Shaping the Shapes

The first time I found out the real meaning of a mathematical equation, I was completely blown.

Everything I had learned about a graph, the x- and the y-axes, a shape, a line, a curve, a point, a slope – everything in my knowledge suddenly gained a new meaning. It was like my world upturned in a matter of a few video lectures. It might sound a little exaggerated, but it actually was such a shift as I make it appear.

Prior to the discovery I made accidentally and unintentionally, an equation meant just a combination of some nonsense variables with constants and something called degrees (which is the highest power of a variable in an expression). Something unconnected I knew was that based on an equation, you could draw something on the graph. If it was a linear equation (with degree one), it meant that the line on the graph would be straight.

It was a useful thing, in fact: you had a sort of relation to find out a variable if a second variable was given. You can buy x pens for y rupees. And you can buy m pens for n rupees. This is all the information you require, given that the relation is linear. Have this much information, and you could effortlessly find out any number of pens for any amount of rupees. How many pens for rupees twenty? Easy to find, just plug ‘20’ in place of ‘y’ and solve for ‘x’. That’s the answer. How much money will you require to buy ten pens? – that’s a piece of cake, nothing more. This time, plug the value in place of ‘x’ and solve for ‘y’. You have the answer.

(1 pen for ₹10. That's all we need.)

This was my knowledge of equations, graphs, and variables and the relations between them until I took Maths in high school. This knowledge was enough, it would do the work, I would be able to solve examination questions – but it was not yet fundamental. It was not as intuitive as it should be. Simply put, it didn’t make my smile go wider at the beauty of the concept.

Things changed when in my class 11, first year in high school, around the middle of the term, we picked up a chapter called Straight Lines. My study at school did not seem enough and so I started to take regular help from YouTube. Around this time, I miraculously stumbled into the channel of this beautiful guy, Ashish Kumar, called “Ashish Kumar – Let’s Learn”.

(listen to him at 1.25x speed, and you will come to love the guy)

The way he defined an equation blew my mind off. He didn’t say that an equation was a collection of variables and constants with an equals-to sign in between somewhere. He didn’t even say that an equation gave the relation between two or more variables. What he said was something novel, something revolutionary.

He said that equation was a way in which you drew a shape in the language of Mathematics.

x² + y² = 4 didn’t just give you a value of x for a value of y: it drew the shape of a circle.

|x| + |y| = 1 was not just a set of random moduli and x and y, it was the way you drew a quadrilateral around the origin in Maths.

The fundamental became clearer when I learned that the x and y in an equation are no different than the x and y on the two axes. A point – a coordinate – meant that for x amount of movement on the horizontal axis, you had to travel a y amount of movement on the vertical axis. It was basic Cartesian geometry, I knew, but now it was one more thing: intuitive.

When you conduct an experiment or collect some data, you usually find a relation between two things: how much distance and how much petrol consumed? How much time and how much distance run? How much investment and how much profit? How much production and how much pollution? When you collect such data and plot it on a graph, you get a set of points. Sometimes when you join them, what you get are irregular shapes that do not make much sense. But at other times, you get special shapes. For example, the relation between distance traveled and petrol consumption might be linear. Some other data might give you a circle. The data of Covid-19 cases might rise exponentially – and might make part of a parabola. The curve might flatten to become a straight line.

Once you knew what the shape was, you required little guesstimation to know how the equation should look. When the YouTuber-teacher Ashish Kumar first mentioned that he could predict how a shape would look just by looking at its equation (although we would study it in the coming chapters) – whether a circle, a parabola, an ellipse, or just a straight line – and he could even be so precise so as to tell if the straight line or the circle would intercept y-axis at (0,2) or (0,5) or both, it was a further double-dyed surprise.

Now, all this is not to say that these things were not what I knew already, or was some theory-of-relativity-ish discovery for humankind – perhaps people before me might have studied and said ‘Ah!’ at it equally: instead what made this a new thing for me was the way everything acquired new meaning after a few lectures of this teacher. It was such a strong knowledge that never again in my life I would be entangled about the fundamentals and the concepts behind a graph, an equation, a shape, a slope. Life would be a much beautiful thing after that little conceptual understanding.

My heartfelt thanks to sir Ashish Kumar, and my friends Larry and Sergey for birthing this YouTube thing. And of course, thanks to René Descartes and fellow men who knew Maths would be too boring if it just had numbers.