The Gateway to Algebra and Geometry

Episode #10 of the course Foundations of mathematics by John Robin

 

What a tour we’ve taken since Lesson 1! I promised you that the core of mathematics is found in numbers, and I hope by this point, I’ve demonstrated that.

There are still more wonderful things you can learn about numbers, so this course is only meant to give you a foundation to move further. But now that we understand what numbers are, how fractions, roots, and radical expressions work, and how to handle exponents and logarithms, we can move to the next level.

This is the world of algebra and geometry, and I will leave you on the doorstep so you can appreciate where you might go next after this course.

 

Replacing Numbers with a Variable

All our examples so far have been concrete. When I taught you about exponents, I gave you concrete expressions: 2⁸x2¹³ = 2²¹. But since the rules for exponents work for all exponents with the same base, we can replace the number with a variable: x⁸x¹³ = x²¹.

Here, x is called a variable. It means the following rule works for any number we want to use for x.

x^ax^b = x^a+b is the law of exponent addition, where a and b are also variables. This is what algebra is all about: taking all the rules we’ve learned but now generalizing them to a variable instead of concrete numbers.

We can write the logarithm as a variable instead of a number: log_cx. This means we can put in any number we want for x, and we can pick any base, c.

We already touched on this yesterday, in fact, with the rules for logarithms: log_cab = log_ca + log_cb, where a, b, and c are all variables. This rule works for any number we choose for a, b, and c.

Here, note that I’ve removed the “x” for “axb,” since as soon as we use variables, the “x” for times can be mistaken for x the variable. In algebra, whenever you see two variables together, like here with “ab,” you can assume they are multiplied together. Whenever you see “x,” assume it, too, is a variable.

Often, when we deal with variables, we must also state what kind of numbers they can be. Usually, variables are real numbers, i.e., any decimal you can imagine, positive or negative.

 

With Variables Come Graphs and Geometry

To close, I want you to appreciate one last thing. When we write log x or 2^x or x⁴, we have the opportunity to make a chart:

 

Value of x	log x	2^x	x^4
1	0	2	1
2	0.30103 …	4	16
3	0.47712 …	8	81
4	0.60206 …	16	256
5	0.69897 …	32	625

 

This is just for natural numbers, 1, 2, 3, 4, 5. You can try any real number for x. Variables in algebra now give us the ability to draw a picture of how as numbers change (x), a specific function calculated (i.e., log x, 2^x, x⁴) changes in turn. This is the world where geometry opens up and calculus lurks on the horizon.

I hope you’ve learned a lot in our ten-day tour of the foundations of mathematics. Congratulations on everything you’ve picked up to add to your skillset. I hope it inspires you to further appreciate the wonderful things you can do with this deep, rich topic.

All the best,

John

 

Other courses by John Robin:

• How to market your book online
• How to begin (and maintain) your career as a writer
• Great math problems for the 21st-century mind

 

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