The Inversion

Note: I use LaTeX code in this piece. The code won’t fit on the page on the mobile app. Read through your email or desktop to access the piece. The LaTeX code feature is still in beta mode, but I wanted to do some mathematics and it presents it in the best way.

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Don’t try to be clever. Try not to be stupid.

I’ll continue to use Charlie Munger’s principles for the rest of my life.

A vital one is the principle of inversion.

Take any problem you currently have. It's easy to consider the problem forward.

Let’s say you want to sleep better. Looking forward, you’ll consider all the strategies you can use to encourage better sleep.

By inverting the problem, we consider all the things that wouldn’t help with sleep. Maybe we don’t stay up late when we have an alarm set for tomorrow, maybe we don’t set an alarm at all.

The biggest problems are often finally solved when considered backwards.

Photo by Jon Tyson on Unsplash

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Proof By Contradiction

In mathematics, many methods exist to prove a statement we believe to be true.

One method uses the art of inversion. Proof by contradiction flips the problem on its head, looking at it backwards.

Take one such example:

\(\text{The square root of two} (\sqrt{2}), \text{is an irrational number.} \)

\(\text{It can’t be expressed as a fraction in the form} \hspace{0.1cm} \frac{a}{b}.\)

\(\text{When using proof by contradiction, we assume the opposite.}\)

\(\text{Assume} \sqrt{2} \hspace{0.1cm} \text{is a rational number, and work until our mathematical proof contradicts us.}\)

Here is the proof below:

\(\text{If} \sqrt{2} \hspace{0.1cm} \text{is rational, it can be expressed as a fraction in its simplest terms. Hence} \sqrt{2} = \frac{a}{b}.\)

\(\text{We can square both sides of the equation.} \)

\(\text{For example, if} \sqrt{49} = 14/2, \text{then} \hspace{0.1cm}49 = \frac{14^2}{2^2} = \frac{196}{4} = 49. \hspace{0.1cm} \text{So} \hspace{0.1cm} 2 = \frac{a^2}{b^2}.\)

\(\text{It follows that} \hspace{0.1cm} a^2 = 2b^2. \text{Anything multiplied by two is even, so a is even.}\)

\(\text{Substitute} \hspace{0.1cm} a = 2k \hspace{0.1cm} \text{into the equation.} \hspace{0.1cm} k \hspace{0.1cm} \text{is another unknown. Now} \hspace{0.1cm} 2 = \frac{(2k)^2}{b^2}.\)

\(\text{Rearrange for} \hspace{0.1cm} b^2. \hspace{0.1cm} 2 = \frac{4k^2}{b^2}.\)

\(2b^2 = 4k^2.\)

\(b^2 = 2k^2\)

\(\text{Any number multiplied by two is even. So} \hspace{0.1cm} b^2 \hspace{0.1cm} \text{is even. Hence b is even.}\)

\(\text{If a and b are even, then the fraction} \hspace{0.1cm} \frac{a}{b} \hspace{0.1cm} \text{is not in its simplest terms.}\)

\(\text{ Because a and b are even, say} \hspace{0.1cm} a = 2m, \text{and} \hspace{0.1cm} b = 2n, \text{then} \hspace{0.1cm} \sqrt{2} = \frac{2m}{2n} = \frac{m}{n}.\)

We can then repeat the above process to prove m and n would be even, and say m = 2g, and n = 2h. This process would repeat forever.

\(\text{But we can’t simplify forever, so there is a contradiction.}\)

\(\text{So} \hspace{0.1cm} \sqrt{2} \hspace{0.1cm} \text{is irrational.}\)

This process is known as Infinite Descent. If we tried to solve the problem going forward through the problem, we would have a tough time, since/sqrt{2} can’t be expressed as a fraction. In this case, it's much easier to invert the problem and assume it is.

Photo by Karla Hernandez on Unsplash

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Concluding Remarks

With any problem you have in your life, consider viewing it backwards. Please don’t focus on how you can solve it, focus on how not to solve it, and avoid those strategies.

In this mathematical example, setting the square root of 2 equal to an irrational number is difficult since it can’t be written as a fraction. So, invert the problem. Write it as a fraction, and prove that isn’t the case.

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Sources:

·         https://www.mathsisfun.com/numbers/euclid-square-root-2-irrational.html

·         https://www.homeschoolmath.net/teaching/proof_square_root_2_irrational.php

·         https://fs.blog/inversion/