Common Errors

Thursday 7 April 2005 at 5:32 pm | In Articles | 4 Comments

I do sometimes tend to go on and on about common errors students make, such as dividing by zero or assuming what they are trying to prove. The Most Common Errors In Undergraduate Mathematics1 is a long-standing page written by Eric Schechter. It probably contains every error that I’ve ever seen (and more) so should be required reading by all maths students.

A few errors taken at random from the page, hopefully they will encourage you to read it in more detail 🙂

Everything is additive

\sin(x+y) & \text{is NOT equal to} & \sin x+ \sin y \\
(x+y)^2 & \text{is NOT equal to} & x^2+y^2 \\
\sqrt{x+y} & \text{is NOT equal to} & \sqrt{x}+\sqrt{y} \\
\frac{1}{x+y} & \text{is NOT equal to} & \frac{1}{x}+\frac{1}{y} \\

Everything is commutative

\log \sqrt{x} & \text{is NOT equal to} & \sqrt{\log x} \\
\sin 3x & \text{is NOT equal to} & 3\sin x

I do like this one, since it gives the right answer via multiple errors

    \displaystyle \int_0^{2\pi}\cos x \;dx=\left[\frac{\sin x}{x}\right]_0^{2\pi}=\frac{\sin 2\pi}{2\pi}-\frac{\sin 0}{0}=\sin - \sin =0

Oh and he has a go at teachers:

    Some teachers are hostile to questions. That is an error made by teachers. Teachers, you will be more comfortable in your job if you try to do it well, and don’t think of your students as the enemy. This means listening to your students and encouraging their questions.

Surely there aren’t many of these type of teachers about are there 😮 ?

1. Undergraduate here is an American term so the page is also highly relevant to A-Level maths students in the UK


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  1. The sections on not noticing that some steps are irreversible and confusing a statement with its converse are also gems! How many students believe that Pythagoras’ Theorem implies a 3-4-5 triangle contains a right angle?

    Comment by Nick — Saturday 16 April 2005 9:24 pm #

  2. “Surely there aren’t many of these type of teachers about are there”

    How long has it been since you’ve been a student? 😉 There are more hostile math profs than you would think. I’ve even had a math prof launch an insult in my direction. Mostly, though, math profs behave in a very anger manner when they see a flaw in a proof. You would think that they would simply point out your error and help you to see the light but no the response is often anger or hostility.

    I honestly believe that these types *do not belong in the teaching profession*. If they want to do research then they should go find a job at some think tank.

    Comment by NP-Hard — Sunday 8 May 2005 10:44 pm #

  3. That sounds terrible.
    I was a student at Warwick University many many years ago and all the staff were always very friendly and helpful.

    Have things changed? Does it depend on the institution or country? What has been others’ experiences?

    Comment by Steve — Sunday 8 May 2005 11:04 pm #

  4. This is in America; however, I think the ratio of people who behave in this manner is probably pretty consistent across all disciplines. I mean you would probably find the same number of people who wouldn’t want to answer your questions in physics, chemistry or engineering.

    It’s more of a side effect of humanity I think.

    Comment by NP-Hard — Monday 9 May 2005 9:45 pm #

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