Solve your problem efficiently with Principle of Interaction of Inverses

Wherever you find inverses interacting, you may be able to use it to your great benefit

Principle of interaction of inverses

The entities involved in interaction of inverses may be a tangible object, such as a large 90 litre toughened plastic container of drinking water.

it may be an object that you can't easily touch and see, such as Oxygen gas, or it can be a strong emotion such as anger.

Or it can even be such a simple thing as a mathematical variable $x$!

If you look around consciously in search of usefully interacting inverses, you would be surprised at how many you would find!

Principle of interaction of inverses

Recognizes first the existence of specific pairs of inverses, elements in each pair being inverses of each other, and then proposes to study the interactions between the inverses in a specific pair to discover results that can be used for solving problems using the special results discovered in the interactions.

What can be a pair of inverses

Like most of the things we have discussed till now, inverses are many. The simplest? Well, $\frac{1}{2}$ as the inverse of $2$ may very well be considered as the simplest. The most esoteric? Perhaps anti-matter as an inverse of mattter!

What about live objects? Can Men be considered as the inverse of Women! No, I am joking. We humans are too important to be considered as objects of obscure analysis just as half of all Animal Beavioural scientists have refused to consider homo sapiens as animals. But just for a while, little while, consider the potential of studying only the interactions between a pair! All types, all kinds of interactions and the results.

Can life itself be considered as an inverse of death? An important question, but out of scope here.

What about the feeling of dry cracked earth when the first raindrop hits it? How do those interact? Those first few moments?

Inverse of light? Great photographers study the subject of interactions between light and darkness or shadow to create art.

Inverse of heavy is light in weight. If you put something light in a heavy liquid or gaseous matter, it floats, cheaply. That's the principle used in ballooning. And Google's project Loon using this very useful natural interaction between inverses coupled of course with many more great innovations, may change the face of internet communication for a long time to come.

I was not aware of this fundamental principle when doing Yogasana myself

For many years I did Yogasanas myself, forming my own routine, totally unaware of the basic driving principles behind creating an effective sequence of Yogasanas. And then I got my Yoga trainer who formed my scientific professionally made effective schedule. Now I can appreciate at least the value of the interaction of inverses in my effective sequence of Yoga positions.

If you twist upper part of your body on the right, next you must make a twist on the left to complete a cycle. If you bend forward in a position, you must compensate it by bending backward. The result of the interactions of these inverse positions? A healthy and supple spinal cord.

If you rotate your neck fully right, next you will rotate it fully left. I know, it should keep my neck muscles flexible and blood flow to the brain better.

Being aware of the principle of inverses now, as I go on analyzing the various asanas I feel amazed at the wisdom of the creators of these positions. 

The same applies to any effective free hand exercise routine also.

My friend didn't have any drinking water system in her house

For some years I noticed appearance of large toughened plastic containers for drinking water, but had no occasion to use it. I always wondered, how water is poured from such a heavy jar without any tap in it.

Then my friend told me, when she occasionally visits her home in Kolkata, she just gets one such large container for her drinking water supply. When it is finished she gets another one. No hassles of installing a costly drinking water system in a house unoccupied most of the time. A very practical decision no doubt.

I couldn't suppress my drive for knowledge, "But how do you use it? I have not seen any tap on it. It is such a large and heavy jar!" She just laughed at me, "You don't know so many things! It comes with another empty jar with a tap. When I buy a container, the man who delivers just inverts the full jar and fits it into the mouth of the empty jar. Those are designed to be coupled."

Later at every ceremonial occasion I had to use it myself. The inverted contraption have become so popular that there is practically no substitute for temporary use of a large supply of high quality drinking water.

I was amazed how my friend quietened down the near stampede

In an occasion a long queue waited for their turn to the free meal during a holy ceremony. The queue was outside the gate of the large compound. The gate was locked and time to time opened to admit the next batch of free guests. My friend was managing the affairs. As the day progressed the queue of the villagers went on growing in length alarmingly. Suddenly there was a commotion, rough words flew. It was a near stampede. My friend calmly unlocked the gate and went out. He was tall and well built no doubt. But his calm reasoned but firm words quietened the angry throng in a few minutes.

I knew the wonderful effect of using calm in an excessively excited situation, but I also knew, with calm, reason has to be used against the unreason. And of course firmness is to be used to counter the breaking apart force. It is not easy but it is the most effective way.

But the awareness of inverses originated in my mind from Maths after all!

While trying to solve awkward looking MCQ based math problems more elegantly, I became aware of the value of studying the property of interaction of inverses for the first time. I will illustrate without any more deviations by directly going into the process of solving three problems.

Problem 1

If $x + \displaystyle\frac{1}{x} = 4$, then $x^4 + \displaystyle\frac{1}{x^4}$ is,

  1. 124
  2. 194
  3. 64
  4. 81


This is a problem involving inverses,  $x$ and $\displaystyle\frac{1}{x}$.

Whenever I meet this pair of inverses, I remember the wonderful result of interaction of these inverses,

When $x$ and $\displaystyle\frac{1}{x}$ are multiplied together, the unknown troublesome variable $x$ vanishes leaving behind the most friendly value of 1.

In other words, $x\times{\displaystyle\frac{1}{x}} = 1$.

This is the useful result of the interaction of inverses in $x$ and is used invariably to unravel the hidden mysteries of quite tricky looking algebraic problems time and again.

For example, in this problem we observe the given power of the inverses is 1, whereas the powers of inverses in the expression to be evaluated is 4. Immediately I know, I have to square the given expression two times.

If we square a sum of such inverse terms first time, the variable $x$ is eliminated in the middle term. This is our rich concept to be used in this type of problems.

First time squaring of the given expression results in,

$x + \displaystyle\frac{1}{x} = 4$,

Or, $\left(x + \displaystyle\frac{1}{x}\right)^2 = 16$,

Or, $x^2 + 2\times{x}\times{\displaystyle\frac{1}{x}} + \displaystyle\frac{1}{x^2} = 16$,

Or, $x^2 + \displaystyle\frac{1}{x^2}  = 16 - 2 = 14$.

Observe how conveniently the middle term is transformed to a pure pristine integer that could be deducted from 16 on the other side of the equation to leave nearly an exact replica of the inverse expression of the given equation, but only the power this time raised to 2 - a very prospective result no doubt.

And we can foresee straightaway that one more similar action will hand us our desired result.

$x^2 + \displaystyle\frac{1}{x^2}  = 14$,

Or, $\left(x^2 + \displaystyle\frac{1}{x^2}\right)^2  = 196$,

Or, $x^4 + \displaystyle\frac{1}{x^4} = 196 - 2 = 194$.

Answer: Option b: 194.

Here we have analyzed the expression to be evaluated against the given expression as a routine of course. That is our End state analysis in action.

Problem 2.

If $2a + \displaystyle\frac{1}{3a} = 6$, then the value of the expression $3a + \displaystyle\frac{1}{2a}$ is,

  1. 12
  2. 9
  3. 4
  4. 8


As a first step, we always examine in a quick scan the similarities between the desired end state (expression to be evaluated) and the initial state (given expression). This is applying our End state analysis approach that has most extensive use in math problem solving.

As a result of this analysis, we form our first conclusion:

The products of the direct and inverse terms of the given expression and the target expression are same, that is, $2a\times{3a}=3a\times{2a}=6a^2$. The coefficient or the power of $a$ of the product in both cases remain same.

This is a different type of interaction of inverses bearing a different result and suggesting a different action.

We understand that the direct term $2a$ is not exactly an inverse of the inverse term $3a$ in the given expression and similarly in the expression to be evaluated, the direct term $3a$ is not exactly an inverse of the inverse term $2a$. But the inescapable pattern looks at your face - the product of the two terms in both the expressions are same.

$2a\times{3a} = 3a\times{2a} = 6a^2$, at least in this interaction then the two sets of terms become very friendly to each other. So we become sure of the process to be followed.

It is a case of transforming coefficients. This is a technique used often, especially in case of terms involving inverses. Thus we transform,

$2a + \displaystyle\frac{1}{3a} = 6$

Or, $a + \displaystyle\frac{1}{6a} = 3$

Or, $3a + \displaystyle\frac{1}{2a} = 9$. 

How lucky! We have directly reached the answer.

Answer: Option b: 9.

Problem 3.

If $x\left(3 - \displaystyle\frac{2}{x}\right) = \displaystyle\frac{3}{x}$, and $x\neq{0}$ then $x^2 + \displaystyle\frac{1}{x^2}$ is,

  1. $2\displaystyle\frac{5}{9}$
  2. $2\displaystyle\frac{4}{9}$
  3. $2\displaystyle\frac{1}{3}$
  4. $2\displaystyle\frac{2}{3}$


Looking at the target expression we identify it as a sum of inverses and feel confident that another sum of inverses must be hidden in the given expression. It is a certainty. We have this feeling of certainty from the knowledge of how starting from a sum of inverses of power 1 of $x$ we could easily get another sum of inverses in power 2 of $x$, all due to the wonderful property of interaction of inverses.

And if you know a certain goal, searching for it becomes that much easier.

In this case we had to be a bit careful but nevertheless the trick could be discovered without any difficulty.

$x\left(3 - \displaystyle\frac{2}{x}\right) = \displaystyle\frac{3}{x}$,

Or, $3 - \displaystyle\frac{2}{x} = \displaystyle\frac{3}{x^2}$

Or, $3 - \displaystyle\frac{2}{x} - \displaystyle\frac{3}{x^2} = 0$

Or, $3x - \displaystyle\frac{3}{x} - 2 = 0$

Or, $x - \displaystyle\frac{1}{x} = \frac{2}{3}$

Now we  are in familiar grounds.

Squaring the last equation,

$x^2 + \displaystyle\frac{1}{x^2} - 2 = \displaystyle\frac{4}{9}$

Answer: Option c: $2\displaystyle\frac{4}{9}$.

Transforming the given expression in the form of sum of inverses led us directly to the solution (the minus sign is also welcome and considered similar in this type of problems).

Just as in real life, there are many forms of inverses in maths, but whenever we see an inverse interaction, we feel comfort in the knowledge that interaction of inverses will surely simplify the problem in no time.


End note: If you decide by chance, to find useful inverse pairs, how would you proceed? Would you just try out the Synonym-Antonym route?