 Calculus. The mathematical study of how things change. It includes differential calculus, rate of change using slope as a function, and integral calculus, determining quantities like areas and volumes under changing conditions. Two people responsible for calculus. Isaac Newton, the guy behind the laws of cartoons. The guy who hypothesized light is made up of particles called carpuscules which allow us to see color when they enter our eyes. Or just the apple guy. And Godfrey Leibniz. Both Newton and Leibniz are independently responsible for the foundations of calculus that we see today. Now, did calculus start with them? Of course it didn't. Let's go back in time. Too far. Babylonia was an ancient Mesopotamian state, comprising of modern Iraq, Syria, and Egypt. The capital city Babylon was about 60 miles from modern Baghdad. Babylonians offered an early intuition on the infinite process. They could determine the square root of a rational number to any number of decimal places. A contributing factor was their sex adjustable number system that allowed for a compact notation. Sex adjustable number system has a base 60. That means they represent all real numbers using 60 symbols. This is different from the base 10 or the decimal number system that we use today. The 10 symbols being the digits zero through nine. Babylonian mathematics recorded their research and observations on clay tablets called cuneiforms. This is a famous cuneiform that shows that the diagonal of a square is root two times its side. They were also able to define other numbers with potentially infinite precision, like the length of the year, or even represent the value of Pi. Fast forward to 300 BC. Babylonian astronomers thought everything that happens in this world, from changes in the weather to rise and fall of tides, was associated with the motion of planets. So the Babylonians used geometry to study the motion of planets, specifically Jupiter, since their god Marduk was associated with Jupiter. Recording velocities at instantaneous times, they could determine how far the planet travels in an interval of time. This was similar to finding the area under a curve with the trapezoid rule, which forms the basis of integral calculus that we see today. Did they do everything? Of course not. Around the same time in ancient Greece were the prominent figures Eudoxus of Nidus and Archimedes. They were able to prove areas of shapes through the method of exhaustion. For neural polygons, say an octagon, we split it up into a number of triangles and find the area of each triangle, then add it up to get the area of the polygon. Circles are a special case since they can't directly be split into triangles. Half a circle is just half a circle. So they inscribe to polygon and gradually increase the number of sides. So the area of the circle would approximately be equal to the area of the polygon. But the area of the circle would only be the same as the inscribed polygon with infinite sides. From this we got the concept of limits and it also incorporated the idea of the infinite process. Onto the medieval era, India had their share of contributions too. In the 12th century, mathematician and astronomer Bhaskara too wrote the Siddhanta Shadomani, which contained texts in astronomy with proofs using calculus. He proved that at its highest point, the instantaneous speed of a planet is zero. Actually I don't know what that means but that's what it literally says in this paragraph of a news article in a language you don't care about. And this paragraph talks about how some English mathematicians called Nudan and Laibnij are known for this discovery even though Bhaskara said it a few centuries earlier. One of the greatest mathematicians and astronomers in the 14th century India, Madhava, linked the idea of infinite series with geometry and trigonometry. He was able to formulate infinite series expansions for sine, cosine, arctan, and even pi. So you see these expansions here? You know the one that we call the power series expansions? But nope, they're actually called the Madhava series. This is the Madhava sine series, this is the Madhava cosine series, and this is the Madhava arctangent series. But if he did find this, then why didn't Madhava get the credit himself? We can't find his work. The only sources were the mathematicians who referenced him. Newton and Laibnitz and other western mathematicians were 300 years too late. The history of calculus and mathematics isn't straightforward. Different countries came up with their own methods around the same time. Using this knowledge, we attribute its flourishing as a promising field to the people who tied it all together. Newton was a physics buff. His calculus was defined out of necessity to lay the foundations of gravitation and the laws of motion. So it's no surprise that he thought of calculus in terms of motion and called his calculus the method of fluxions. A variable of interest was called a fluent and its velocity was called a fluxion. While Newton thought of his calculus in terms of motion, Laibnitz had a more mathematical interpretation. To him, calculus was more of sums of infinitesimal distances. The same meaning behind the notation we use today. And so most of the notation we use is actually from Laibnitz. So who took the credit for calculus? Newton conceived his ideas in the 1660s while Laibnitz came to similar conclusions a decade later, but Laibnitz published his findings nearly a decade before Newton. Well initially it was Newton, but after their deaths both were attributed the title Inventors of Calculus. Regardless, as we have seen, calculus wasn't invented overnight. It's a branch in mathematics that has multiple roots from thousands of years ago, and it continues to power newer fields like machine learning and AI. If not for all these people, some of us would be doing different things. Hope you guys like this little tension from the hardcore math videos I've been dishing out lately. Subscribe for more content on data sciences and I will see you in the next one.