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 carpiscules, which allows to see color when they enter our eyes. Or just the apple guy. And God-free 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-agestimal number system that allowed for a compact notation. Sex-agestimal 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 used today. The 10 symbols being the digits 0 through 9. 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 Udoxas of Nidus and Archimedes. They were able to prove areas of shapes through the method of exhaustion. For normal polygons, say an octagon, we split it up into a number of triangles and find the area of each triangle, then added 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 inscribed a polygon and gradually increased 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 Bosch Grattoo wrote the Siddhanta Shadomani, which contained texts in astronomy with proofs using calculus. He proved the attitized 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 New Dunn and Leibnich are known for this discovery, even though Bosch Grattoo 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, arc-ten, and even pie. 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 sign series, this is the Madhava cosine series, and this is the Madhava arc tangent series. But if he did find this, then why did Madhava get the credit himself? We can't find his work. The only sources were the mathematicians who referenced him. Newton and Leibnits 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 fluxes. A variable of interest was called affluent, and its velocity was called affluxen. While Newton thought of his calculus in terms of motion, Leibnits had a more mathematical interpretation. To him, calculus was more of sums of infinitesimal distances. The same meaning behind the notation we used today. And so most of the notation we use is actually from Leibnits. So who took the credit for calculus? Newton conceived his ideas in the 1660s while Leibnits came to similar conclusions a decade later, but Leibnits 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 liked this little tangent from the hardcore math videos I've inditioned out lately. Subscribe for more content on data sciences, and I will see you in the next one.