I started learning LaTeX a couple of years ago, but it wasn’t until last year when I started studying for actuarial exams 4/C and 3/MFE that I really started to become proficient at using it. If you are not familiar with what LaTeX is, it’s a markup language that lets you easily (although it takes some effort to learn) write mathematical formulas on a computer screen by using the keyboard. For example, \[ f(x) = \frac{1}{\sigma\sqrt{2\pi}}\mathrm{e}^{-\frac{(x-\mu)^2}{2\sigma^2}} !\], which is the code written in the WordPress editor, produces the formula for the Normal Distribution:
\[ f(x) = \frac{1}{\sigma\sqrt{2\pi}}\mathrm{e}^{-\frac{(x-\mu)^2}{2\sigma^2}} \]
You can try to write the above formula by using Microsoft Equation editor (like many of us did as highschoolers), but you’ll quickly realize that it takes an extremely long time, and you’ll be wishing that you had a faster, more efficient way of writing mathematical notation – this is where the usefulness of LaTeX becomes apparent.
I started using LaTeX while posting in the actuarial message boards, which are popular amongst candidates who are trying to study for exams. The bulletin board system has a LaTeX compiler installed, so you can easily consult other students from all across the world. For example, if I’m studying at 2:00 AM in the morning, I can post a question on the message board and there will most likely be someone who is awake at that time in Europe or China who would be willing to answer that question.
There are some more well-known message boards as well, such as StackOverflow and MathOverflow, where people (mostly from technical backgrounds) ask each other questions. Oftentimes they’ll use LaTeX to write technical notation – which greatly facilitates communication. I’ve found StackOverflow to be very helpful from time to time. On the other hand, I can’t even understand most of the questions being asked in MathOverflow, which is an online community of mathematicians asking each other research-level questions pertaining to mathematics. Fortunately, there’s another site under the StackExchange umbrella called Mathematics Stack Exachange, which caters to undergraduate and early graduate-level students, and is much more accessible. These websites are only 4 years old and have already made a huge impact on the way people collaborate on technical projects. I’m not sure if Don Knuth imagined this when he invented TeX way back in 1978, but if he did, he had tremendous foresight.
I’ll close by demonstrating a problem on matrices, which I started studying last week. I covered the basic row operations on matrices and today I just went over spanning and matrix equations in the Ax=b form. It’s been very interesting going back to material that I first learned 5 years ago – I have a different perspective now and it’s much like watching a movie – you always pick up something new the second time around.
Problem:
Solve the system of equations:
\[\begin{aligned} x_1 – 3x_3 &= 8 \\ 2x_1 + 2x_2 +9x_3 &= 7 \\ x_2 + 5x_3 &= -2 \end{aligned}\]
Solution:
We’ll start by writing the augmented matrix of this system of equations:
\[\left[ \begin{array}{rrrr} 1 & 0 & -3 & 8 \\ 2 & 2 & 9 & 7 \\ 0 & 1 & 5 & -2 \end{array} \right] \]
Replace row 2 with the sum of row 2 and negative 2 times row 1:
\[\left[ \begin{array}{rrrr} 1 & 0 & -3 & 8 \\ 0 & 2 & 15 & -9 \\ 0 & 1 & 5 & -2 \end{array} \right] \]
Interchange rows 2 and 3:
\[\left[ \begin{array}{rrrr} 1 & 0 & -3 & 8 \\ 0 & 1 & 5 & -2 \\ 0 & 2 & 15 & -9 \end{array} \right] \]
Replace row 3 with the sum of row 3 and negative 2 times row 2:
\[\left[ \begin{array}{rrrr} 1 & 0 & -3 & 8 \\ 0 & 1 & 5 & -2 \\ 0 & 0 & 5 & -5 \end{array} \right] \]
Scale row 3 by 1/5:
\[\left[\begin{array}{rrrr} 1 & 0 & -3 & 8 \\ 0 & 1 & 5 & -2 \\ 0 & 0 & 1 & -1 \end{array} \right] \]
Replace row 2 with the sum of row 2 and negative 5 times row 3:
\[\left[\begin{array}{rrrr} 1 & 0 & -3 & 8 \\ 0 & 1 & 0 & 3 \\ 0 & 0 & 1 & -1 \end{array} \right] \]
Replace row 1 with the sum of row 1 and three times row 3:
\[\left[\begin{array}{rrrr} 1 & 0 & 0 & 5 \\ 0 & 1 & 0 & 3 \\ 0 & 0 & 1 & -1 \end{array} \right] \]
This final matrix is equivalent to the following system of equations:
\[\begin{aligned} x_1 &= 5 \\ x_2 &= 3 \\ x_3 &= -1 \end{aligned} \]
Thus the solution set is \((5,3,-1)\)
The example above is how I would typically ask or answer a question posted on a bulletin board. Actually, giving out the entire solution (as above) is typically frowned upon and most people just give enough hints so that the person who asked the question can figure it out on their own. However, you can definitely see from the example that LaTeX allows you to cleanly print the matrices, which makes it much easier to understand. I remember back in high school when my friends and I would struggle trying to help each other out via AIM or some other chat client. I only wish I’d found out about LaTeX sooner.