Approximation Methods: When Algebra Doesn’t Work

The problems students see in class are usually only those that can be solved by the methods they have been taught. Too many students conclude that algebra can solve anything! But the reality is that if you just wrote an equation at random, it probably could not be solved algebraically. When students ask us about …

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Properties as Axioms or Theorems

To close out this series that started with postulates and theorems in geometry, let’s look at different kinds of facts elsewhere in math. What is commonly called a postulate in geometry is typically an axiom in other fields (or in more modern geometry); but what about those things we call properties (in, say, algebra)?

Finding the Radius of a Sphere

(An archive question of the week) An interesting question came to us in 2016, where rather than using a well-known formula, it was necessary to work out both what data to use, and how to calculate the desired radius.

Principles for Solving a Formula

(An archive question of the week) Last time I discussed issues that arise in solving a simple algebraic equation. In researching that, I found a discussion of solving a formula for a variable (which in some countries is called “making x the subject”, that is, changing an equation involving x into the form “x = …

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Principles for Solving an Equation

Questions about solving algebraic equations are common. Here I will bring together several answers where we discussed the basic principles for solving relatively simple equations, which are important to learn well before moving on to quadratic equations and beyond.

Partial Fractions: How and Why

I have often noted that calculus class is where you really learn algebra. Certain techniques in calculus demand algebraic skills that either were not taught in algebra classes (because they are not needed until you get to calculus), or have been forgotten. Chief among these is the method of partial fractions. I have here put …

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