Mistakes regarding algebra

A lot of mistakes that people make in Calculus originate in simple algebraic mistakes. Most calculus textbooks have a section in the beginning dedicated solely to the properties of real numbers. One is not required go through each of the proofs. However it's important to read them at least once because mistakes made when operating with functions often arise from mistakes when operating with the real numbers themselves.

There is some problem at school, which I don't know the cause, that is to give properties without proving anything. Some teachers are aware of this and give the proofs, but many others don't. Then when it comes to calculus and linear algebra people are faced with proofs and have no idea where to begin with. This is exactly what happened to me and around 75% of every other colleague that I knew.

Note: I'm using more words than mathematical symbols because it's faster to type and I'm not expecting everyone to be familiar with all the symbols.

Regarding radians

• Radian versus degrees. I believe that most of this confusion is caused by the sole fact that radians and degrees are numbers. Angles are numbers. In all other functions we think on [math]\displaystyle{ x }[/math] as the distance from the origin. Degrees are more common because they are integers and easier to read than irrational numbers. The thing is, one degree is a completely arbitrary measure. A circle can be subdivided in any number of slices that one wants. 10, 50, 100, it doesn't matter. [math]\displaystyle{ \sqrt{1^{\circ}} }[/math] is meaningless for example. What a radian does is to make the relationship that angles can be measured as multiples of some fundamental unit, [math]\displaystyle{ \pi }[/math] in this case. You can think of it as being similar to what [math]\displaystyle{ \sqrt{-1} = i }[/math] is to complex numbers. There is the problem. Every calculator or software has a standard built-in to read numbers as either degrees or radians. In numerical classes teachers often go mad because people forget this and input numbers to calculate trig functions thinking that the calculator is reading that number in degrees, when the machines are most commonly reading it as radians.

• Radians vs. Degrees vs. arcminutes. This is not so important for calculus because in physics and calculus we rarely use arcminutes. Arcminutes are more common when we have problems regarding astronomy or navigation. The thing is, clocks rotate clockwise, while the unit circle is anti-clockwise. Not only that, one full turn is either 60 seconds, 60 minutes or 12 hours in a regular clock. One full turn is 360°. By definition, 1° = 60 arcminute. There is the problem, 360° is not 60 arcminutes and 1 arcminute is not the same as the angle of 1 minute in a regular clock.

Regarding log and exp

• [math]\displaystyle{ e^{2x^2} \neq (e^{2x})^2 }[/math]. This mistake is common. Be careful!

• [math]\displaystyle{ a^n \cdot b^m \neq (ab)^{m \ + \ n} }[/math]. If the base is equal, yes. But if there are two different bases, then no. It also worth mentioning that [math]\displaystyle{ a^x + a^y \neq a^{x \ + \ y} }[/math].

• [math]\displaystyle{ b^n \neq b \times n }[/math]. I pretty sure that almost everyone made this and teachers everywhere see this mistake very often. I don't know the cause but my theory is that we often memorize that [math]\displaystyle{ 2^2 = 2 \times 2 }[/math].

• We learn that exponentiation, with natural numbers, is an operation that is a series of repeated multiplications. Such as [math]\displaystyle{ 2^3 = 2 \cdot 2 \cdot 2 }[/math]. What's the reversed operation? The inverse of multiplication is division, therefore some people assume that the inverse of repeated multiplications is repeated divisions! I think that the origin of the mistake I just mentioned is due to the fact that we all know that the log of some number yields a smaller number. Therefore, one natural conclusion is to think that we did divide the number by something to obtain a smaller number.

• The next mistake is related to reading the log itself. [math]\displaystyle{ \log_b{x} = a }[/math]. It's very common to misread it as "b to the power x is equal to a". Because that's how we write powers, we write [math]\displaystyle{ b^x }[/math]. I remember making this mistake many times. The correct way to read it is "log of x, in the base b, is equal to a ". Alternatively "b to the power a is equal to x" and this is where I think many confusions arise, because it's feels very unnatural to read a log in the same way as an exponential.

• [math]\displaystyle{ \frac{\log_c{a}}{\log_c{b}} \neq \log_c\left({\frac{a}{b}}\right) }[/math]. The formula of change of base is a quotient of two logs, not the log of the quotient! I think one argument for this is that when we write functions (log is a function) with parenthesis we can clearly see that [math]\displaystyle{ \frac{\log}{\log}c\frac{(a)}{(b)} }[/math] does not exist.

Regarding intuition

• [math]\displaystyle{ \sqrt{1/10} }[/math] vs. [math]\displaystyle{ (1/10)^2 }[/math]. When we calculate [math]\displaystyle{ x^2 }[/math] the number always become larger, except for [math]\displaystyle{ 0 \lt x \lt 1 }[/math]. When we calculate [math]\displaystyle{ \sqrt{x} }[/math] the number always become smaller, except for [math]\displaystyle{ 0 \lt x \lt 1 }[/math]. This causes some problems when evaluating limits and/or plotting the graph of functions because we can be mislead and think that the function is increasing or decreasing when it's not.

Regarding identities

• [math]\displaystyle{ \sqrt{x \pm y} \neq \sqrt{x} \pm \sqrt{y} }[/math]. The root of the sum | difference is not equal to the sum | difference of the roots

• [math]\displaystyle{ \sqrt{x^2 \pm y^2} \neq \sqrt{(x \pm y)^2} }[/math]. Sometimes people cancel out the squares with the roots.

• [math]\displaystyle{ (x + y)^2 = x^2 + 2xy + y^2 }[/math].

• [math]\displaystyle{ (x - y)^2 = x^2 - 2xy + y^2 }[/math].

• [math]\displaystyle{ x^2 - y^2 = (x + y)(x - y) }[/math]. This identity shows up a lot when calculating limits. Do not confuse it with the above two!

• [math]\displaystyle{ \sqrt{x^2} = |x| }[/math]. Be careful with this property. Often we cancel the power with the root. But [math]\displaystyle{ \sqrt{(-x)^2} \neq -x }[/math]. Any number squared can't be negative. First we square, then calculate the root. With functions it would be a composite function, a square nested in a root.

• [math]\displaystyle{ x^2 = a \iff x = \pm \sqrt{a} }[/math]. Very often we forget the minus sign and assume that the root is always positive. Two ways to explain this: the first is the previous identity; the second is to think that both [math]\displaystyle{ x^2 }[/math] and [math]\displaystyle{ |x| }[/math] are even functions. It's not uncommon to confuse "root of a negative number" with "negative root".

• [math]\displaystyle{ -x^2 \neq x^2 }[/math] but [math]\displaystyle{ -x^3 = (-x)^3 }[/math]. This quirk is a mistake that happens just about everywhere.

• [math]\displaystyle{ \left( \sqrt{\frac{a}{b}} \right)^2 = \sqrt{\frac{a^2}{b^2}} }[/math]. This is false if the fraction is negative.

• [math]\displaystyle{ |x^2| = |x|^2 = (-x)^2 }[/math]. Careful with this property! It does not mean that parenthesis and vertical bars are the same thing!

Regarding inequalities

• [math]\displaystyle{ 2 \gt 1 }[/math]. Now multiply by -1 and we have [math]\displaystyle{ -2 \lt -1 }[/math]. It's common for people to forget this fact and make mistakes in calculus because of this. When we have numbers it's easy to quickly see that [math]\displaystyle{ -2 \gt -1 }[/math] is wrong. However, when we have functions on both sides we can easily be fooled and make this mistake.

• [math]\displaystyle{ \frac{3x \ - \ 1}{x \ + \ 2} \geq 5 }[/math]. The inequality reads "For which values of [math]\displaystyle{ x }[/math] we have a function whose graph is above or equal to 5?". First [math]\displaystyle{ x + 2 }[/math] cannot be zero, else we have a division by zero. Now be careful with the comparison! When we have an equality we can multiply both sides and keep the equality the same. That's not the case with inequalities.
  
  Let's multiply both sides by [math]\displaystyle{ (x + 2) }[/math] and solve for [math]\displaystyle{ x }[/math]. [math]\displaystyle{ 3x - 1 \geq 5x + 10 }[/math]. We should get [math]\displaystyle{ x \geq -\frac{11}{2} }[/math]
  
  Our analysis is not over yet! We have a quotient of functions. What happens if both the numerator and denominator are negative? That means that for a certain range of negative values we also have that the result is greater than or equal to 5. So [math]\displaystyle{ (x + 2) }[/math] is negative for any [math]\displaystyle{ x \lt -2 }[/math]. [math]\displaystyle{ (3x -1) }[/math] is negative for any [math]\displaystyle{ x \lt 1/3 }[/math]. Conclusion, both are negative when [math]\displaystyle{ x \lt -2 }[/math].
  
  Now combine both results in this form [math]\displaystyle{ \{x \in \mathbb{R} | -\frac{11}{2} \leq x \lt -2\} }[/math] or x is in [math]\displaystyle{ [-5.5, -2[ }[/math]

Regarding vectors

• We can add and subtract vectors from vectors. But we cannot add or subtract points from each other! [math]\displaystyle{ (x_1,y_1) \pm (x_2,y_2) }[/math] is an operation that only makes sense with vectors. That's why [math]\displaystyle{ \log(\overrightarrow{a}) }[/math], [math]\displaystyle{ \sin(\overrightarrow{a}) }[/math] and [math]\displaystyle{ (\overrightarrow{a})^2 }[/math] are meaningless operations with vectors. We can have [math]\displaystyle{ \overrightarrow{r} = (\log(a), \sqrt{b}) }[/math] for example. The operation [math]\displaystyle{ \sqrt{\overrightarrow{r}} = (\sqrt{a}, \sqrt{b}) }[/math] does not exist. What exists is [math]\displaystyle{ \sqrt{||\overrightarrow{r}||} = \sqrt{\sqrt{a^2 + b^2}} }[/math].
  
  To add or subtract a vector and a point means to displace the point from its initial position to a different position.
  
  If we have functions of two or three variables, to add or subtract one from another is not the same thing as adding or subtracting points from their respective domains!

• Physics teachers like to call vectors arrows and vice-versa. Careful! They are not wrong in doing that. Is just that, from the point of view of (abstract) mathematics, vectors aren't arrows. We learn this in linear algebra. An arrow is just a graphical depiction to make a clear association between vectors and motion or change. However, in mathematics, vectors are an abstract object with no shape at all.