What was Newton’s beautiful solution? │ The History of Mathematics with Luc de Brabandère

Sir Isaac Newton was much more than a physicist. Some of his maths is amazing too.

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During our journey through mathematics we've already met some geniuses. Pythagoras, Aristotle, Descartes, Galileo.

This video is about Newton.

He said himself: ‘I built my science on the shoulders of giants.’ Newton of course is just another giant. Newton is famous for his discovery of the gravity law and also he studied light.

But those people were really multi-dimensional. And in mathematics Newton wanted to solve, to find an easy way to calculate a + b ‘to the power of something’. So, let's start, (a + b)^2, ‘square’. There is an elegant graphic solution to that. Let's build a square. And each side is built with a + b, a + b. You build the square and you can organise the square in 4 pieces. A little square here, a^2, a bigger square here, b^2. And Z rectangles. The 2 rectangles have the same surface a x b, so the answer to the question (a + b )^2 is equal to a^2 + 2.ab + b^2.

Now let's do it for the ‘3rd square.’ There is no visual possible, because you should do that in three dimensions. But believe me, the answer (a + b)^3 will be something with a^3 +… +…+  b^3. And if you look at the coefficients, you find 1 3 3 1. You remember the first was 1 2 1? Now, 1 3 3 1. And so, Newton connected his problem, the (a + b)^3 named, I don't know why, Pascal's triangle. And the triangle is built this way. You'd start with 1 and each digit is the sum, each number, is the sum of the two upper digits 1 2 1, 1 3 3 1, 1 4 6 4 1.

And this connection between Newton and Pascal immediately gave a solution to his problem

(a + b)^n because you just need the ^n line. And this triangle has lots of properties. For example, each line, if you sum the numbers of the line, you have ‘a power of’ 2.

Let's take line number 4. 1 + 4 + 6 + 4 + 1 is equal to 16, and 16 is 2^4. And this formula designed by Newton was a tremendous breakthrough in probability. Why? Because imagine an experiment with two outcomes, 3 chances out of 10 to get this result. 7 chances out of 10 to get that result. The sum of the probability must be 1. 3/10 + 7/10 is equal to 1.

Now let's take those two numbers and you put in the formula of Newton and of course you have (3/10 + 7 /10) ^n. 1^n is always 1 and that's how you can easily connect this work of Newton to the science of probability and what we call those distribution curves. Did you know that it’s faster to travel in a cycloid than a straight line?

Find out more in the next episode of our series ‘The History of Maths’.

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