April 21, 2021
Positive real numbers positively did a real number on me
Every so often you come up with a brilliant idea that you don’t know what to do with. Today seems to be one of those days for me.
For some reason, I was reminded of a mathematical paradox. It has to do with real numbers. For our discussion here let’s just limit ourselves to positive real numbers; that is any number you can think of that is greater than 0.
First of all, we know there is no limit on how many such numbers there are. If you don’t believe me, let’s play a game.
If I am wrong, there must be a highest number with no possible number larger than it.
You tell me what that highest number is. I will then tell you a higher number by adding 1 to it. Simple stuff.
There are infinite positive real numbers. I am positive about that. So should you be.
Reciprocal pairs tell us a story
Things start to get real interesting when you think about inverses; specifically multiplicative inverses.
If you are not sure, let’s just refresh the concept. A number and its multiplicative inverse, when multiplied together, always yields 1. So, multiplicative inverse of a number is 1 divided by that number. This makes 1 its own multiplicative inverse.
For example, if you think of 5, it’s multiplicative inverse is 1/5, or 0.2. So, 5 and 0.2 are multiplicative inverses of each other. You can come up with many other examples.
Without going into nuances, let’s just say a multiplicative inverse can also be called a reciprocal for our purposes. We are only considering positive real numbers.
Thinking further on this relationship, you would realize that reciprocal numbers are always in pairs where one of them is less than 1 and the other is greater than 1. (In addition, the number less than 1 is always greater than 0, because if it’s 0, then the product will be 0, not 1. Also, if it’s less than 0, the product would be negative, and not positive 1.)
Do you realize what this means? For every number you can think of that is larger than 1, I can counter with a number that is between 0 and 1 that will be its reciprocal. More interesting, that small number is the reciprocal of just that one number you thought of. No more, no less!
This is huge.
This means that the interval 0 to 1 packs the same punch as the entire set of numbers larger than 1! That’s right. There are as many numbers in that teeny interval as the entire upper side of 1 has.
This made me start to think philosophically. There must be the meaning of life lurking there somewhere.
I can think of one.
Self is a reflection of the universe, and vice versa
You may be familiar with that teaching. Different people present it in different forms, but they all come down to essentially the same idea.
Don’t go looking for answers all over the universe; just look inward and seek your answers within yourself. It may be called meditation, penance, or simple introspection. Whatever the vehicle, it is known to lead to calm and inner peace, and solutions that work.
I am seeing a parallel between this life principle and what our little math is telling us.
I see profundity in simplicity.
Do you? Hit ‘Reply’ and let me know what you think.
P. Venkat Raman
P.S. Are you wondering where the paradox is? Thought you’d never ask!
For every number in the interval between 0 and 1, add 1 to it. The result must be in the interval between 1 and 2. Simple addition, right? This also means there are as many numbers between 1 and 2 as there are between 0 and 1.
But we had just established earlier using reciprocals that there are as many numbers between 0 and 1 as there are above 1, unbounded. This unbounded set includes all the numbers between 1 and 2 (which are themselves equal in number to the number of positive real numbers between 0 and 1) and many more.
Aaaaaarrrrrrgh!
Do you see?
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