Numbers are just tools, how we bend them to articulate reality is upto us.

From ancient golden Islam age of prosper math and chemistry to upto 15th century, mathematician couldnt solve cubic equation. However, they managed to solve quadratic n even degree equation, because till then, x + 1 = 2 so, x = 2-1 algebra hadnt been invented.

For eg, if they had to solve x^2 =4, they would visualize square of side-length 2 only would give area of 4. hence x must be 2 logic.

So, cubic equation on solving with this logic would always give sq.root(-ve numbers). and what does that even -they wondered, as square with negative side would make no sense in reality.

Long story short, one guy standing on earlier giants showed that you have to pretend sq.root(-ve number) exists ie. i. Hence came imaginary number.

He said sthg like, “My genious math ancestors, I humbly request you to not see numbers not something intrinsic meaning to reality. Reality is what it is. Numbers - Number system etc etc are just tools how we decide to present or write language to articulate reality..”

My own story - I had to deal with principal component. -in data science problem solving. I visualized- ok, principal component PC can be two if we see each data point as (x,y,z) space. But I was first little puzzled, for nth degree polynomial, there are n PC. How could that be - I tried to visualize. But that my hell-bent trial to visualize was the problem. I reminded - “Numbers are just tools, how we bend them to articulate reality is upto us. On this my visualization difficulty case, which is vice versa visioning process, we cant even visualize it. Its just like I am forcing to see how could square of negative side exists to give sq.root(-4). Lesson is- lot of times, visualization easifies but all cases, visualization is not possible coz we cant visualize beyond 3 dimensions.

Little more to this story - funny it is though. I initially thought when given graph was about T,P, density graph by matlab- oh, these are how gas particles are dispersed in actual xyz space. NOOO, they are just visualizations. But funny, sometimes mind is funny.

Recently, I was dealing with vectors. 3Blue1Brown YT channel kills it on explaining. He was saying, basically co-ordinated system had limitations. When we had to do sophisticated maneveur, then no such tool is possible w co-ordinated system n (co-ordinate math that comes w it). One genius famously writes sthg like, “Its a crime to hard glue bind number to just x-, y- and z-axes, Co-ordinate system.” He has got a point btw- how do you represent light illuminating from bulb central source w orthogonal system, we have to switch to radial system.

Anyway, tho I couldnt properly reason out for now (time for 3Blue1Brown - I guess), vector system is the answer because it had got that sense of direction in its lingo. Sense of direction becomes critical in lingo when we have to do sophisticated maneveur - like for eg., we want to answer, what happens if say system’s axes rotates by 90 deg or gets skewed. Believe me, such skewing, rotating needs to be done for higher analysis.

Many years ago, in College, I was intrigued by how Rene Descartes found confidence that deriving just plainly on + x and + y axes derivation would suffice for all 4 quadrant scenario. I was having casual talk w Anish. Achut Ji, Dinesh Ji came. Tho simple it was, I didnt forward my conversation not because I didnt want to share for oneupmanship, but simply because what I was talking, would make no sense in academic context, as one in university table would argue, 'What are you talking about, we just have to, its way things are done, its way book does?'

Why I switched to total Matrix n f**k co-ordinate system whenever possible? -Cause its P O W E R F U L.

Also, I was taken aback by compactness of matrix tool. For eg., if we have to solve say n eqn of n degree polynomial, matrix is built in such a way that all those 10 lines of Gaussian elimination etc are just 2 3 steps.

Also, I hadnt visualized that determinant is infact way of saying that (lets stick to plain paper 2 axes system for explanation simplicity) if my axes system gets transformed according to matrix rows cols number, what would area of 1 unit area (ie. 1 unit square) become?

Dont believe me? wait writing.

I now am firm believer the power of matrix. In simple situation, matrix is overkill tool but in higher analysis, no other way.