I must have been 15 years old when that happened. Fortunately, the next year I was taught the secret “all numbers different from zero have exactly n different n-th roots”. In particular, the roots of $-1$:
\[z^n-1=0\quad\Leftrightarrow\quad z\in\left\{\exp\left(\frac{2\pi k i}{n}\right):0\leq k\leq n-1\right\}\]
where, of course, $i$ is the elementary component of the imaginary numbers, the square root of $-1$
\[i^2=-1\]
At my 17 years old, I was happy. I had one last shot at imaginary operations in the last year of preparatory school. I thought I knew everything there was to know about complex numbers... I couldn’t have been more wrong!
Before I turned 18, I was completely sure I was going to study a degree in computing engineering. Fortunately, I found a course in Applied Maths. When I went home with the good news that I have found the first love of my life (that is, my degree) my mum freaked out. It is now clear to me why she freaked out and I understand her. She didn't know, and still doesn't know, what doing maths is about... but that's another story. The point here is that she proposed me to look at the syllabus of the Actuarial Sciences programme.
I only understood the word Calculus, and that was enough for me to take a leap of faith and study actuarial sciences. In my second Algebra course, while studying monoids, groups, rings, and fields... the complex numbers visited me again.
This time it was a fellow classmate who asked the question "can we raise a complex number to a power that is itself a complex number?" Wow! That was completely ridiculous! But what really blew my mind was the answer of the lecturer: "Yes, sure. But that will be seen only by those taking the subject on Complex Analysis".
I had to take that class! I realised only mathematicians took that class as an elective unit. My solution: change my degree from Actuarial Sciences to Applied Maths. However, while doing the transfer I had to pay some validations of my previous units and I couldn't afford it. The option to delay the payment was to actually sign in for both courses at the same time and make the payment when I finished both degrees... so I did. Anything to know what it even means something like $i^i$.
The moment arrived. I signed in to the Complex Analysis course and my favourite lecturer was giving the class. I was so happy! I didn't just learn how to evaluate $i^i$, but to find logarithms of complex numbers, trigonometric functions, etc. In fact, I even wrote an article for the maths uni mag that you should be able to read here (that is if you can read Spanish).
But I liked something even more! There were real integrals which could be easily evaluated if considering the complex version of it and a convenient path. I remember perfectly the problem on my final exam and I remember I was able to solve beautifully \[\int_0^{\infty}\frac{dx}{1+x^{2n}}\quad\text{for }n\in\mathbb N\]
This time the course finished and I knew I didn't know everything. But I knew what I wanted. By that time I discovered that my vocation was not with the Complex Analysis even though by then I had a fascination with the Riemann Hypothesis (which I bet it is true!) but with Probability.
Every single time I found an integral, I tried to apply Complex Analysis techniques. Most of the times without very good results. It was usually easier to just do the integrals by the regular means.
When I finished my degrees, I was already working in a market research company as a statistician. I really liked my job. But I got to believe that my life has got rid of the complex numbers for good... along with other things I used to like. The epsilons and the deltas. They were no more in my life. It was like that for 5 long years...
I managed to travel to the UK for an MSc in Applied Maths and I saw them again. Fourier transforms, inverse Fourier transforms, and oscillatory integrals. It was cool that after so long they keep visiting me. After all I don't like when friends just disappear... and the complex numbers and I see each other as we've been friends all the time regardless of the time we last saw each other. Those are the good friends.
As of today, the last time I saw them was when I tried a real integral via a saddle point approximation that if possible, I would have been able to find a large deviation principle on the boundary of a semi-infinite totally asymmetric simple exclusion process. I don't remember quite well what went wrong.
Chances are we will meet again this week. I hope this is not the last time...
Before I turned 18, I was completely sure I was going to study a degree in computing engineering. Fortunately, I found a course in Applied Maths. When I went home with the good news that I have found the first love of my life (that is, my degree) my mum freaked out. It is now clear to me why she freaked out and I understand her. She didn't know, and still doesn't know, what doing maths is about... but that's another story. The point here is that she proposed me to look at the syllabus of the Actuarial Sciences programme.
I only understood the word Calculus, and that was enough for me to take a leap of faith and study actuarial sciences. In my second Algebra course, while studying monoids, groups, rings, and fields... the complex numbers visited me again.
This time it was a fellow classmate who asked the question "can we raise a complex number to a power that is itself a complex number?" Wow! That was completely ridiculous! But what really blew my mind was the answer of the lecturer: "Yes, sure. But that will be seen only by those taking the subject on Complex Analysis".
I had to take that class! I realised only mathematicians took that class as an elective unit. My solution: change my degree from Actuarial Sciences to Applied Maths. However, while doing the transfer I had to pay some validations of my previous units and I couldn't afford it. The option to delay the payment was to actually sign in for both courses at the same time and make the payment when I finished both degrees... so I did. Anything to know what it even means something like $i^i$.
The moment arrived. I signed in to the Complex Analysis course and my favourite lecturer was giving the class. I was so happy! I didn't just learn how to evaluate $i^i$, but to find logarithms of complex numbers, trigonometric functions, etc. In fact, I even wrote an article for the maths uni mag that you should be able to read here (that is if you can read Spanish).
But I liked something even more! There were real integrals which could be easily evaluated if considering the complex version of it and a convenient path. I remember perfectly the problem on my final exam and I remember I was able to solve beautifully \[\int_0^{\infty}\frac{dx}{1+x^{2n}}\quad\text{for }n\in\mathbb N\]
This time the course finished and I knew I didn't know everything. But I knew what I wanted. By that time I discovered that my vocation was not with the Complex Analysis even though by then I had a fascination with the Riemann Hypothesis (which I bet it is true!) but with Probability.
Every single time I found an integral, I tried to apply Complex Analysis techniques. Most of the times without very good results. It was usually easier to just do the integrals by the regular means.
When I finished my degrees, I was already working in a market research company as a statistician. I really liked my job. But I got to believe that my life has got rid of the complex numbers for good... along with other things I used to like. The epsilons and the deltas. They were no more in my life. It was like that for 5 long years...
I managed to travel to the UK for an MSc in Applied Maths and I saw them again. Fourier transforms, inverse Fourier transforms, and oscillatory integrals. It was cool that after so long they keep visiting me. After all I don't like when friends just disappear... and the complex numbers and I see each other as we've been friends all the time regardless of the time we last saw each other. Those are the good friends.
As of today, the last time I saw them was when I tried a real integral via a saddle point approximation that if possible, I would have been able to find a large deviation principle on the boundary of a semi-infinite totally asymmetric simple exclusion process. I don't remember quite well what went wrong.
Chances are we will meet again this week. I hope this is not the last time...
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