Hi, I'm Vladmir Podolskii and today we're going to discuss invariants.

Looking at the right property is

important in general and in particular with medical problems.

Sometimes we have to look at numbers,

sometimes it's the sum equality or inequality or something else.

We will see examples later.

The properties that do not change after

some operations or during some processes are called invariants.

Double counting can be viewed as a special case,

though it be artificial.

We look at some number and note that it doesn't change no matter how we count.

Let's look at the following problem.

There are two cups, one with coffee and another with milk.

And we do the following process.

We take a spoon of coffee from the first cup and add it to the cup with milk.

After that we take a spoon of a drink,

of the resulting drink in the second cup and put it back in the first cup.

What is larger, the amount of milk in the cup with coffee

or the amount of coffee in the cup with milk in the second cup?

So it might seem from the beginning that some serious calculations are needed.

We have to count proportions.

What happened after the first stage

of the separation then what happened after the second stage?

Besides that, we don't know the sizes of the cup and the size of the spoon.

So, maybe the results depends on these parameters. So they do not know.

But it turns out that no calculation is needed here so the problem can be solved without

any calculations at all. Let's look at the picture.

In the left column we have what we had before the process and

the right column represents what we have after the process.

And let's observe the following: the size of the first drink is invariant.

Indeed in the beginning we take one spoon out of the first drink and in the end,

in the second part of the process we put it back.

Maybe there's some milk but still the size of the drink doesn't change.

So, this means that the amount of coffee that is missing in cup

one is the same as the amount of milk we added to cup one.

So since the size of the drink didn't change then

we substituted it with the coffee that we removed from cup one by

exactly the same amount of milk.

But note that the coffee that is missing in cup

one is exactly the same as the coffee that was added to cup two.

So the amount of milk that we added to cup one is the

same as the amount of coffee we added to cup two.