In my first article “The most enlightening Calculus books”, I argued the importance of maintaining high standards for mathematics education and suggested deep and inspiring calculus books for those of you who are interested in pursuing the joy of learning mathematics. The feedback has been overwhelming and I wish to follow up with an article that addresses a couple of remarks that I’ve received by email.

One person commented on the blog, and another wrote me privately, to express their concern that “harder books are not necessarily better books” and that teaching which is geared towards only the smartest kids is a mistake. I want to point out that I’m in no way advocating teaching for the brightest minds only. Wide access to mathematics is something that should be encouraged all over the world, and I’m pretty sure it will help take society in a better direction. In fact, education in general – and mathematics, technical and scientific education in particular – are key for the development of every country and ultimately for good of humankind.

However my point was that with wider access to higher education mathematics, we should not reduce the expected and established testing standards. In other words, there is a fair level of understanding that we should expect from people who major in math or from students who strongly depend on mathematics for their future careers. Furthermore, the textbooks adopted should be mathematically sound and provide the right intellectual stimulation for those who could use it. That said, there is nothing wrong with teachers trying to use different styles of teaching to reach a wider audience, or for students who struggle with the level of math presented in the textbook, to supplement it with simpler books in order to get an easier start. Hence, it’s perfectly OK for a student (who for example is taking an undergraduate class in programming in C) to read C For Dummies if The C Programming Language by K&R is too hard for them off the bat. But that does not imply that the class should adopt “C for Dummies” as their textbook nor that the examination should be based on such a book. So to summarize this point, feel free to study any number of introductory books, as long as you know that if you plan to be serious about mathematics, you should be able to eventually read and understand standard books and be able to solve most of the exercises put forward in them.

Having clarified the first concern, I’d like to provide an answer for the second point, which actually interests me the most. A few readers wrote me emails about how they feel enthusiastic about the post and the opportunity to study mathematics again, but how those books are way too advanced for them, because they simply forgot all the mathematics taught at a high school level. So I’ve received a few “how can I get a refresher of high school math?” type of questions. The mathematics that you learned in high school is classified as *pre-calculus*, and as you can expect it is propaedeutic to learn math at an higher level. It is normal that you forgot quite a few formulas, but having a good grasp of the essentials of precalculus can make a big difference when trying to master calculus. You should have a decent knowledge of basic algebra, trigonometry, exponential, logarithmic, and analytic geometry. Calculus itself will provide you with a refresher of some of these topics and give you a deeper understanding not only of “how” but rather “why”. That said, Calculus without a decent precalculus base can be a big challenge for most people. Before proceeding to suggest a few resources, let’s try to establish if you actually need a refresher course or not. Here is a (simple and of course incomplete) list of some basic exercises. If you haven’t a clue or struggle to find a lot of the solutions for them, a refresher may be in order.

**Simple Precalculus Questions:**

1) Factor the following polynomials:

- [tex]\displaystyle x^{2}-6x+9[/tex]
- [tex]\displaystyle x^{2}+x-6[/tex]
- [tex]\displaystyle x^{3}-27[/tex]

2) Solve for x:

- [tex]\displaystyle 3x^{2}+5x-2=0[/tex]
- [tex]\displaystyle |x^{2}-x|=3[/tex]
- [tex]\displaystyle x^{4}-8ax^{2}+16a^{2}=0[/tex]
- [tex]\displaystyle \frac{x^2+x-6}{x+3}=0[/tex]
- [tex]\displaystyle 2\sqrt{x} = x – 15[/tex]

3) Find the values of x for which:

- [tex]\displaystyle x^{2}>9[/tex]
- [tex]\displaystyle |2x-3| \leq 5[/tex]
- [tex]\displaystyle |2x-1| > 9[/tex]
- [tex]\displaystyle |x-1| + |x-3| \geq 8[/tex]

4) Evaluate:

- [tex]\displaystyle \log_{2}{1}[/tex]
- [tex]\displaystyle \ln{e}[/tex]
- [tex]\displaystyle \log_{2}{1024}[/tex]
- [tex]\displaystyle \frac{4^{8}2^{4}}{2^{12}}[/tex]

5) Solve for x:

- [tex]\displaystyle 5^{x}=10[/tex]
- [tex]\displaystyle \log_{3}{7x} = 2[/tex]
- [tex]\displaystyle \log_{x}{9}=2[/tex]
- [tex]\displaystyle \ln(3x-2)=0[/tex]
- [tex]\displaystyle 3^x+x=4[/tex]

6) Solve for x, where [tex]\displaystyle 0\leq x \leq 2\pi[/tex]:

- [tex]\displaystyle 2\sin{x} = 1[/tex]
- [tex]\displaystyle \tan{2x} = \frac{\sqrt{3}}{3}[/tex]
- [tex]\displaystyle \sin{3x} = 1[/tex]
- [tex]\displaystyle \cos^{2}{x} – x = 2 -\sin^{2}{x}[/tex]

7) Write the equations of the following curves in the Cartesian plane:

- Parabola
- Hyperbola
- Circle
- Ellipse

8 ) Find the vertex, focus, and directrix of the parabolas given by the equations:

- [tex]\displaystyle x^{2}=16y[/tex]
- [tex]\displaystyle y^{2}+4y+12x=-16[/tex]

9) Find the center, vertices, foci, and eccentricity of the hyperbola given by the equation:

[tex]\displaystyle \frac{x^{2}}{4}-\frac{y^{2}}{36}=1[/tex]

10) Find the equation of a circle whose center is at [tex](2, -3)[/tex] and radius [tex]3[/tex].

11) Determine the center and radius of the circle with equation:

[tex]\displaystyle x^{2} -4x+ y^2-18y = -4[/tex].

How did it go? Did you experience many struggles and the feeling that “I used to know this stuff”? If so, then it is a good idea to go for a refresher before attempting calculus right away. The following are two books that you may find useful to respectively learn and refresh basic math in a well organized manner:

- Precalculus by Michael Sullivan: a big book, which is quite extensive and thorough. If you want an all-in-one book that covers all you need to know about precalculus and more, in a clear but college oriented manner, than this is without doubt an excellent choice. It will likely make the step up to Calculus quite easy.
- Schaum’s Outline of Precalculus: it has a less prosaic approach but it’s still very clear and easy to read. If you were pretty good at math in high school and you just forgot a few things because you haven’t touched these topics in a while, then pick this book up. It is adequate for already mathematically inclined people who are in a rush to brush up the skills they once had.

If you feel entirely clueless and would like a “for dummies” type of book, the following two titles seem to have a good table of contents and excellent reviews:

If you would like to use some free resources available online instead, here are some lessons:

- Precalculus Tutorial
- Topics in PRECALCULUS
- OJK’s Precalculus page
- Exploring Precalculus
- Precalculus problems
- Prof. Ward’s lecture notes (PDF, 23 pages)
- Dave’s Short Trig Course
- Precalculus Lessons
- Collection of links related to Precalculus
- Wikipedia entry on Precalculus

If you know of any other resources that are available for free, or if you successfully used other books for these purposes, please feel free to use the comment section to add to the discussion.

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Is (x^3 – 9) a typo?

The typical thing to go there is to factor the difference of two squares. This is the difference between a cube and a square.

Hi Dan,

it was meant to be a trick question, but I’ve changed it in order to avoid confusion. ๐

“Factorize”? Do you mean “Factor”? =)

Joel, both forms are accepted, even if “factorize” is less common. I’ve changed it though, so it won’t bug you. ๐

I factored 1.1 and 1.2 Fine. I am having trouble with 1.3. 27 factors into either 1×27 or 3×9 so to factor it I did this:

x^3 – 27 [27 factors to 1×27 or 3×9]

(x + 3)(x^2 – 9) [9 factors to 1×9 or 3×3]

(x + 3)(x – 3)(x + 3) [is it right? let’s work backward]

(x^2 – 9)(x + 3) [-3x + 3x = 0]

x^3 + 3x^2 – 9x – 27 [so I was wrong… or did I screw up the multiplication?]

I found Cliffs Math Review for Standadized Test by Jerry Bobrow, Ph.D to be emensly helpful as a review guide.

Scratch my last post, I got it.

Hi Drew,

your first step is mistaken. In fact:

[tex](x+3)(x^{2}-9) = x^{3} + 3x^{2}-9x-27 \neq x^{3}-27[/tex]

The solution to this problem becomes immediate if you remember how to factor the difference of two cubes:

[tex]a^{3}-b^{3}=(a-b)(a^{2}+ab+b^{2})[/tex]

Antonio, that’s exactly what I realized after I made my first post. My problem was the I didn’t immediately recognize 27 as 3^3 ๐

Is 2.3 also a trick question?

x^4-8ax+16a^2=0 is tough

but x^4-8ax^2+16a^2=0 looks easier.

2.3 had an actual typo Dave. Thanks for pointing this out. ๐

Jeez, I think you might be aiming too high. I used to teach introductory programming classes to college-age adults, I always spent the first 2 weeks teaching the Associative and Distributive Laws. This helped tremendously in improving the students’ ability to parse basic equations, but hell, it isn’t even high-school level MATH, let alone algebra.

My academic background is math and IT – your examples are the kind of stuff I was doing when I was in school.

But I have a younger brother who didn’t study a lot of Math but is now entering the world of business and is realizing he needs to improve his math.

Not everyone was an A-grade Math student at high-school and I was wondering what resources you might recommend for someone like my brother (an adult) who would have some difficulty working out what 75% of $1200 was, etc.

I know for us that’s trivially easy for us but actually it’s probably only a small %age of American adults that would know what to do with any of the examples you gave above as ‘high school math’ in your post.

What might you suggest for the population?

Thanks, Ben

I have a BS in math and obviously don’t use it because I FORGOT ALMOST EVERYTHING THERE IS TO KNOW!

I can’t solve 1.3, 2.5, 3.4 and everthing past 4.3 !

Readers of this blog post might be interested in free high school and university video lectures in mathematics to get a refresher on the subject.

I have collected links to most of the free math video courses on my blog. One of the posts: link.

Lots and lots of other lectures:

https://freescienceonline.blogspot.com/

And I just launched a new free science education website:

https://www.freesciencelectures.com

Hope you find them useful! ๐

I just wanted to comment on C for Dummies. I read C for dummies over the winter break before I was goint to take my first programming class. I was told that basically everyone failed the first time they took it. I bought the textbook that was assigned for the course and thought it was to difficult and so I returned it. I spent a lot of time working with my code and went to all the lectures, and ended up with an A.

I guess what I am trying to say is that students should take the initiative to get the book that works best for them. Also I think an easy textbook, and rigourous lectures is not a bad combination.

Also at least at the University of Wisconsin, all the topics on conic sections are covered in calculus 2.

I immediately noticed the one solution of 5.5 by inspection, but is there a way to analytically solve the equation a^x + x = b for constants a and b in general other than here where you have the luxury of knowing that b = a + 1?

John, 5.5 is a Transcendental equation. In this case it was obvious that 1 is a root but generally speaking you can’t solve arbitrary transcendental equations with algebraic techniques. The most common approaches are graphical and numerical methods, which generally yield approximated answers. If you plot both sides of the equation, the intersecting points of the two curves provide you with the solution. Common numerical methods are the Bisection method, Regula Falsi method, Fixed Point Iteration and the famous Newton-Raphson method.

Hi there!

I was thinking you would like to know a better way of doing high school math faster and much more easier than the normal way.

check out the indian system of vedic maths at http://www.vedicmathsindia.org

It is much more applicable to students today as it helps them todo mental calculations.

Check it out.

Gaurav

Gosh…. I can imagine those were the days when I had to handle trigonometry, algebra, geometry, etc. and so on. A lot has been lost since for a long time. Great refreshment crash course ๐

High school mathematics is very important and the problems you give are very suitable to refresh our high shool math skills.

Great, thx for knowledge

monsieur,

i like this website for mathematics. this is to much intresting for me.

thank you

To solve 3^x + x = 4, the standard method used by high school students is:

1) rewrite the equation as 3^x + x = 3 + 1, and find by guessing its solution x = 1.

2) show that x = 1 is the only solution.

PLEASE PUT SOME MORE EXAMPLES OUT FOR THE PUBLIC.

do you know a class i can take in NY city? i rather go to class than study alone from a book. i gratuated from highschool 7 years ago, so i dont remember much. i need a refresh class for high shool math.

can u tell me what is the last digit of

(((((((7)^7)^7)^7)^7)…….there are 2001 7s?

I can heartily recommend “Engineering Mathematics” by KA Stroud & DJ Booth. It has a major section on Foundation topics that covers precalculus in great detail as well as being a complete undergrad engineering mathematics textbook. It is also significantly less expensive than your recommendation. They also have a textbook called “Foundation Mathematics” that appears to be pure precalculus but “Engineering Mathematics” seems to be the better deal.

I am now retired and refreshing my math knowledge from MANY years ago and the Stroud books are vastly superior to what I had to put up with as an undergrad.

Nothing good can be learned without doing. Thanks for all the questions.